List Price and Discount in A Stochastic Selling Process

Z. Eddie Ning ∗

Updated: June 2020

Abstract

From B2B sales to AI-powered ecommerce, one common pricing mechanism is “listprice - discount”: the seller first publishes a (committed) list price, then during inter-actions with a buyer, offers (non-committed) discounts off of the list price. Some B2Bsellers never sell at their list prices. In such cases, what role does a list price play andhow to choose the optimal list price and discounts? In this paper, I study a stochasticsales process in which a buyer and a seller discover their match value sequentially. Theseller can adjust its price offers over time, and the buyer decides whether to accepteach offer. I show that this discovery process creates a hold-up problem for the buyerthat results in inefficient no-trades. The seller alleviates this problem by committingto an upper bound in the form of a list price. But in equilibrium players always reachagreement at a discount. I show that the seller prefers this mechanism to having nolist price or committing not to discount. When the cost of selling is high, the seller’sability to offer discount is necessary for trade to happen. There exists reverse pricediscrimination when buyers are heterogeneous, and list price can serve as a signalingdevice when sellers are heterogeneous. Extensions with alternative pricing or matchingmechanisms are discussed.

Keywords: dynamic pricing, information acquisition, sales process, bargaining, pricediscrimination, matching, continuous-time game, optimal stopping.

∗Cheung Kong Graduate School of Business. Email: zhaoning@ckgsb.edu.cn. I’m indebted to J. MiguelVillas-Boas, Yuichiro Kamada, Brett Green, Ganesh Iyer, Philipp Strack, and Willie Fuchs for their inputand guidance. All errors are my own. I thank Aaron Bodoh-Creed and seminar participants at MarketingScience Conference, IO Theory Conference, European Winter Meeting of Econometric Society, Berkeleyseminar, and Yale seminar for helpful discussions and comments.

1

1 Introduction

In many sales contexts, the final pricing agreement has two components: a list price anda discount. The seller first sets its list price internally, then after interacting with a buyer,quotes a discount. This is common in business-to-business transactions such as purchasingof industrial equipment or enterprise software, big-ticket consumer items such as real estateor automobile, or AI-powered e-commerce where sellers offer personalized discounts basedon buyers’ browsing behaviors. These two types of prices are different in their timing andcommitment power. In the above examples, list price is set by the seller long before a buyerarrives, and the same list price is used regardless of who the buyer is. From a buyer’sperspective, the list price is visible (either through public information or direct inquiry withthe seller), and will remain constant during their interactions. On the other hand, theseller chooses discounts only after interacting with the buyer through activities such as salespitch and product demonstration. The size of the discount depends on the information thatseller acquires during these interactions, and can vary from buyer to buyer. From a buyer’sperspective, the seller makes no commitment on whether, when, or what discount will beoffered. Even after a discount is offered, the quote often has a short deadline, after which theseller can retract the discount or offer a bigger one. Thus the list price promises an upperbound, while the seller maintains the flexibility to offer any price below it through dynamicdiscounting.

Articles from management consulting firms suggest that this is a common B2B prac-tice. In an article by Boston Consulting Group, Schurmann et al. (2015) state that “[i]nmost B2B industries, discounts represent a company’s largest marketing investment, oftenamount to 30% or more of list-price sales”. A Mckinsey article on B2B pricing suggeststhat firms should focus on transaction pricing, which is “for each transaction - starting withthe list price and determining which discounts ... should be applied” (Marn et al. 2003).PricewaterhouseCoopers states that “discounting off of the ‘list’ price of as much as 100% iscommon” in the software industry (PwC 2013). L.E.K. Consulting states that “[d]iscountsare a universal feature of any buying process” (Pierre and Ossmann 2017), with similar state-ments found in a Bain article (Mewborn et al. 2014). Furthermore, Bain and McKinsey eachreveals data on final prices as percentages of list prices for one of their clients (Kirmisch andBurns 2018, Marn et al. 2003). Interestingly, both of these firms offer discounts to 100% oftheir customers, so no client pays the full list price. Other anecdotal evidence suggests thatthese two components of pricing perform more than just their sum. Bain documents a casein which a firm’s strategy of setting high list price and quoting deep discounts hurt sales,“even though the final prices (after discounts) would have been very competitive” (Burnsand Murphy 2018).

This “list price - discount” mechanism has not received sufficient academic attention. Indynamic bargaining or pricing models where players can adjust offers over time, there lacksan explanation for why a seller wants to self-restrain its future offers by establishing a listprice ex-ante. For example, if one allows the seller to set an upper bound in the repeated-offers model of Fudenberg et. al. (1985) and Gul et al. (1986), then the upper bound isoptimal as long as it is higher than the valuation of the highest type. Thus providing theseller with the ability to set a list price does not affect the equilibrium outcome, making“list price” and “discount” meaningless terms in such a model. One may also view the use

1

of list price and discount as a method of price discrimination, where the list price is thehighest price tier. However, this cannot account for cases in which no buyer pays the listprice, as documented in Kirmisch and Burns (2018) and Marn et al. (2003). If the highestprice tier receives no demand, then neither its existence nor its size has any impact. Onemay also hypothesize that the never-used list price serves as a signaling tool to separatesellers of different qualities. But for there to be separation in equilibrium, there needs to bean adverse effect of increasing list price, otherwise all sellers can mimic the highest qualityseller. The micro foundation for such adverse effect is unclear.

If nobody pays the list price, what purpose does this upper bound serve? If the sellerchanges this list price, how does this affect the equilibrium price that players agree to?What factors should be considered when selecting the optimal list price? How does thismechanism compares to ones where seller does not use a list price or commits to never offera discount? This paper studies these questions in a setting where the total surplus fromtrade between a buyer and a seller is uncertain ex-ante due to heterogeneity, and playersspend time discovering the size of this surplus prior to transaction. In many industries,sellers differ in attributes of the products/services that they offer, and customers differ inthe attributes that they need. A buyer may not know how well the seller can address hisneeds, and has to acquire information to guide his purchasing decision.

But the buyer is not the only party that learns. In practice, many B2B firms have anexplicit ”discovery” step in their sales processes, in which the salesperson inquires about thebuyer’s situations and needs.1 Industry studies find that a well-executed discovery plan playsan important role in securing sales (Zarges 2017). The information about the buyer’s needshelps the seller to fine-tune her selling strategy, such as whether to continue pursuing a buyeror what price to quote. For example, firms often delegate some pricing power to salespersons,because the salespersons know more about a client’s willingness-to-pay than the firm does,through their communications with the client (see Mantrala et al. 2010 and Coughlan andJoseph 2011 for surveys on the literature of price delegation). A survey of sales forces byHansen et al. (2008) find that 72% of companies give their salespersons pricing authority.The number rises to 88% for industrial-goods companies. The two parties discover how wellthey match with each other which determines their total surplus from trade. This view isconsistent with writings from practitioners (see, e.g., Nick 2017 and Mehring 2017).

In e-commerce, this “list price - discovery - discount” phenomenon begins to emerge aswell. Using AI-powered tools, online retailers can rate a customer’s purchase intent in realtime and intervene with discount offers. One company providing this technology is Appier,whose product AiDeal allows retailers to track a customer’s likelihood of purchase through“real-time activities across the site, such as how they tap and swipe on their phones, thecursor position and the amount of scrolling.” Retailers can then choose the right moment tooffer time-limited deals during a browsing session to trigger purchases based on the inferredlikelihood of purchase (Appier 2020). Research has shown that a user’s behavior during abrowsing session can be used to predict her purchase intent. For example, a user’s clickstreamon a site contains information about her purchase horizon (Moe 2003). On computers, cursormovements correlate with eye gaze, which highlights user attention (Guo and Agichtein

1See, for example, Hubspot’s sales process in Skok (2012) and Talview’s sales process in Jose (2017) withmore details. Both companies offer Software-as-a-Service to other businesses.

2

2010). Guo et al. (2019) use data from Toubao that include users’ mobile taps and swipes topredict a user’s purchasing intent in real time. They show that the use of touch-interactivedata significantly improve prediction accuracy. Their A/B testing during “Double 11”, theChinese shopping holiday, shows that a coupon allocation strategy that targets users whosereal-time purchase intent falls into an intermediate range outperforms a non-targeted strategyby about 40% in GMV. A simple use of real-time intervention that is common in ecommercealready is exit-intent popup. When a user moves cursor towards certain parts of the browser,or highlight product names, the website can infer from these movements that the user is aboutthe leave, and intervene with a popup offering discount (Webtrends Optimize 2020). As theunderlying technology matures, the practice of real-time personalized discounting based onpredicted purchase intent may become more widely adopted. This paper analyzes its effecton equilibrium prices and welfare.

We study a dynamic model where a buyer and a seller discover how well they match witheach other over time while negotiating on price. The seller chooses a list price ex-ante, andcan offer dynamic discounts over time as players receive gradual information on their levelof match. The buyer decides whether to accept an offer or continue the process, and bothplayers can choose to leave because waiting is costly for both players.2

In this model, discovery and pricing are interdependent. Information on product fit affectspricing strategy and vice versa. For example, intuitively, a good match benefits the buyer.But if the seller observes that the match is good, the seller may end up charging a higherprice (with a smaller discount), which reduces the buyer’s incentive to discover their matchin the first place if doing so is costly. This presents a hold-up problem, where the expectationof a future “bargaining” outcome affects the players’ current choice of “investment” (whetherto continue or leave). Another important feature of the model is that we allow the seller tomake offers at any moment during the discovery process, hence endogenizing the amount oflearning and the timing of agreement.

I show that list price acts as an instrument to reduce the buyer’s concern for hold-up. Ifthe list price is too high (or there is no list price), then the buyer leaves immediately. Thebuyer does not expect to get surplus ex-ante because the seller can charge a high price oncethe good match is revealed. In order to encourage the buyer to gather information, the sellerneeds to set a low enough list price to limit the impact of such hold-up. On the other side,a lower list price decreases the seller’s ability to exploit the buyer’s discovery of high matchvalue, because it increases the buyer’s continuation value. The optimal list price thus has tobalance these two effects. More importantly, I find that the parties always trade below thelist price, regardless of what the list price is.3 The reason is that it is not efficient for theseller to wait until the buyer is willing to pay the full list price. Instead, the seller prefers toclose the sale earlier by offering a price discount. The model provides a rational explanationfor the “list price - discount” pattern that is observed in practice.

Should the seller commit to not offer a discount? I find that using a fixed price leadsto a higher transaction price and a lower ex-ante probability of trade. Having the abilityto discount always benefits the seller and also increases overall welfare, but its effect on the

2The buyer incurs cost from processing information or opportunity cost when dedicating employees totalk to the salesperson, while seller’s cost can come from the salesperson’s salary and product demonstrationcost.

3This is before considering consumer heterogeneity in costs.

3

buyer’s utility depends on the ratio of the players’ costs. When the seller’s cost of selling islow, buyers are better off under fixed price. But, when the seller’s cost is high, having theability to discount is necessary for trade. The flexibility in price improves overall efficiencyby saving time and cost and by increasing the ex-ante success rate. In such a case, theability to discount is welfare-enhancing for both parties. Comparisons to other mechanismto reduce hold-up, including non-retractable offers and direct subsidy, are explored.

The model extends to cases with asymmetric information. When buyers have privateoutside options, I show the existence of reverse price discrimination, where the buyer witha higher valuation for the product pays a lower price. If sellers have private information intheir quality, the model finds that a high type seller can set a higher-than-optimal list priceas a way to signal. Thus a list price can have signaling power even though it is never tradedin equilibrium, through its indirect impact on equilibrium price. The main results can alsobe replicated in a 3-period model with asymmetric learning, where a seller only observesnoisy signals of a buyer’s preferences.

Overall, the model provides novel insights on the role of list price and price discountsin facilitating information discovery. The analyses give implications on pricing and sellingstrategies in a stochastic environment. On the theoretical side, the paper expands existingrepeated-offers literature by introducing a hold-up problem to the stochastic bargainingframework.

1.1 Literature Review

The paper is related to the literature on sequential information acquisition. Similar toRoberts and Weitzman (1981), Moscarini and Smith (2001), Branco et al. (2012), and Keet al. (2016), this paper uses a Brownian motion to capture the effect of gradual arrivalof information on product value. Whereas previous papers focus on single-agent decision-making, this paper features a two-player game and allows price to be bargained. Kruse andStrack (2015) look at a problem in which a principal tries to influence an agent’s stoppingdecision through a transfer. The price discount in this paper can be seen as a transfer, butunlike the principal in Kruse and Strack (2015), the seller in this paper cannot commit tofuture transfers.

The paper follows Fudenberg et al. (1985) and Gul et al. (1986) in studying a repeated-offers model where one party makes all the offers, which can be interpreted as either bargain-ing or dynamic pricing of a durable goods monopoly. Other papers have studied repeated-offers models with stochastic payoffs. Daley and Green (2020) look at a repeated-offers gamewith asymmetric information and gradual signals. One party knows the true quality of theproduct, and the other party receives noisy signals of the quality over time. In contrast, bothplayers acquire information in the present paper. Ortner (2017) studies dynamic pricing ofa durable goods monopoly whose marginal cost changes over time. Fuchs and Skrzypacz(2010) look at bargaining with a stochastic arrival of events that can end the game. Ishiiet al. (2018) studies wage bargaining with both sequential learning and stochastic arrival ofcompetitor.

Merlo and Wilson (1995) present a general framework for stochastic bargaining in whichthe total surplus fluctuate over time and the order of proposing is random. However, theydo not allow players to quit for outside options, as in this paper. Rubinstein and Wolinsky

4

(1986), Gale (1987), and Binmore and Herrero (1988) expand the alternating-offers model ofRubinstein (1982) with a different type of stochastic matching. In these papers, buyers andsellers have fixed values for the product, but can only bargain when matched with a player ofthe opposite type. The matching process is exogenous, so players are automatically in searchof new partners as long as they are in the market. This literature explores the difference inequilibrium prices between centralized and decentralized markets. In comparison, the presentmodel focuses on the hold-up problem created by the combination of stochastic surplus andendogenous quitting, both absent in the aforementioned papers.

The idea that list price can reduce hold-up relates to consumer search literature whichshows that sellers can use published price to encourage search. Wernerfelt (1994b) showsthat, when product quality is uncertain and requires search/inspection, the seller wants toinform the buyer about the price before search. Similarly, to encourage visits, multi-productretailers advertise the prices of loss-leaders to put an upper-bound on the total price of abasket (Lal and Matutes 1994). However, in these papers, buyers learn all information in oneshot while sellers do not learn. Sellers cannot, and do not have incentives to, give discountsafter buyers inspect the products. Thus they do not capture the ”list price - discount” patternstudied in this paper. The current model bridges this gap by adding stochastic arrival ofinformation that is observed by both parties which allows seller to offer dynamic discounts.We show that setting a list price reduces hold-up even if seller always offer discounts inequilibrium. Other papers have discussed the use of list price in different contexts. Xu andDukes (2017) show that the list price can be used to convince uninformed buyers of theirtypes when the seller has superior information. Yavas and Yang (1995) and Haurin et al.(2010) discuss the signaling role of list price in the real estate market. Shin (2005) showsthat a non-committal price can signal true prices when the sales process is costly.

The paper also relates earlier works that study the role of sales in providing informationto consumers, such as Wernerfelt (1994a) and Bhardwaj et al. (2008). Shin (2007) studies afirm’s decision to provide pre-sales service that reveals the match between a customer andthe product, when competitor can free-ride on such service. This paper also studies a firm’sdecision to delegate pricing authority, but with a different approach from the principal-agentmodels of Lal (1986), Bhardwaj (2001), and Joseph (2001).

The rest of the paper is organized as follows. Section 2 presents the baseline model.Section 3 solves the equilibrium outcome. Section 4 to 6 presents various extensions to themodel. Concluding remarks are offered in Section 7.

2 The Model

We first describe the model in discrete time, then present the solution to the continuous-time limit of the game.

2.1 Description of the Model

Consider two players, a Buyer and a Seller. Seller has a product with T attributes. Eachattribute is worth vh

√dt to Buyer if it matches Buyer’s needs or vl

√dt if it does not, where

dt measures the size of each attribute. The total value of the product is the sum of its

5

attributes. For simplicity, I assume that the probability of match for each attributes is 12

and independent across attributes.4 Outside options and the cost of producing the productare normalized to 0 without loss of generality. Note that this means that the value of theproduct can be negative if it is less than the sum of the outside options and the productioncost.

The game is illustrated in Figure 1.

Figure 1

Discovery of Match Each period, players simultaneously discover their match on thenext attribute. The expected value of the product after observing t attributes can be writtenas xt = x0 +

∑t0 zs, where x0 is the expected value of the product at time 0, and zt is the new

information from period t. Define σ = vh−vl2

, then we have E[zt] = 0 and V ar[zt] = σ2dt.

Pricing Before the game starts, Seller can set a list price P . Buyer can always buy theproduct at P in any future period. Denote P = ∞ if Seller choose not to set a list price.Each period, Seller can offer a discounted price Pt with no future guarantee. Buyer decideswhether to accept the offer. The game ends if Buyer accepts. Notice that Seller cannoteffectively offer any price above P . Also, if Seller does not offer a discount, Buyer can stillbuy at P , so there is a standing offer at each period.

Quitting Buyer incurs flow cost each period, and can choose to quit at the end of eachperiod. If Buyer quits, the game ends and players receive their outside options. In Section4, I study the case where Seller has flow cost and can quit too.

Stationary Baseline As the length of each period (or size of each attribute) approaches0, the game approaches continuous time. As the mass of total attributes T approach infinity,the product value xt becomes a Brownian motion. I examine this limiting case in the baselinefor its tractability. Non-stationary extensions are discussed in Section 6 and Online Appendixto check that results from the baseline model are robust.5

4Using other probabilities increase the analytic complexity without providing additional insights. Ifattributes are correlated, then attributes that are discovered first reveal more information about the totalsurplus, while the additional information revealed by subsequent attributes must still be independent frominformation revealed before. This case is discussed in Online Appendix

5One qualitative implication of infinite horizon is that there is always more information to discover so thatplayers can never finish learning before trade. In reality, though the amount of information is not infinite,there is often too much information available so that buyers cannot feasibly learn everything before making

6

Figure 2

Figure 2 presents a sample evolution of the product value in the stationary baseline.Number of attributes discovered, t, is on the horizontal axis, and the expected productvalue, x, is on the vertical axis. First assume that Seller never offers a discount. This isBuyer’s single agent optimization problem as he decides between discovering and buying.In the beginning, Buyer optimally chooses to wait and find out more about the product.Waiting is optimal even some x above the list price because the option value of learning.Buyer’s optimal strategy is to buy when product value reaches a threshold. However, Sellermay prefer to close the sale earlier at a discounted price. Trading earlier saves time and cost,while also reducing the possibility of Buyer leaving upon discovering a bad match. This isshown in Figure 3.

Solving the equilibrium in Figure 3 is difficult. Because Seller cannot commit to futurediscounts, Buyer’s decision depends on Seller’s entire pricing strategy Pt in all future states.Conversely, Seller’s optimal offer depends on what Buyer is willing to pay in all future states.An equilibrium then should be the solution to a two-sided stopping problem. Buyer’s strategygives him the optimal stopping time given what Seller is willing to offer in the future, andSeller’s pricing strategy gives her the optimal stopping time given what Buyer is willing toaccept in the future.

2.2 Formal Definitions

The game has an infinite horizon. Product value xt is observable to both players andfollows dxt = σdWt with initial position x0, for some Wiener process Wt.

Equilibrium I look for stationary SPE with pure strategy, henceforth referred to as“equilibrium”.6 An equilibrium strategy profile θ(xt) = (P (xt), a(xt, Pt), q(xt)) maps each

a decision. Toman et al. (2017) observe that B2B buyers face “open-ended learning loops by the deluge ofinformation. With each iteration they work harder to ensure that they fully understand the requirements andthe alternatives. More information begets more questions”. For consumer products, there is often a wealthof consumer reviews or third-party reviews on internet so that buyers often do not exhaust all informationbefore making a purchase.

6As shown by Simon and Stinchcombe (1989), a strategy profile may not produce a well-fined outcome

7

Figure 3

state xt to Seller’s price offer, Pt = P (xt), Buyer’s acceptance decision for each price offer,at = a(xt, Pt) ∈ {0, 1}, and Buyer’s quitting decision, qt = q(xt) ∈ {0, 1}. I focus on station-ary behavior because product value xt is stationary and time is payoff-irrelevant conditionalon xt. An equilibrium outcome can be described by a quadruple (A,Q,U(x), V (x)), whereA = {x|a(x, P (x)) = 1} is the region where players reach agreement, Q = {x|q(x) = 1} isthe region where Buyer quits, U(x) is Seller’s equilibrium value function, and V (x) is Buyer’sequilibrium value function.

Utility The game ends when Buyer buys or quits. Let τθ = inf{t | at + qt > 0} denotethe stopping time of the game given a strategy θ. Buyer incurs flow cost cb during the game.Players are risk neutral and perfectly patient.7 If Buyer and Seller agree to trade at time τ ,Buyer receives utility xτ − Pτ and Seller receives Pτ . If Buyer quits at time τ , then playersget their outside options, normalized to 0 without loss of generality. Thus Seller’s utility attime t from a strategy θ is defined as:

ut = Eτθ[Pτθ1{aτθ = 1}

]And Buyer’s utility at time t from a strategy θ is defined as:

vt = Eτθ[ Buyer’s cost︷ ︸︸ ︷−cb(τθ − t) +

utility from purchase︷ ︸︸ ︷(Xτθ − Pτθ)1{aτθ = 1}

]Key Assumptions and Extensions This stylized model contains a number of features

in continuous time. Subsequent utility functions are well-defined if and only if τθ is a measurable function.Thus when considering profitable deviations, only strategies that produce a measurable stopping time τθ isallowed. In the Online Appendix, I construct an alternative equilibrium concept that does not restrict thestrategy space.

7In Online Appendix, I discuss how using time discounting instead of flow costs does not affect resultsqualitatively, as long as outside options are positive so that the players have an incentive to quit. The paperuses flow cost to illustrate the “investment effort” in the hold-up problem more directly.

8

Table 1: List of Extensions

Baseline Model Extensions Section

Seller has no flow cost Seller has flow cost and can quit 4Seller can subsidize Buyer’s cost 4.1

Symmetric information Heterogeneous Buyers and price discrimination 5.1Heterogeneous Sellers and signaling 5.2Seller observes match with noise 5.3

Seller can retract offers Seller can never retract past offers 6Seller makes offers Buyer makes offers 6Infinite horizon Finite horizon 6Additive attributes Bayesian signals on the true value 6

that help to simplify the problem. In various extensions, I explore what happens if thesefeatures are altered. For example, in the baseline model, product value xt is assumed to befully observable by both players. Seller observes Buyer’s preference and Buyer observes eachattribute accurately. In reality, potential buyers may hide their preferences in order to barterfor a lower price.8 In Section 5, I extend the model by giving players private information.Buyer can have private information on his outside option, Seller can have private informationon her quality, and Seller may learn about Buyer’s preference with noise. The full list ofextensions are summarized in Table 1.

3 Equilibrium Outcome

When Buyer and Seller reach agreement, Buyer’s utility should be equal to his contin-uation value, otherwise Seller should raise the price offer. Because the product is alwaysavailable at the list price P , Buyer cannot have a lower continuation value than if Sellernever gives a discount. Thus, solving for Buyer’s value function facing a fixed price of Pprovides a lower bound on Buyer’s continuation value in equilibrium. This is simply Buyer’ssingle-agent optimal stopping problem. Each moment, Buyer decides between buying at P ,continuing, or quitting. Such problems have been studied for investment under uncertainty(eg., Dixit 1993), R&D funding (Roberts and Weitzman 1981), experimentation (Moscariniand Smith 2001), and consumer search (e.g., Branco et al. 2012).

For states in which players trade, Seller receives U(x) = P (x) and Buyer receives V (x) =x−P (x). For states in which no agreement is reached and a player quits, U(x) = V (x) = 0.For a state x such that players choose to continue negotiating, we can write recursively:

U(x, t) = −csdt+ EU(x+ dx, t+ dt)

V (x, t) = −cbdt+ EV (x+ dx, t+ dt)(1)

8We can motivate truthful revelation in a simple model. Suppose that Buyer can choose whether to revealhis preference for each attribute. Buyer always chooses to reveal if he does not like the attribute. ThenSeller can infer Buyer’s preference when he does not reveal. Thus Buyer’s preference becomes unravelled.

9

Under stationarity, and by taking Taylor expansion and applying Ito’s Lemma on EU andEV terms, these expressions can be reduced to the following equations:

0 = −cs +σ2

2U ′′(x)

0 = −cb +σ2

2V ′′(x)

(2)

The solutions to the equations must be of the form:

U(x) =csσ2

(x− P )2 + As(x− P ) +Bs

V (x) =cbσ2

(x− P )2 + Ab(x− P ) +Bb

(3)

Let V (x) denote the Buyer’s value function facing a fixed price of P . Let x denote theBuyer’s buying threshold, and let x denote the Buyer’s quitting threshold. By Equation (3),

Buyer’s value function must be of the form V (x) = cbσ2 (x − P )2 + α(x − P ) + β for some

coefficients α and β. The function has to satisfy the following boundary conditions:{V (x) = x− P and V (x) = 0

V ′(x) = 1 and V ′(x) = 0(4)

The first two conditions ensure that the value function must match the stopping valuewhen Buyer buys or quits. The last two conditions, often referred to as “smooth-pasting”conditions, ensure that the stopping time is optimal. See Dixit (1993, pg.34-37) for moredetails.

Solving this system of equations shows that Buyer buys at

x = P +σ2

4cb(5)

and quits at

x = P − σ2

4cb(6)

Buyer’s value function of facing a fixed price of P is

V (x|P ) =

0, if x− πb ≤ P − σ2

4cbcbσ2 (x− P )2 + 1

2(x− P ) + σ2

16cb, if x ∈

(P − σ2

4cb, P + σ2

4cb

)x− P , if x− πb ≥ P + σ2

4cb

(7)

Lemma 1 shows that there exists an equilibrium where Buyer’s continuation value isV (x), and all equilibrium in which Buyer receives V (x) must yield the same outcome. Inthis equilibrium outcome, Buyer acts as if future prices are fixed at P . He accepts anoffer if the price makes him indifferent between continuing or stopping, and rejects the offerif the price is higher. In Online Appendix, we show that if one solves the discrete-time

10

game described in Section 2.1, then Buyer’s equilibrium value functions must converge toV (x) as the game approaches continuous time. As a result, the equilibrium outcome withV (x) = V (x) represents the continuous-time limit of the discrete-time equilibrium outcomes.For this reason, the equilibrium outcome with V (x) is our outcome of interest.

Lemma 1. There exists an equilibrium in which V (x) = V (x). If V (x) = V (x), then thereare unique thresholds x and x such that

• If x ≥ x, then Seller offers P (x) = x− V (x), and Buyer accepts.

• If x < x, then Seller offers P (x) > x− V (x), and Buyer rejects.

• If x ≤ x, then Buyer quits.

Even though Seller cannot commit to future prices, she can still enforce Buyer’s continu-ation value to be V (x). Consider the following strategy: Seller offers price P (x) = x−V (x) ifshe wants to trade, and offers P (x) > x−V (x) if she does not want to trade. If Seller followsthis strategy, Buyer never receives more than V (x) utility. Buyer’s optimal response then isto buy if P (x) = x− V (x), which is the price that makes Buyer indifferent between buyingnow and continuing. If P (x) > x− V (x), Buyer rejects because he gets less utility than hegets from continuing, since continuation value must be at least V (x). Thus by following thispricing strategy, Seller can control when they trade. In the proof we show that x < x, sothat

Intuitively, at every moment in time, Seller is choosing between two options. She cancontinue the sales process, or she can close the sale right now by offering a discount. If Sellerchooses to close the sale, she offers a discounted price of P (x) = x− V (x). This transformsthe game into Seller’s optimal stopping problem. Solving Seller’s optimal stopping problemalso solves the equilibrium, since Buyer’s optimal action is to comply with Seller’s stoppingdecision. The questions become when Seller should make the final offer, and what offer Sellershould make.9

Figure 4 illustrates Seller’s problem graphically. Product value x is on the horizontal axis,and shifts stochastically as the games continues. Buyer’s expected value from continuing thesales process given product value x, V (x) = V (x), is shown on top. The bottom graph showsP (x) = x− V (x), which is the highest offer that Buyer is willing to accept given x. Seller’svalue function U(x) is the straight line (due to zero flow cost) that hits 0 at x when Buyerquits, and hits P (x) = x − V (x) when they trade at x. To maximize U(x), the agreementthreshold x must make U(x) tangent to P (x). Otherwise, Seller can profitably deviate bymaking the offer earlier or later.

Note that Buyer’s continuation value, V (x), increases in product value x. However, thisdoes not mean that Buyer has stronger incentive to delay purchase when product valueincreases. Because product value x increases faster than V (x), Buyer is willing to pay ahigher price, P = x− V (x), or accept a lower discount, when x is higher.

Proposition 1 gives the closed-form solution of the equilibrium outcome given a list priceP .

9Note that Seller can only control the stopping decision for states between thresholds x and x, which are

Buyer’s stopping thresholds facing a fixed price of P . At x, Buyer quits and ends the game. At x, Buyerbuys the product even if there is no discount, so Seller cannot delay trade beyond these thresholds.

11

Figure 4

Proposition 1. Buyer and Seller trade at x = P + σ2

cb

[√14− P cb

σ2 − 14

]at price P (x) =

σ2

cb

[√14− P cb

σ2−2(

14−P cb

σ2

)], Buyer quits at x = P−1

4σ2

cb, and players continue for x < x < x.

The size of the price discount is strictly positive.

Seller’s ex-ante utility is

U(x0|P ) =

0 if x0 < x

(14σ2

cb+ x0 − P )(1− 2

√14− P cb

σ2 ) if x ≤ x0 ≤ x

P if x0 > x

(8)

Proposition 1 shows that the list price plays a crucial role in facilitating discovery, andthere exists an optimal list price for Seller. A higher list price discourages Buyer fromdiscovering but allows Seller to extract a bigger share of the pie. Figure 5a illustrates thispoint.

As P increases, x = P − 14σ2

cbincreases, which means that Buyer leaves the sales process

earlier when he receives unfavorable information about product match. Note that any listprice above x+ 1

4σ2

cb, including a list price of∞, leads to immediate breakdown of the process.

Buyer wants to quit at time 0 because x0 < x. Its expected gain from future trade is notenough to justify the cost of staying in the sales process. This is conceptually similar tothe hold-up problem in Wernerfelt (1994b). It is costly for Buyer to find out about productmatch (or quality), and Seller can hold Buyer up by charging a price equal to the productvalue after Buyer incur the cost. This gives Buyer negative utility ex-ante, and as a result,Buyer chooses to not spend any effort in the first place. Thus in order to encourage Buyer toparticipate, Seller has to impose a low enough list price to raise the option value of discoveryfor Buyer. If the product fit is bad, Buyer does not have to buy the product, but if theproduct fit is good, Buyer is guaranteed to pay no more than the list price. On the downside,

12

Figure 5

(a) (b)

a lower list price restricts Seller’s ability to extract rent. Buyer’s higher continuation valuemeans that Seller has to offer lower price in order to close the sale, because P (x) = x−V (x)decreases as P increases. The list price can be viewed as an instrument that balances Seller’stwo incentives: inducing Buyer to discover and exploiting surplus post discovery.

The second finding is that the final trading price is always lower than the list price,regardless of what the list price is. Thus, Proposition 1 predicts that the sale must come ata discount.10 Figure 5b illustrates why it is never optimal for Seller to sell at the list price. IfSeller never offers a discount, Buyer will continue until product value reaches x = P+ σ2

4cb. As

Buyer gets close to this threshold, Buyer’s value function V (x) becomes increasingly tangentto x−P . As a result, P (x) becomes increasingly tangent to P . So, Buyer is willing to acceptoffers with discounts that approach 0. Seller can close the deal earlier by sacrificing verylittle on price. Closing the sale earlier is beneficial because it increases the ex-ante successrate and saves cost (in this case, only Buyer’s cost is saved, but Seller can extract Buyer’ssavings through price). The amount of discount required to close the sale increases at afaster rate as product value decreases, so it is not optimal to close the deal too early. Inother words, having some discovery of product match is valuable to Seller, because discoveryhas the possibility of increasing total surplus by resolving uncertainty.

Proposition 2 provides the optimal list price as a function of the initial value, x0.11

Proposition 2. (Optimal List Price)

10If there are different types of buyers with different costs and starting positions, then some buyers couldbuy at the list price in equilibrium.

11Buyers in this model arrive with the same x0 and face the same list price. Suppose Buyers arrive withdifferent x0 drawn from a distribution, and Seller observe x0 after setting P . Proposition 1 shows that, givena list price, equilibrium thresholds do not depend on initial position. But final price can be different. Ifa buyer’s initial position is already above x, then Seller’s price offer, P = x − V (x), must be higher thanx− V (x). Price for x > x is shown in Figure 5(b). One can solve the optimal list price by maximizing

U0 = maxP

∫U(x0|P )dF (x0)

13

• For intermediate values of x0 (−14σ2

cb< x0 < 1

16σ2

cb), the optimal list price is 1

3x0 −

118σ2

cb

√1− 12x0

cbσ2 + 7

36σ2

cb, and the game continues beyond t = 0.

• For x0 ≥ 116σ2

cb, any list price above x0 + σ2

4cb(including no list price) is optimal. Players

trade at price P = x0 at t = 0.

• For lower x0, any non-negative list price (including no list price) is optimal. Buyerquits at t = 0.

Seller’s ex-ante utility at optimal list price is

U(x0) =

x if x0 >

116σ2

cb23x0 + 1

54σ2

cb+ ( 1

54σ2

cb− 2

9x0)√

1− 12x0cbσ2 if − 1

4σ2

cb≤ x0 ≤ 1

16σ2

cb

0 if x0 < −14σ2

cb

(9)

Proposition 2 shows that the selling process only takes place if the initial surplus is nottoo big or too small. If the initial surplus is big enough (x0 ≥ 1

16σ2

cb), then Seller does not

want Buyer to learn anything about the product. Seller publishes a list price high enoughto deter Buyer from discovering, and offers a monopoly spot price that extracts all existingsurplus. On the other hand, if the initial surplus is too low (x0 ≤ −1

4σ2

cb), matching is socially

inefficient. Buyer will leave the game even if the list price is set to 0.12

When coming to the market, should Seller commit to offering a fixed price? The followingcorollary compares the equilibrium outcome to the outcome under an optimal fixed price.In the context of e-commerce, this comparison provides a prediction on what would happenif more retailers can monitor customers’ browsing behaviors and make dynamic offers basedon their real-time purchase intent. We find that this would improve profit, reduce consumerwelfare even though final prices are lower, and reduce the amount of information discoveredas retailers use discounts to shorten sales cycle.

Corollary 1 (Comparison to Fixed Price). Social welfare and Seller’s utility are higherwith discounting. Buyer’s utility is lower with discounting. The optimal list price underdiscounting is higher than the optimal fixed price, and the final price after discount is lowerthan the optimal fixed price. Expected length of the game is shorter if discount is allowed.

Proof. Let T denote the total length of the game. Buyer’s ex-ante utility must satisfyV (x0) = x0−x

x−x (x − P (x)) − cbE[T ], where x0−xx−x is the ex-ante probability that players reach

agreement. Expected length of the sales process is thus calculated as E[T ] = 1cb

[x0−xx−x (x −

P (x))− V (xn,0)]. The comparisons are straightforward.

Figures 3 from Section 2 illustrates the value of Seller’s ability to offer dynamic discounts.A small discount significantly decreases Buyer’s buying threshold from P + σ2

4cbto x. Seller’s

ability to offer discount significantly decreases the length of the sales process. The cost

where U(x0|P ) is given in equation (8) and F (x0) is the CDF of x0. If Seller does not observe x0, thenBuyers may receive different discounts in equilibrium. This latter case is explored in Section 5.1.

12Since only Buyer has cost, Buyer’s action is socially optimal when the list price is 0.

14

savings are captured by Seller through price and increases her ex-ante utility. If playerssubsequently discover that the product match is bad, then the game ends without a trade.Thus lowering trading threshold increases the ex-ante success rate.13

Corollary 2. (Comparative Statics w.r.t cb and x0 for −14σ2

cb< x0 <

116σ2

cb)

Optimal list price and final price decrease in Buyer’s cost cb; size of price discount decreasesin Buyer’s cost cb; ex-ante probability of trade is unaffected by Buyer’s cost cb; expected lengthof the game decreases in Buyer’s cost cb, and has inverse U-shape in initial value x0.

Interestingly, the expected length of the game is non-monotonic in initial position x0.Note that the initial position is normalized with respect to outside options and cost ofproduction. Intuitively, one would expect that worse outside options make players moreinterested in matching with each other. However, as outside options become increasinglypoor, surplus from trade gets higher. As a result, Seller is more inclined to close the dealearly. Thus the length of the sales process is shorter for both very good and very bad outsideoptions.

Another finding is that the ex-ante probability of trade is unaffected by Buyer’s cost. Fora given list price, the probability of trade decreases in cb. However, a higher cost for Buyermakes the hold-up problem more severe, which then pushes Seller to set a lower list price.In equilibrium, these two effects negate each other. Note that if discount is not allowed, onecan show that the ex-ante probability of trade is monotonically decreasing in Buyer’s cost,even if Seller set the price optimally.

Furthermore, when Buyer’s cost of continuing the sales process increases, the price dis-count he receives actually decreases. The reason is that with a higher cost for Buyer, Sellerhas to set a lower list price to begin with. Buyer still receives a lower final price even thoughthe size of the discount is smaller. In the context of e-commerce, this means that as onlineshopping and information search become more convenient, we may observe higher list pricesbut with deeper personalized discounts.

4 Costly Selling

Selling can be a costly effort so that Seller may want to quit before Buyer does. Followingnotation from the last section, let indicator functions qb(x) and qs(x) denote Buyer andSeller’s quitting decisions and x = sup{x|

∑i qi(x) > 0} denote the threshold where the

“earliest” quitting happens. There are many trivial equilibria in which both players quitsimultaneously. If the opponent quits at x, then a player is indifferent between quitting andnot quitting at x. As a result, quitting at any state x can be supported in an equilibriumby having both players quit simultaneously.14 To avoid this triviality, I restrict attention toequilibrium outcomes that satisfy the following condition.

13Note that the graph does not show the full price path. There are infinite price paths leading up to timeτ that produce the same equilibrium outcome. Price strategies before time τ only need to be high enoughso that Buyer does not want to buy. Keeping the price at the list price before time τ , for example, wouldwork.

14Due to the nature of continuous time, a strategy profile can achieve the effect of simultaneous quittingwithout having players quitting at the same x. For example, suppose Seller quits in a set B except a singlepoint x′. Then on x′, Buyer cannot extend the game whether he quits or not. The game ends immediately

15

Condition 1. Either U(x) = U ′(x) = 0 or V (x) = V ′(x) = 0.

The condition implies that the quitting decision is optimal for at least one of the player.In a single agent optimal stopping problem, the quitting threshold must satisfy the value-matching condition, ui(x) = 0, and smooth-pasting condition, u′i(x) = 0. The value-matching condition ensures that the player does not quit if continuation value is positive,and the smooth-pasting condition ensures that the timing of quitting is optimal (Dixit 1993,pg.34-37). This condition is satisfied for Buyer’s quitting threshold in Section 2. In OnlineAppendix, I show that if one requires the players to quit if and only if quitting is strictlypreferred, then the limit of discrete-time equilibrium outcomes satisfies Condition 1. ThusCondition 1 eliminates the simultaneous quitting triviality.

As in Section 3, I first derive the lower bound on Buyer’s equilibrium value function.One can solve for Buyer’s value function facing a fixed price of P , subject to a game-endingstate at x. This gives a lower bound on the Buyer’s equilibrium value function x. Denotethis lower bound as V (x|x). The closed-form solution for this lower bound is presented inAppendix A.2. Then one can construct equilibrium strategies that give such payoff to Buyerand satisfy Condition 1. As before, this lower bound payoff is the limit of Buyer’s equilibriumpayoff in the discrete-time game, so this is the outcome of interest in this paper.

Lemma 2. There exists an unique equilibrium outcome that has V (x) = V (x|x) and satisfiesCondition 1.

It is unclear which player quits first in equilibrium.15 Intuitively, the player with a higherflow cost is more likely to quit earlier. However, Proposition 3 below shows that this is notthe case. If Seller chooses the list price optimally, then Buyer always quits before Seller does.

Proposition 3. If Seller quits earlier than Buyer does, i.e., sup{x|qs(x) = 1} >sup{x|qb(x) = 1}, then the list price P is sub-optimal (too low) for the equilibrium outcomein Lemma 2.

Consider the effect of raising the list price. As Figure 5a shows, increasing the list pricehas both positive and negative effects for Seller. The negative effect is that Buyer’s lowercontinuation value pushes him to quit earlier. However, if Seller is quitting earlier thanBuyer does anyway, then pushing Buyer to quit earlier has no effect. Thus, raising the listprice is strictly beneficial to the Seller. The original list price must be sub-optimal.

Proposition 3 implies that, to find the outcome under optimal list price, one can ignoreSeller’s quitting decision, then maximizing U(x0) over P , and finally verifying that Sellerdoes not want to quit for x > x. For tractability, I restrict attention to the case of x0 = 0.

Proposition 4 (Costly Selling with x0 = 0). Let k = cscb

. The optimal list price is P =

σ2

cb(1

4− 18+10k−6

√9+10k+k2

16k2). Buyer and Seller trade at x = P + σ2

cb

[3−

√9+k1+k

4k− 1

4

]at price

P (x) = P − σ2

cb

(3−

√9+k1+k

4k− 1

2

)2

. Buyer quits at x = P − 14σ2

cb.

even if Buyer does not quit, making him indifferent between quitting at x′ or not. This example illustratesthat it it not sufficient to simply restricting players to not quit at the same x.

15That is, whether x is the optimal quitting threshold for Buyer or for Seller. If x is optimal for Buyer(Seller), then we must have V (x) = V ′(x) = 0 (U(x) = U ′(x) = 0), due to the discussion above.

16

The ex-ante utility of Seller in equilibrium is:

U(x0) =σ2

cb

[kz2 + z − 2

√1 + kz3/2

]where z =

18 + 10k − 6√

1 + k√

9 + k

16k2(10)

If Seller has the option to commit to a fixed price, should Seller do it? Corollary 3 showsthat, in some cases, having the ability to discount is necessary for the sales process to takeplace. Under fixed price, when Seller’ cost is high, Seller needs to set a high price to make upfor her expected cost of selling. But a higher fixed price pushes Buyer to wait longer beforebuying, which makes the process even more costly for Seller, who in turn has to charge aneven higher price. If Seller charges too high a price, Buyer quits immediately. Thus theredoes not exist a fixed price such that both find it worthwhile to “sit down” at time 0. Thisproblem is avoided if Seller can offer discounts. The flexibility in price lowers both player’sexpected costs from engaging in the sales process. This flexibility is particularly beneficialwhen Seller’s flow cost is high. When Seller’s cost is low, Buyer prefers fixed price, but whenthe ratio k = cs

cbexceeds a certain threshold, both Buyer and Seller gain from Seller’s ability

to discount.

Corollary 3 (Importance of Discounting). If k(= cscb

) ≥ x04cbσ2 + 1, then the game ends

at t > 0 with positive probability of trade if and only if price can be discounted. Seller’sex-ante utility is higher under discounting for all k. Buyer’s ex-ante utility is higher underdiscounting if k > k, for some k ≤ x0

4cbσ2 + 1.

This result helps to explain the prevalence of “list price - discount” in many B2B in-dustries. Firms have to incur significant cost in employing and training salespersons, andsalespersons often spend a significant amount of resources on each client. Corollary 3 showsthat, if this cost is high relative to the customer’s cost of participating in the sales process,then the firm must be open to discounting. Otherwise, trade cannot take place.

In practice, a firm that wants to offer individual discounts has to delegate pricing au-thority to its salespersons. The question of whether to delegate has been studied underprincipal-agent models. See, e.g., Lal (1986), Bhardwaj (2001), and Joseph (2001). Theprincipal-agent models used in these papers highlight the disadvantage of delegating pricingauthority when selling cost is high. Salespersons have the incentive to give customers toomuch discount so they can shirk on selling efforts. This problem is intensified when theselling efforts are more costly. On the other hand, the information acquisition approachof this paper emphasizes the advantage of delegating pricing authority when selling cost ishigh. Using a survey of 270 companies from different industries, Hansen et al. (2008) findthat firms that need more calls to close a sale are more likely to delegate pricing authorityto salespersons. Corollary 3 provides one explanation for their observation.

Corollary 4 (Comparative Statics w.r.t cs and cb at x0 = 0).Optimal list price decreases in Buyer’s cost, cb, and increases in Seller’s cost, cs; final pricedecreases in both players’ costs, cb and cs; size of price discount decreases in Buyer’s cost, cb,and increases in Seller’s cost, cs; and expected length of the game decreases in both player’scosts, cb and cs.

A higher cs leads to a higher P but a lower P (x). When Seller’s cost increases, Sellerposts a higher list price, but gives a bigger discount quickly and sells at a lower price than

17

before. The lower final price helps to reduce the length of the game and saves cost. Notethat, if Seller is not able to give a discount (as under a fixed price), Seller would want tocharge a higher price when cost increases. This eventually leads to the no-trade result fromCorollary 3.

4.1 Subsidy to Buyer

In the previous analysis, Seller’s cost is exogenous. In practice, this cost may be en-dogenous as sellers often subsidizes buyers’ costs by providing certain perks such as meals,resort conferences, gifts, and coupons. These activities are intended to encourage buyers toparticipate and stay.

Assume Seller’s flow cost without subsidy to be 0, and Buyer’s cost without subsidy tobe c. For simplicity, let the initial value x0 = 0. At time 0, Seller promises a “per-period”subsidy, g, to be paid to Buyer. This transfer of utility is not free. I assume that Buyeronly receives θg, where θ ≤ 1. This is a parsimonious way to capture the idea that the useof subsidy involves efficiency loss. Typically, sellers cannot directly pay buyers monetarily.Instead, sellers provide perks to buyers through hosted events or free meals. These activitiesrequire administrative costs, and buyers may not value these perks at their monetary facevalue. These losses make up 1 − θ. For a period of dt, Seller incurs cost gdt while Buyerincurs cost (c− θg)dt.

No shirking is allowed. Seller cannot change per-period subsidy after time 0, and Buyercannot receive subsidy g but avoid paying cost c. Note that if shirking is allowed, bothplayers have incentives to shirk. A more realistic model may consider letting Seller promisefuture subsidy with a finite duration. Such model is beyond the scope of this paper.

As before, k = gc−θg denotes the ratio between Seller’s cost and Buyer’s cost. One can

solve the equilibrium with subsidy by calculating Seller’s ex-ante utility for a given subsidyg, then maximizes with respect to g, to find the optimal size of subsidy. From Proposition(4), Seller’s ex-ante utility in equilibrium must be:

U0 =σ2

c(1 + θk)

[kz2 + z − 2

√1 + kz3/2

]− F (g) (11)

where z =1

4− P c− θg

σ2=

18 + 10k − 6√

1 + k√

9 + k

16k2(12)

The first order derivative with respect to k gives:

dU0

dk=( 0︷︸︸︷∂U0

∂z

dz

dk+∂U0

∂k

)(13)

=σ2

cz3/2[√z − θ − 1

θ

1√1 + k

] (14)

=σ2

cz3/2[

3√

1 + k −√

9 + k

4k− θ − 1

θ

1√1 + k

] (15)

All interior solutions to dU0

dk= 0 have d2U0

dk2> 0. As a result, there is no intermediate size of

18

subsidy that maximizes Seller’s utility. In equilibrium, Seller either chooses to not providesubsidy at all, or subsidizes away all of Buyer’s cost.

What happens if Seller chooses to subsidize all of Buyer’s cost? Buyer will always havea utility of 0 and be indifferent between quitting and not quitting. If we assume that Buyerdoesn’t quit, then Seller acts like a social planner as it incurs all cost and receives all welfare.Solving Seller’s stopping problem with a cost of c/θ and a stopping value of x gives Seller anex-ante utility of U0 = θ σ

2

16c. Seller does not need a list price here, as a full subsidy eliminates

the hold-up problem.Compare this to Seller’s ex-ante utility without subsidy from Proposition 2, 1

27σ2

c, one

can see that Seller prefers to subsidize Buyer’s cost when θ > 1627

. Thus if the efficiency lossfrom subsidy is high, Seller prefers to not subsidize and use a list price to deter Buyer fromquitting. If the efficiency loss is low, Seller prefers to subsidize away Buyer’s cost, ratherthan using a list price.

Note that in this example, there’s no interior solution for the optimal subsidy becausethe cost of subsidizing is linear. If there is increasing cost of subsidizing, or if there’s afixed cost that is increasing in the size of subsidy, Seller may only subsidize part of Buyer’scost. However, such equilibrium would give the same qualitative findings as having exoge-nous costs. Subsidy lessens Buyer’s concern of hold-up but does not eliminate it. Giventhe optimal subsidy, the equilibrium list price and discount must still follow Proposition(4). The list price, though never executed, provides a lower bound on Buyer’s utility andencourages discovery. Given a list price, Seller always prefers the ability to discount ratherthan committing to a fixed price. Corollary (3) also has a new interpretation in such a case.Seller is restricted to providing low subsidy if she commits to a fixed price. The flexibilityto offer discount allows Seller to offer higher subsidy.

5 Asymmetric Information

5.1 Heterogeneous Buyers and (Reverse) Price Discrimination

The previous Sections assume that Buyer’s expected value for the product, xt, is fullyobservable to Seller. In many settings, though, the buyer may have private informationregarding his willingness-to-pay. Also, even if the seller observes the buyer’s preference foreach attribute, the seller may not know what the buyer’s outside option is. This sectionexpands the model to incorporate this scenario.

There are two types of Buyer, H and L, with different outside options. This is Buyer’sprivate information. L type has a better outside option than H type. A better outside optiondecreases Buyer’s maximum willingness-to-play. If we normalize Buyer’s outside option intohis product value, then at each moment, Seller is facing a distribution of Buyers with differentlevels of net product values. Seller observes how this distribution shifts over time, but doesnot observe where Buyer falls within this distribution.

Nature draws H type with probability λ and L type with probability 1− λ. Define ε asthe different in outside options between the two types. Let xH and xL denote the H typeand L type’s product values minus their respective outside options. Denote x = 1

2(xH + xL)

so that xH = x + ε2

and xL = x− ε2. Both H type and L type incur flow cost cb > 0. Seller

19

is assumed to have no cost and never quits.I look for stationary PBE with pure strategies. Equilibrium utilities and strategies now

depend on two state variables, x and µ, where µ ∈ [0, 1] denote Seller’s belief that Buyer isof type H. Type i’s value function is Vi(x, µ). Note that with pure strategies, Seller’s beliefµ can only take 3 values: 0, 1, or λ. If µ = 0 or µ = 1, then there is no private information.This happens on the path of a separating equilibrium. I assume that Seller does not updateher belief off the equilibrium path.

Similar to earlier sections, I first look for the lower bound on Buyer’s equilibrium utility.Let V i(x, µ) denote the lower bound on type i’s equilibrium value function in state x andunder belief µ. If only one type of Buyer remains in the game (µ = 0 or µ = 1), then thelower bound must be V (x) from Section 3. One can show that L type’s lower bound is V (x)even when both types are still in the game. This then pins down H type’s lower bound. Theproof and closed-form solutions for V i(x, µ) are presented in Appendix A.3.

Seller then faces the new optimal stopping problem regarding when to sell to H type andat what price. Proposition 5 describes the equilibrium outcome in which each type of Buyerreceives the lower bound of his equilibrium value functions. We only need to solve for thecase of P < x0 − ε

2+ σ2

4cb, otherwise types unravel immediately.16

Proposition 5 (For P < x0 − ε2

+ σ2

4cb). Buyer with type i ∈ {H,L} quits at xi = P − 1

4σ2

cb,

and buys at xi = P + σ2

cb

[√14− P cb

σ2 − 14

]. H type buys earlier and pays less than L type.

Figure 6a presents a sample equilibrium path. H type buys earlier, and pays a lowerprice. Price increases from time τH to τL. Figure 6b shows xH and xL separately. Bothtypes of Buyer buy when their net product value reach x. H type reaches the thresholdearlier, because his worse outside option translates to a higher net value from buying theproduct.

The finding that H type pays less than L type runs counter to previous repeated-offersmodels, in which the seller makes declining offers to screen through buyers’ reservationprices, so types that buy earlier pay higher prices. Why does L type not take H type’s offer,if waiting is costly and price is rising? The reason is that, when Seller makes an offer toH type, L type’s net product value is lower by ε, and this difference makes L type strictlyprefer to wait and collect more information. On the flip side, Seller has to offer H type alower price because H type can pretend to be L type. H type can wait until xL reaches thebuying threshold, and then pay the same price as L type. However, Seller prefers to closethe sale with H type earlier, which saves cost as well as locking in the surplus. Intuitively, ifa Buyer already comes into the sales process with a good intention to buy, then discovery isless valuable, and closing the sale earlier is more efficient. This creates a binding incentivecompatibility constraint that forces the seller to concede more utility to H type. Thus inthis game, Buyers self-sort into processes of different lengths: a shorter process for H type

16By Lemma 3 from Appendix A.3, L type should act the same way as in Proposition 1. Thus, when

P ≥ x0 − ε2 + σ2

4cb, L type buys immediately if x0 − ε

2 > 0 or quits immediately if x0 − ε2 ≤ 0. By Lemma 3,

H type should buy immediately if L type buys immediately. If L type quits immediately and H type stays,then there is only a single type of Buyer remaining, which is again solved in Proposition 1. So we do not

need to solve for the case of P ≥ x0 − ε2 + σ2

4cb.

20

Figure 6: Equilibrium Path for Heterogeneous Buyer

(a)(b)

and a longer process for L type. The price cut for H type represents H type’s informationrent.17

5.2 Heterogeneous Seller and List Price Signaling

If sellers have private information about the quality of their products or services, theycan publish different list prices as a way of signaling. In this extension, I show that thereexists separating equilibria in which the high quality firm sets a higher-than-optimal listprice to reveal its type.

There are two types of sellers, H and L, who are different in their qualities and in theirscosts of delivering the product. I assume that quality is a vertical component of productutility that is additive to the match value xt. Conditional on the same level of match,purchasing from H type Seller delivers q more utils than purchasing from L type. In themodel, this translates into different starting positions. Let x0 = 0 for L type, then x0 = q forH type. Consumers observe the incremental changes of xt, but has to form belief about thestarting position from the list price. L type’s cost of delivering the product is normalizedto 0, and H type’s cost of delivering the product is K > 0. I look for separating PBE withstationary strategies. If L type Seller mimics H type’s list price, it may still has incentivesto deviate from H type’s subsequent discounting strategy. Thus on off-path nodes, I assumethat Buyer infers that Seller is L type. That is, if L type mimics H type’s list price butengages in different discounting behavior afterward, Buyer believes that the Seller is actuallya L type upon seeing such deviation.

17Some factors ignored in this model can cause a downward trend in price. For example, if Seller cancommit to each price offer for a short period of time and if players have time discounting, then Seller canuse declining prices to screen different types. This is common in the repeated-offers bargaining literature.Section 6 shows a few extensions in which price can decrease over time - for example, if the product has afinite mass of attributes, or if players observe signals of the true match value. However, the core intuitionshould still remain. The fact that H type has a worse outside option makes the product more appealingex-ante and decreases the value of information acquisition. This encourages Seller to trade with him earlier,which puts downward pressure on H type’s price in order for the offer to be incentive compatible.

21

Let PH and PL denote H type and L type’s list prices, respectively. Also let P∗H and

P∗L denote H type and L type’s optimal list prices if quality is public information. In a

separating equilibrium, Buyer correctly infers Seller’s type at t = 0 upon observing the listprice. That means L type must set its list price at PL = P

∗L = 5

36σ2

cbfrom Proposition (2).

From Section 3, we know that Seller’s ex-ante utility for a given list price, with the costof product normalized to 0 is:

U(x0|P ) = (1

4

σ2

cb+ x0 − P )(1− 2

√1

4− P cb

σ2) (16)

Then L type’s equilibrium utility must be U(0| 536σ2

cb) = 1

27σ2

cb. H type’s equilibrium utility

can be written as U(q − K|PH − K). If L pretends to be H type by using H’s list price,then L has to subsequently follow through with H’s offer strategy, otherwise Buyer infersthat Seller is actually L type. But comparing to H type, L type saves the cost of product Kwhen a sale is made. Thus L’s deviating utility is U(q−K|PH−K)+K ∗Pr(q−K|PH−K),where Pr(q − K|PH − K) is the ex-ante probability of a sale. If H pretends to be L byusing L’s list price, H does not have to follow through with L’s offering strategy, becauseBuyer already believes Seller to be L type. H’s optimal offering strategy given Buyer’sbelief gives a deviating utility of U(−K|P ∗L − K). Combining these, we get the followingincentive-compatibility constraints:

(ICL)1

27

σ2

cb≥ U(q −K|PH −K) +K ∗ Pr(q −K|PH −K)

(ICH) U(q −K|PH −K) ≥ U(−K|P ∗L −K)

H type’s list price PH is the only unknown in these equations. Any PH that satisfiesthese two conditions constitute a separating equilibrium. There is no analytical solution toPH , but given any q and K, one can solve for PH numerically. Figure 7 shows the range ofPH and H type optimal list price P

∗H under public information as functions of q −K.

There are a few observations to make. First, H type’s list price is higher than its list priceunder public information. A higher list price decreases the ex-ante probability of sale, whichhurts L type more than H type because L type’s cost advantage. So a higher-than-optimallist price is used by H type to deter L type from imitating. Second, separating equilibriaexist if q − K is not too high. If H type’s quality advantage is much higher than its cost,then H cannot stop L from imitating. Third, the IC condition for L type implies that Ltype must earn higher profit than H type in a separating equilibrium, because L type canset its list price optimally, whereas H type must a higher-than-optimal list price to deterimitation.

Other papers, including Xu and Dukes (2017), Yavas and Yang (1995), Haurin et al.(2010), and Shin (2005), have discussed how list price can signal the seller’s private informa-tion. The current model is different from previous papers in several way. In Xu and Dukes(2017), some consumers buy at the list price, whereas all trades happen below list price inthe current model. Both Yavas and Yang (1995) and Haurin et al. (2010) directly assumethat a higher list price leads to less matched customers. The current model provides a microfoundation for their assumption. In Shin (2005), actual price cannot be changed over time,

22

Figure 7: H type’s list price in separating equilibria with K = 120σ2

cb

as is studied here.

5.3 Noisy Learning by Seller

In the baseline model, the discovery process is symmetric. Buyer observes its match withSeller’s attributes, and Seller learns Buyer’s preferences on attributes perfectly. However, inreality, one can imagine the discovery process to be asymmetric. Seller may not know exactlyhow Buyer feels about each attribute, but only get a noisy signal of Buyer’s preference.

Fully allowing asymmetric learning in the baseline model with continuous time and con-tinuous state space is intractable. Instead, to capture the idea, I study the discrete modelmotivated in Section 2.1 with 3 periods (0, 1, and 2). The product is consisted of 2 at-tributes. Depending on the fit, each attribute delivers vl or vh utility to Buyer, and thevalue of the product is the sum of the two attributes. As in the baseline model, the ex-anteprobability of match for each attribute is 1

2and independent across attributes. Starting in

t = 0, Buyer incurs a small cost of c for moving to the next period, and learns the value ofone attribute in t = 1 and the other in t = 2. When Buyer learns, Seller receives a signal onthe match with an accuracy of θ. That is, if Buyer learns that attribute 1 is a good match,Seller observes that the match is good with probability θ and observes that the match is badwith probability 1− θ.

Let ∆v = vh − vl. Outside options and production cost are normalized to 0. Let v0

denote the expected value of the product at t = 0. To make learning socially relevant, weassume v0 − ∆v < 0. This means that a product with two mismatched attributes has lessvalue than its cost (production cost plus the opportunity cost of outside options).

One can solve SPE of this game using backward induction. For θ not too high, theequilibrium outcome is summarized below:

Proposition 6 (For θ ≤ ∆v−6c32

∆v−8c).

23

• For intermediate v0 (6c−∆v < v0 <12∆v−2c), Seller sets a list price of v0 + ∆v−6c.

At t = 1, Seller offers a discounted price of v0 + 12∆v − 2c. Buyer buys at t = 1 if the

first attribute is a match, and leaves at t = 1 if the first attribute is a mismatch.

• For lower v0, Buyer leaves at t = 0.

• For higher v0, Seller does not set a list price and charges P0 = v0 at t = 0. Buyer buysimmediately.

Proposition 6 closely resembles Proposition 1 and 2. The “list price - discount” patternreplicates. If the ex-ante value of the product is too high or too low, games ends at time 0.But for an intermediate range, the Seller sets a list price to encourage Buyer to learn, butalways close the sale with a discount before all information is revealed.

One may be interested in a model with two-sided noise. However, such models aregenerally difficult to solve due to the explosion of states and possibility of signaling. Pastmodels of repeated-offers with stochastic value, such as Daley and Green (2020), focus onone-sided learning in which only one player updates his belief about the value. I argue that itis important to understand what happens if both parties learn. The baseline model exploresthe special case in which the arrival of information is symmetric. This 3-period extensionshows that the findings can replicate even when Seller does not observe Buyer’s preferencesperfectly.

6 Other Extensions

This section discuss key results from other extensions of the model. Details can be foundin Online Appendix.

Non-retractable Offers

In the baseline model, an offer expires instantly if Buyer does not accept it. Seller makesno promise regarding future discounts and can actually raise prices over time. However, onemay imagine circumstances in which social norms prevent Seller from doing so. If Buyercan recall all previous price offers, then each new lowest offer becomes the new list price,so the standing list price can only decrease over time. We can show that Seller prefers tomaintain the highest price possible, so the only time that Seller decreases list price is whenBuyer is about to leave. Figure 8 gives an example of the equilibrium path. Furthermore,there exists a threshold, x, such that for x0 < x, Seller prefers to use non-retractable offers,and for x0 > x, Seller prefers the offers to be retractable. The intuition is that, using non-retractable offers allow Seller to lower list price over time to prevent Buyer from quitting.This is beneficial when ex-ante product value is low. The disadvantage of non-retractableoffers is that it takes away Seller’s ability to close sale with discount. Any new discountlowers the list price, which increases Buyer’s continuation value. This limits Seller’s abilityto extract utility when product value is high.

24

Figure 8: Non-retractable Offers

Finite Mass of Attributes

The baseline model assumes an infinite mass of attributes. In reality, a product mayhave only a limited number of features, or consumers may care about only a finite subset ofa product’s features. As a result, players exhaust all the information they can learn fromthe other party if the sales process continues long enough. Suppose that the product has afinite mass of attributes, T . The product value xt follows a Brownian motion between time0 and time T . One can solve the equilibrium numerically. Figure 10 presents the solutionfor the case of T = 1, cb = 0.25, σ2 = 1, and x0 = 0. We recover the “list price - discount”pattern. List price cannot be too high, otherwise Buyer quits at time 0. Also, players alwaysreach agreement at a discount. The difference between finite horizon and infinite horizon isthat all three thresholds shrinks over time. This happens because as t approaches T , theremaining game is just a smaller version of the game started at t = 0. Also note that thethresholds collapse before T , so players don’t wait until they discover all attributes.

Bayesian Learning

Instead of matching on attribute, players observe signals on the true value of the productand update their beliefs using Bayes’ rule. Suppose that, before the negotiation begins, Buyerand Seller have a common prior on the true value x∗ that follows a normal distribution withmean ν and variance e2. Let St denote the cumulative signal up to time t. The signal isassumed to follow the process dSt = x∗dt + ηdW . Bayesian updating on the normal priorimplies that the expected value of the product after observing t signals, xt, can be writtenas

xt =ν/e2 + St/η

2

1/e2 + t/η2(17)

Figure 9 illustrate the numerical solution for the case of cb = 0.25, ν = 0, e2 = 1, η2 = 1,at a list price of 0.5. Similar to the previous extension to finite attributes, we replicate the“list price - discount” pattern, and the game must end before Buyer’s quitting threshold and

25

Figure 9

agreement threshold collapse at 0.Note that if Buyer has evolving preferences so that the true value of x∗t evolves with

a random walk of variance ρ2, then the posterior variance follows dσtdt

= −σ2t /η

2 + ρ2. Ifσ0 = ρη, then σt = σ0 remains constant for all t. Thus the baseline model can be interpretedas a Bayesian learning model with an evolving preference.

Buyer Offers and Price Floor

If Buyer, instead of Seller, makes all the offers, list price becomes useless. In a subgame-perfect equilibrium, Buyer never offers any price above 0. A price ceiling in the form of listprice cannot prevent Seller from being held-up, and if Seller has any cost, Seller would quitimmediately. Instead, Seller may benefit from a price floor that bound offers from below.

The model is modified as follows: (1) at time 0, Seller chooses a price floor, P , suchthat all future offers below P are automatically rejected; (2) Buyer makes offers, and Sellerdecides whether to accept or reject. The optimal lower bound is P = σ2

8cb. Buyer’s offers P

when xt reaches P + σ2

4cband quits when xt reaches P − σ2

4cb.

One can compare the equilibrium outcome with Buyer offers to the equilibrium outcomewith Seller offers. Final price is actually higher when Buyer makes offers. The sales process isexpected to be longer when Buyer makes offers. Buyer’s utility increases and Seller’s utilitydecreases when Buyer makes offers. However, social welfare is lower when Buyer makesoffers.

7 Conclusion

This paper studies the pricing mechanism where a seller sets a list price upfront, thendecides what discounts to offer during interaction with a buyer. Such mechanism is commonin B2B sales but also begins to emerge in AI-powered e-commerce. A given pair of buyer and

26

Figure 10

seller do not know their value from trading with each other ex-ante due to heterogeneity, butthey can learn about their match over time. The seller chooses a list price ex-ante that actsas a price ceiling, but can adjust offer discounts dynamically as the two parties learn. Thematching process causes the expected total surplus to be stochastic. The process is costlyso that the parties have incentives to quit.

The model shows that the combination of stochastic surplus and endogenous quittingcreates a hold-up problem for the buyer. A list price that puts an upper bound on seller’sfuture offers serves as an instrument that reduces the buyer’s concern for being held-up afterspending efforts to match. The optimal choice of list price has to be low enough to encouragethe buyer to stay, without being too restrictive on the seller’s ability to extract surplus later.Not setting a list price leads to an immediate breakdown of trade as the buyer becomesunwilling to engage in the process. On the other hand, the model shows that trade alwayshappen strictly below the list price. The seller always prefers to shorten the sales cycle byoffering discounts. This result provides a rational explanation for the “list price - discount”pattern observed in practice, especially in industries where buyers rarely pay the list prices.

Sellers always prefer this mechanism to committing to a fixed price. When the seller’scost of matching is high relative to the buyer’s, having the ability to discount is necessaryfor both parties to participate in the sales process, which explains the prevalence of pricenegotiation in B2B sales. On the other hand, under this mechanism, comparing to fixedprice, buyers face higher list prices, receive bigger discounts that lead to net price drops,spend less time discovering product information, and receive lower utility. We may observethese changes in e-commerce if more retailers use AI to track buyers’ real-time purchaseintentions.

List price and discounts play additional roles when firms have private information. Ifsellers have private information in their quality, a high type seller can set a higher-than-optimal list price as a way to signal, even though the list price is never used. When buyershave private information on their outside options, the paper finds a case of reverse price

27

discrimination, where the seller screens different types of buyer by reducing discounts overtime.

The model highlights how a non-binding price ceiling can facilitating information ac-quisition and trade. The analyses provide implications on selling and pricing strategies in astochastic environment. However, the paper has several important limitations. It ignores im-portant aspects of sales, such as manager-salespersons relationship. The interaction betweenagency and hold-up can bring additional insights. The model also does not fully explorethe effects of asymmetric learning and alternating offers due to tractability concerns. Theyremain as important topics for future research.

28

References

Appier (2020). AiDeal’s Features, https://www.appier.com/aideal-feature/, accessed onApril 2, 2020.

Bhardwaj P (2001). Delegating Pricing Decisions. Marketing Science, 20(2), 143-169.

Bhardwaj P, Chen Y, David G (2008). Buyer-Initiated vs. Seller-Initiated Information Rev-elation. Management Science, 54(6), 1104-1114.

Branco F, Sun M, Villas-Boas JM (2012). Optimal Search for Product Information. Man-agement Science, 58(11), 2037-2056.

Binmore KG, Herrero MJ (1988). Matching and Bargaining in Dynamic Markets. The Reviewof Economic Studies, 55(1), 17-31.

Burns D, Murphy J (2018). The Secret to B2B Pricing in a Digital World. BainInsights, https://www.bain.com/insights/the-secret-to-b2b-pricing-in-a-digital-world/, ac-cessed on April 2, 2019.

Coughlan A, Joseph K (2011). Sales Force Compensation: Research Insights and ResearchPotential. Handbook of Business-to-Business Marketing, ch.26, 473-495.

Daley B, Green B (2020). Bargaining and News. American Economic Review, 110(2), 428-474.

Dixit A (1993) The Art of Smooth Pasting. Harwood Academic.

Fuchs W, Skrzypacz A (2010). Bargaining with Arrival of New Traders. American EconomicReview, 100(3), 802-836.

Fudenberg D, Levine D, and Tirole J (1985). Infinite-Horizon Models of Bargaining withOne-Sided Incomplete Information. Game Theoretical Models of Bargaining. Cambridge Uni-versity Press.

Gale D (1987). Limit Theorems for Markets with Sequential Bargaining. Journal of EconomicTheory, 43, 20-54.

Gul F, Sonnenschein H, Wilson R (1986). Foundations of Dynamic Duopoly and the CoaseConjecture. Journal of Economic Theory, 39, 155-190.

Guo Q, Agichtein E (2010). Ready to buy or just browsing? Detecting web searcher goalsfrom interaction data. Proceedings of the 33rd international ACM SIGIR conference on Re-search and development in information retrieval, 130-137.

Guo L, Hua L, Jia R, Zhao B, Wang X, Cui B (2019). Buying or browsing?: predictingreal-time purchasing intent using attention-based deep network with multiple behavior. Pro-ceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery &Data Mining, 1984-1992.

29

Hansen A, Joseph K, Krafft M (2008). Price Delegation in Sales Organizations: An EmpiricalInvestigation. Business Research, 1(1), 94-104.

Haurin D, Haurin J, Nadauld T, Sanders A (2010). List Prices, Sale Prices and MarketingTime: An Application to U.S. Housing Markets. Real Estate Economics, 38(4), 659-685.

Ishii Y, Ory A, Vigier A (2018). Competing for Talent. Cowles Foundation Discussion PaperNo.2119.

Jose ST (2017). How I Finally Got My Sales Team to Follow a Sales Process. Hubspot Blog,https://blog.hubspot.com/sales/get-sales-team-to-follow-sales-process, accessed on Novem-ber 11, 2017.

Joseph K (2001). On the Optimality of Delegating Pricing Authority to the Sales Force.Journal of Marketing, 65(1), 62-70.

Ke TT, Shen ZJM, Villas-Boas JM (2016). Search for Information on Multiple Products.Management Science, 62(12), 3576-3603.

Kermisch R, Burns D (2018). Is Pricing Killing Your Profits? Bain & Company.http://www.bain.com/publications/articles/is-pricing-killing-your-profits.aspx, accessed onJune 16, 2018.

Kruse T, Strack P (2015). Optimal Stopping with Private Information. Journal of EconomicTheory, 159, 702-727.

Lal R (1986). Delegating Pricing Responsibility to the Sales Force. Marketing Science, 5(2),159-68.

Lal R, Matutes C (1994). Retail Pricing and Advertising Strategies. The Journal of Business,67(3), 345-370.

Mantrala Mm, Albers S, Caldieraro F, Jensen O, Joseph K, Krafft M, Narasimhan C,Gopalakrishna S, Zoltners A, Lal R, Lodish L (2010). Sales Force Modeling: State of theField and Research Agenda. Marketing Letters, 21(3), 255-272.

Marn MV, Roegner EV, Zavada CC (2003). The Power of Pricing. McKinsey Quarterly,February 2003.

Mehring J (2017) Understanding the Modern B2B Sales Process. SnapApp Blog,https://www.snapapp.com/blog/understanding-modern-b2b-sales-process, accessed onSeptember 17, 2017.

Merlo A, Wilson C (1995). A Stochastic Model of Sequential Bargaining with CompleteInformation. Econometrica, 63(2), 371-399.

Mewborn S, Murphy J, Williams G (2014). Clearing the Roadblocks to Better B2B Pric-ing. Bain & Company. http://www.bain.com/publications/articles/clearing-the-roadblocks-to-better-b2b-pricing.aspx, accessed on May 16, 2017.

30

Moe WW (2003). Buying, searching, or browsing: Differentiating between online shoppersusing in-store navigational clickstream. Journal of consumer psychology, 13(1-2), 29-39.

Nick M (2017). Success Comes Down to Great Sales Discovery. ROI4sales.http://www.roi4sales.com/sales/it-is-all-about-discovery/, accessed on October 5, 2017.

Ortner J (2017). Durable Goods Monopoly with Stochastic Costs. Theoretical Economics,12(2), 817-861.

Pierre A, Ossmann R (2017). Optimizing B2B Pricing - One of the Greatest Levers onProfitability. L.E.K Insights, XIX(64).

PwC Technology Institute (2013). The Future of Software Pricing Excellence:Transaction Pricing Management. https://www.pwc.com/gx/en/technology/software-pricing/assets/pwc-the-future-of-software-pricing-excellence-transaction-pricing-management.pdf, accessed on October 5, 2017.

Rubinstein A (1982). Perfect Equilibrium in a Bargaining Model. Econometrica, 50, 97-109.

Rubinstein A, Wolinsky A (1985). Equilibrium in a Market with Sequential Bargaining.Econometrica, 53(5), 1133-1150.

Shin J (2005). The Role of Selling Costs in Signaling Price Image. Journal of MarketingResearch, 42(3), 302-312.

Skok D (2012). Inside Sales Best Practices: HubSpot A Case Study. For Entrepreneurs.https://www.forentrepreneurs.com/hubspot-saas-inside-sales/, accessed on October 5, 2017.

Schurmann J, Voller S, Petzke A, Langkemp D (2015). Three Steps to CreatingValue From B2B Discounts. BCG Perspective by The Boston Consulting Group.https://www.bcgperspectives.com/content/articles/marketing-sales-pricing-three-steps-creating-value-from-b2b-discounts, accessed on October 2, 2017.

Simon L, Stinchcombe M (1989). Extensive Form Games in Continuous Time: Pure Strate-gies. Econometrica, 57(5), 1171-1214.

Toman N, Adamson B, Gomez C (2017). The New Sales Imperative. Harvard BusinessReview, March-April 2017 Issue, 118-125.

Webtrends Optimize (2020). Exit Intent That Reduces Site Abandonment,https://www.webtrends-optimize.com/exit-intent/, accessed on May 18, 2020.

Wernerfelt B (1994a). On the Function of Sales Assistance. Marketing Science, 13(1), 68-82.

Wernerfelt B (1994b). Selling Formats for Search Goods. Marketing Science, 13(3), 298-309.

Xu Z, Dukes A (2017). Personalized Pricing with Superior Information on Consumer Pref-erences. Working paper.

31

Yavas A, Yang S (1995). The Strategic Role of Listing Price in Marketing Real Estate:Theory and Evidence. Real Estate Economics, 23(3), 347-368.

Zarges D (2017). Three Sales Demo Disasters and How to Avoid Them. SBIInsights, https://salesbenchmarkindex.com/insights/three-sales-demo-disasters-and-how-to-avoid-them/, accessed on November 10, 2017.

32

Appendix A Proofs

A.1 Section 3

Proof of Lemma 1First we assume the existence of an equilibrium with V (x) = V (x), and show that the

equilibrium outcome must have the stated properties.Suppose V (x) = V (x). Buyer’s quitting strategies must be the same as if price is fixed

at P . So Buyer quits for all x < x = x. Given the continuation value V (x), Buyer buys at xif and only if P (x) ≤ x− V (x). So the maximum price that Seller can extract is x− V (x).This is Seller’s optimal stopping problem. Seller can stop and receive a payoff of x− V (x),or continue. If Seller wants to continue, she must charge P (x) > x− V (x). The solution toSeller’s stopping problem is a threshold x, such that Seller stops for all x ≥ x.

Next we prove that, in equilibrium, Buyer buys at all x ≥ x, so that A = [x,∞].Suppose there exists x > x such that a(x, P (x)) = 0 for an open neighborhood around x.Let xleft = sup{x|x ∈ A & x < x}, and xright = inf{x|x ∈ A & x > x}. Because there’sno trade between xleft and xright, U(xn) is on a strictly convex function connecting U(xleft)and U(xright). But Buyer is willing to accept P (x) = x − V (x) which is a weakly concavefunction connecting U(xleft) and U(xright), so we must have U(x) < x − V (x, P (x)). ThusSeller should stop at x. If there’s no open neighborhood around x, then when xt = x, thegames stops at τ = 0 and pays utility of 0 by definition of utilities in Section 2. Thus Sellershould stop at x, a contradiction.

Finally, we confirm that the outcome above is indeed an equilibrium. Buyer’s continuationvalue is V (x) for all x. Given the continuation value, buying at x if and only if P (x) ≤x−V (x) is optimal. Seller does not have a profitable deviation either, since A is the solutionto the optimal stopping problem with stopping payoff of x− V (x).

Note that by equation (7) from Section 3, we know that V (x) is a convex C1 function.Thus Buyer’s quitting threshold x is internal and satisfies smooth-pasting condition. Seller’stopping value x− V (x) is then a concave C1 function which equals to a constant, P , for allx ≥ x. This insures that Seller’s optimal stopping threshold x must be strictly lower thanx. Thus both equilibrium thresholds must be internal solutions and satisfy smooth-pastingconditions.

Proof of Proposition 1 and 2Take any P ≤ x0 + 1

4σ2

cb. Given Lemma 1, V (x) = V (x) from equation (7).

The trading threshold x must solve Seller’s optimal stopping problem with stoppingvalue x − V (x, P ). By Taylor expansion and Ito’s Lemma, Seller’s value function satisfiesrU(x) = −cs+ σ2

2U ′′(x). Having r = 0 and cs = 0 imply a linear value function for the Seller

U(x) = αs(x−P ) +βs. We have three boundary conditions. The first two conditions matchvalues at quitting and trading points. The third condition is a first-order “smooth-pasting”

33

condition ensuring the stopping time is optimal (Dixit 1993).αs

(− 1

4σ2

cb

)+ βs = 0

αs(x− P ) + βs = x− cbσ2 (x− P )2 − 1

2(x− P )− σ2

16cb

αs = 1− 2cbσ2 (x− P ) + 1/2

(18)

Solving the system of equations, we get the trading threshold

x = P +σ2

cb

[√1

4− P cb

σ2− 1

4

]and trading price

P (x) = x− V (x, P ) =σ2

cb

[√1

4− P cb

σ2− 2(1

4− P cb

σ2

)]The optimal list price is found by

argmaxP

U(x0) = argmaxP

αb(x0 − P ) + βb = argmaxP

(1

4

σ2

cb− P

)(1− 2

√1

4− P cb

σ2

)If P > 1

4σ2

cb, then x = P − 1

4σ2

cb> 0. At x, Seller would offer P (x) = x and Buyer buys.

The only difference now is that Seller receives utility of x instead of 0 at Buyer’s quittingthreshold. We can re-solve the above system of equations by swapping out the first conditionof Equations (18) with

αb

(− 1

4

σ2

cb

)+ βb = P − 1

4

σ2

cb(19)

and we get x = x = P − 14σ2

cb. Thus players trade immediately for x ≥ x. Furthermore, for

0 < x ≤ x, Seller would offer P (x) = x and Buyer accepts, otherwise Buyer would quit. Forx ≤ 0, Buyer quits immediately. Thus the game must end immediately for all x ∈ R.

A.2 Section 4

Lower Bound on Buyer’s Value FunctionWe calculate the lower bound on Buyer’s value function for a given list price P and a

given quitting threshold x = sup{x|∑

i qi(x) > 0}.Treating x as an exogenous stopping point, let V (x, x) denote the lower bound. By Ito’s

Lemma, V (x, x) = −cb+ σ2

2ddx2V (x, x|P ), which implies V (x, x|P ) = cb

σ2 (x−P )2+α(x−P )+βfor some α and β. Let x denote the point at which Buyer chooses to buy if price is fixed atP . Then we have following value-matching and smooth-pasting conditions.:

cbσ2 (x− P )2 + α(x− P ) + β = x− P2cbσ2 (x− P ) + α = 1cbσ2 (x− P )2 + α(x− P ) + β = 0

(20)

34

Solving these 3 conditions and combining with the fact that V (x) = 0 ∀x ≤ x, we get:

V (x, x) =

0, x ≤ xcbσ2 (x+ η − P )2 + 1

2(x+ η − P ) + σ2

16cb, x ∈

(x, x)

x− P , x ≥ x

(21)

where η = 14σ2

cb− (x− P )−

√−σ2

cb(x− P ) and x = P + 1 +

√1− (x− P )2 − σ2

cb(x− P ).

Note that the Costless Seller lower bound in equation (7) is a special case of this, whereη = 0 and his value function does not depend on Seller’s cost. Buyer’s quitting thresholdxb is smaller than P , so V (x, x) = 0 for x ≤ x. If Seller quits first, then η is positive andBuyer’s value function is shifted down and left from a positive η.

Proof of Lemma 2First, we claim that for any x, there must exist x′ < x such that some player quits at

x′. Suppose not, then players’ utilities must approach −∞ as x→ −∞, and players shouldquit, a contradiction. Thus x = sup{x|

∑i qi(x) > 0} exists.

Suppose V (x) = V ′(x) = 0, then the case is identical to the case of a costless seller. Sowe can use the proof directly from Lemma 1.

Suppose U(x) = U ′(x) = 0 and V ′(x) 6= 0, first we see that in this case, we must havex > P − 1

4σ2

cb, otherwise, V (x, x) < 0 for x < P − 1

4σ2

cb, and Buyer would quit at P − 1

4σ2

cb,

a contradiction. An equilibrium outcome should be Seller’s optimal stopping solution, withstopping payoff of x− V (x, x).

Let x denote the point where Buyer would buy if price is fixed at P , and let x denote thepoint where trade happens in equilibrium. From earlier analysis, we know that Buyer’s valuefunction is in the form of V (x) = cb

σ2 (x − P )2 + αb(x − P ) + βb and Seller’s value function

is in the form of U(x) = csσ2 (x − P )2 + αs(x − P ) + βs. Then x, x, and x must satisfy the

following seven value-matching and smooth-pasting boundary conditions - the first 3 for theBuyer and the last 4 for the Seller:

cbσ2 (x− P )2 + αb(x− P ) + βb = x− P2 cbσ2 (x− P ) + αb = 1cbσ2 (x− P )2 + αb(x− P ) + βb = 0csσ2 (x− P )2 + αs(x− P ) + βs = 0

2 csσ2 (x− P ) + αs = 0csσ2 (x− P )2 + αs(x− P ) + βs = x− V (x)

2 csσ2 (x− P ) + αs = 1− d

dxV (x)

(22)

Solving for x, x, αb, βb, αs, and βs then generates the equilibrium outcome.Lastly, we show that the game cannot simultaneously have an equilibrium outcome with

V ′(x) = 0 and an outcome with U ′(x) = 0 and V ′(x) 6= 0. Using the last six conditionsfrom the system of equations above we can derive x = − cb

csP . One can then compare this

threshold to the quitting threshold for V ′(x) = 0, which is x = P − 14σ2

cb. Only the higher

35

of the two thresholds can be supported in an equilibrium. If quitting happens at the lowerthreshold, then in the region between the two thresholds, one of the player must have anegative continuation value. That player can profitably deviate by quitting at those states.

Proof of Proposition 3We prove by showing that, if Seller quits earlier than Buyer, then there’s a higher P

such that the equilibrium outcome in Lemma 2 generates a higher ex-ante utility for Seller.From the proof of Lemma 2, we know that x = − cs

cbP > P − 1

4σ2

cb, otherwise Buyer’s quitting

threshold must be higher than x, a contradiction. Note that x = − cscbP decreases in P .

Increasing P by ∆P = x + 14σ2

cb− P is strictly better for the Seller ex-ante, because (1)

x is unchanged and (2) V (x, x) is strictly lower. Fixing an x in equilibrium, we can think ofSeller as facing an optimal stopping problem regarding when to sell, with a lower boundaryat x. Her utility from stopping is P (x) = x−V (x, x) = x−V (x+η, P )+η. Now suppose thelist price is increased from P to P+∆P , where ∆P = x+ 1

4σ2

cb−P . Seller’s quitting threshold

is decreased as argued above, and Buyer’s quitting threshold increases to P +∆P − 14σ2

cb= x,

so the lower boundary of the game, x = max{xb, xs}, is unchanged. With the new list priceP + ∆P , η becomes 0, and Seller’s payoff from trading is x−V (x, P + ∆P ), which is strictlyhigher than her selling price under P : x−V (x+η, P )+η, for all x > x. Thus Seller is facingthe same lower boundary under the two list prices, and a stopping utility under P +∆P thatstrictly dominates P . Thus regardless of where she wants to trade, Seller’s ex-ante utility isstrictly higher under P + ∆P than under P . Thus P is sub-optimal.

Proof of Proposition 4From Proposition 3, we know that Buyer quits first in equilibrium. Thus given P , Buyer’s

value function must be

V (x) = V (x) =cbσ2

(x− P )2 +1

2(x− P ) +

σ2

16cb

and Buyer’s quitting threshold is x = P − σ2

4cb.

Because cs > 0, by U ′′(x) = 2csσ2 , we know that U(x) = cs

σ2 (x− P )2 + As(x− P ) + Bs forsome coefficients As and Bs. At x, we must have (1) U(x) = 0. Since x is Seller’s optimalstopping points, x must satisfy: (2) U(x) = P (x) = x − V (x), and (3) U ′(x) = 1 − V ′(x).From these three conditions, we can solve x, As, and Bs by solving the following system ofequations:

csσ2 (− σ2

4cb)2 + As(− σ2

4cb) +Bs = 0

csσ2 (x− P )2 + As(x− P ) +Bs = x− cb

σ2 (x− P )2 − 12(x− P )− σ2

16cb

2 csσ2 (x− P ) + As = 1− 2cb

σ2 (x− P )− 12

(23)

36

From which we get x = P − σ2

4cb+ σ2

cb

1√1+k

√14− r

As = 2+k2− 2√

1 + k√

14− P cb

σ2

Bs = (14

+ k16

)σ2

cb−√

1+k2

σ2

cb

√14− P cb

σ2 .

(24)

Plugging As and Bs into U(x0), and letting r = P cbσ2 , we get

U(x0 = 0) =σ2

cb

[k(

1

4− r)2 + (

1

4− r)− 2

√1 + k(

1

4− r)3/2

](25)

Maximize U(0) with respect to r to get the optimal r∗ and P∗

= r∗ σ2

cb. For x0 = 0, we

get r∗ = 14− 18+10k−6

√9+10k+k2

16k2. Plugging P into x = P − σ2

4cb+ σ2

cb

1√1+k

√14− r produces x.

Plugging P into P (x) = x− V (x) produces P (x). It’s easy to check that limx→x+ U′(x) > 0

and U(x) > 0 ∀x > x, which confirms that Seller does not want to quit before Buyer does.

Proof of Corollary 3 and 4To make the comparison, we need to first solve for the equilibrium under a fixed price.

Let P ∗ be the optimal fixed price. We solve separately for the cases of (1) Buyer quits firstand (2) Seller quits first.

Suppose Buyer quits first. We know that Buyer’s value function is V (x) = cbσ2 (x−P ∗)2 +

12(x − P ∗) + σ2

16cb, and Seller’s value function is U(x) = cs

σ2 (x − P ∗)2 + As(x − P ∗) + Bs forsome coefficients As and Bs. Let x and x denote Buyer’s buying and quitting thresholds,respectively. Since both buying and quitting are Buyer’s decision, we get x = P ∗ + σ2

4cband

x = P ∗ − σ2

4cbfrom Section 3. Solving U(x) = P ∗ and U(x) = 0 simultaneously give us

As = P ∗ 2cbσ2 and Bs = −k σ2

16cb+ 1

2P ∗. Thus

U(x0) =kcbσ2

(x0 − P ∗)2 + P ∗2cbσ2

(x0 − P ∗) +−k σ2

16cb+

1

2P ∗ (26)

Taking derivative with respect to P ∗ and setting to zero, we get P ∗ = 12−k

σ2

4cb+ 1−k

2−kx0.

To make sure x < x0, we need k < x04cbσ2 + 1, otherwise Buyer quits immediately.

Now suppose Seller quits first. Since now quitting is Seller’s decision, we do not knowcoefficients Ab and Bb in Buyer’s value function V (x) = cb

σ2 (x−P ∗)2 +Ab(x−P ∗) +Bb. Westill have U(x) = cs

σ2 (x−P ∗)2 +As(x−P ∗) +Bs. To solve Ab, Bb, As, Bs, x, and x, we solvethe following system of equations:

U(x) = P ∗ and U(x) = 0

V (x) = x− P ∗ and V (x) = 0

U ′(x) = 0 and V ′(x) = 1

(27)

This gives x = (1 − 1k)P ∗, and U(x0) = kcb

σ2 (x0 − P ∗)2 + 2cbσ2 P

∗(x0 − P ∗) + kcbσ2

(P ∗

k

)2.

Then maximize U(x0) with respect to P ∗. If k ≤ x04cbσ2 + 1, then d

dPU > 0 for all P .

37

This implies Seller will raise the price till Buyer quits first, i.e., P ∗ − σ2

4cb≥ (1 − 1

k)P ∗. If

k > x04cbσ2 + 1, we get P ∗ = k

k−1x0 = x. Thus Seller quits immediately. Thus there does not

exist an equilibrium with positive length such that Seller quits strictly before Buyer. Fork < x0

4cbσ2 + 1, the length fo the game is positive and Buyer quits first. For k >= x0

4cbσ2 + 1,

one player stops immediately regardless of the price.Now we can compare the equilibrium outcome under bargaining to the outcome under

a fixed price. Particularly, for x0 = 0, we can compare P = σ2

cb

[14− 18+10k−6

√9+10k+k2

16k2

]with

P ∗ = 12−k

σ∗

4cb. There exists a k < 1 such that P > P ∗ for k < k and P < P ∗ for k > k.

A.3 Section 5

Lower Bound on Buyer’s Value FunctionLet V i(x, µ) denote type i’s lower bound in state x with Seller’s belief µ.First, if µ = 0 or µ = 1, then Buyer’s type is revealed. Only one type of Buyer remains

and the problem is identical to the costless seller model in Section 3. Thus the lower boundis the same as in equation (7). Thus we must have V i(x, µi) = V (xi) from equation (7).

It remains to solve for V i(x, λ). The following Lemma proves that, if type L receivesV (xL) in equilibrium when his type is reveled, then H type must buy (weakly) earlier thanL type when the type has not been revealed, and the existence of H type does not change Ltype’s lower bound.

Lemma 3. If VL(x, 0) = V (xL) from equation (7), then H type must buy (weakly) earlierthan L type does, and there exists an equilibrium outcome in which VL(x, λ) = V (xL).

Proof. Suppose both types are still in the game. Since L type cannot be worse than if price isfixed at P , we must have VL(x, λ) ≥ V (xL), where V is from Equation (7). If L type buys atstate x, then we must have P (x, λ) ≤ xL − V (xL). If H types also buys, he gets utility of atleast xH−(xL−V (xL)) = V (xL)+ε. If H type does not buy at x, then he will be the only typeremaining after L type buys, and will get continuation value VH(x, 1) = V (xH) = V (xL + ε).It is straightforward to show that V (xL) + ε ≥ V (xL + ε), hence H type must buy no laterthan L type does.

Suppose that H type buys strictly earlier (xH < xL + ε), then we have a separatingequilibrium. After H type buys, L type buys immediately if xL ≥ xL. Thus H type mustbuy strictly before xH reaches xL + ε, otherwise there cannot be separation. After H buys,L type is revealed, so the outcome must follow Proposition 1. The existence of H type doesnot affect L type’s equilibrium payoff.

Suppose instead we have a pooling equilibrium where H and L types buy together (xH =xL+ ε). If Seller cannot separate the two types, then it is as if she’s only dealing with L typeonly. Seller can charge up to P (x, µ) = P (xL + ε

2, µ) = xL − V (xL) at the point of trade.

The optimal price and thresholds follow Proposition 1.

By Lemma 3, L type buys and quits at the same thresholds as in the case without privateinformation. The existence of H type does not affect L type’s lower bound value function,

38

nor does it affect L type’s equilibrium strategies. Thus we must have:

V L(x, λ) = V (xL)

=cbσ2

(xL − P )2 +1

2(xL − P ) +

σ2

16cb, xL ∈

(P − σ2

4cb, P +

σ2

4cb

) (28)

The lower bound on H type’s value function when µ = λ must be higher than V (x) fromequations (7). Seller cannot “threaten” to never make discount, because H type rationallyexpects Seller to make L type an offer when xL reaches xL.

When x drops to xL + ε2, L type quits and H type is revealed with product value xH .

When x increases to xL + ε2, L type buys, and by Lemma 3, H type buys too. When L quits

at xL = xL, H’s utility becomes V (xL + ε). When L buys, H’s utility becomes V (xL) + ε.These two conditions bound H type’s value function from below. By Taylor’s expansion andIto’s Lemma, we have

VH(x, λ) =cbσ2

(xH −ε

2− P )2 + αH(xH −

ε

2− P ) + βH ∀xH ∈ [xL + ε, xL + ε] (29)

The two conditions translate to:{cbσ2 ( ε

2− σ2

4cb)2 + αH( ε

2− σ2

4cb) + βH = cb

σ2 (ε− σ2

4cb)2 + 1

2(ε− σ2

4cb) + σ2

16cbcbσ2 (xL + ε

2− P )2 + αH(xL + ε

2− P ) + βH = cb

σ2 (xL − P )2 + 12(xL − P ) + σ2

16cb+ ε

(30)

Solving these two equations produces αH , βH , and consequently VH(x, λ). The solutionis:

αH =1

2+

ε2− cb

σ2 ε2 − ε

√14− P cb

σ2

σ2

cb

√14− P cb

σ2

βH =3cb4σ2

ε2 +σ2

16cb−

ε2− cb

σ2 ε2 − ε

√14− P cb

σ2

σ2

cb

√14− P cb

σ2

(ε

2− σ2

4cb)

(31)

Proof of Proposition 5The only case we need to prove is when Buyer is H type and Seller’s belief is µ = λ. Given

H type’s lower bound V H(x, λ), which is calculated in Appendix A.3, it’s easy to check thatV H(x, λ) > V L(x + ε, λ) for xL + ε

2< x < xL + ε

2. Thus H type has a higher utility than L

type in every state, even after compensating L type for the difference in outside options, ε.It’s also easy to check that V H(x, λ)− ε < V L(x+ ε, λ) for xL + ε

2< x < xL + ε

2; thus L type

would not buy when Seller makes an offer to the H type. Now, given VH(x, λ) = V H(x, λ),Seller receives P (x, λ) = xH − VH(x, λ) when she trades with H type. Seller’s decision ofwhen to trade with H type must solve the following optimal stopping problem, with two

39

value matching boundary conditions and one smooth-pasting boundary condition:αs(xH − ε

2) + βs = −(xH − ε

2− P )2 + (1− αH)(xH − ε

2− P ) + P − βH

αs = −2(xH − ε2− P ) + (1− αH)

αs(ε2− σ2

4cb) + βs = (1− 2

√14− P cb

σ2 )ε

(32)

where αH and βH are solutions to equations (30). Using the system of equations, we canderive the following condition

(xH − P −σ2

4cb)2 + P − σ2

4cb= 0

which implies that

xH = xL = P +σ2

cb

[√1

4− P cb

σ2− 1

4

]Note that even though they buy at the same threshold, H type arrives at the threshold earlierthan L type does because xH = xL + ε. Also, this common threshold is the same tradingthreshold as when there is no private information.

When L type quits, H type’s value function becomes V (xL + ε) > 0, thus H type doesnot quit as long as L is present. After L quits, H has the same quitting threshold as inProposition 1. So

xi = x = P − 1

4

σ2

cb

At their respective times of trade, L type pays price

PL = x− VL(x+ε

2)

and H type pays price

PH = x− VH(x− ε

2)

Thus PL − PH = VH(x − ε2) − VL(x + ε

2). Since we showed above that VH(x) > VL(x + ε),

this proves that PL > PH .

Proof of Proposition 6First we show that Proposition 6 gives the unique equilibrium outcome conditional on

P . If there is no list price, then Seller charges v0 + ∆v at period 2, and charges v0 + 12∆v

at period 1. Buyer’s continuation value is then negative in all periods, so it must buy orquit immediately. If P = v0 + ∆v− 6c, then, at period 2, given θ ≤ ∆v−6c

32

∆v−8c, Seller’s optimal

price is P = v0 + ∆v − 6c regardless of its signal. This implies that at period 1, Buyer’scontinuation value is 1

2(v0 +∆v−P )−c = 2c if the first attribute is a match, and is −c if the

first attribute is a mismatch. Seller’s optimal response in period 1 is to charge v0 + 12∆v−2c,

which makes Buyer with a matched attribute indifferent between buying and continuing,and let Buyer with a mismatched attribute quit. Buyer’s continuation value is 0 at period0. Seller can either sell at t− 0 with price v0, or wait till period 1 for an expected profit of

40

12(v0 + 1

2∆v−2c). Seller prefers to wait for v0 ≤ 1

2∆v−2c. So Proposition 6 gives the unique

equilibrium conditional on P .Now consider alternative P . For higher P , Buyer’s continuation value at time 0 is neg-

ative, so it must buy or leave immediately. For lower P , Buyer continuation value must beweakly higher in all history. In either case, Seller cannot be more profitable.

41

Online Appendix

1. Discrete-time Analog

In this section, I show that the continuous-time equilibrium outcomes represent the limitof discrete-time equilibrium outcomes. Specifically, because the equilibrium outcomes pre-sented in the paper is identified through Buyer’s value function, we want to prove that, asthe period length in the discrete-time game approaches 0, Buyer’s equilibrium value functionmust approach V (x) from equation (7) or V (x, x) from equation (21) and satisfies Condition1.

I first formally state the discrete-time game, then introduce a refinement that eliminatestrivial equilibria in which players quit simultaneously, then prove the uniqueness of the limitas the game approaches continuous time.

The discrete-time game follows the model described in Section 2.1. Let G(∆t) denotethe game with period length of ∆t. Players act at time t ∈ {0,∆t, 2∆t, · · ·}. There aretwo players, a Seller (s) and a Buyer (b). Product value is denoted as x, with x = x0 attime t = 0. The state variable xt evolves as a Markov chain, with xt+∆t = xt + σ

√∆t with

probability 12

and xt+∆t = xt−σ√

∆t with probability 12. Let X(∆t) = {x0+ασ

√∆t | α ∈ N}

denote the grid on x spanned by σ√

∆t from x0. State xt ∈ X(∆t) is observable to bothplayers.

Before the game starts,a list price P is chosen. Then at t ≥ 0:

1. Seller chooses price Pt subject to Pt ≤ P .

2. Buyer chooses whether to buy, at ∈ {0, 1}.

3. If at = 0, then both players choose whether to quit, qi,t ∈ {0, 1}. If neither playerquits, then players incurs cost cs and cb and the game moves to time t+ ∆t. If eitherplayer quits, the game ends and players receive their outside options.

I look for stationary SPE with pure strategy, henceforth referred as equilibrium. Givena list price, Seller’s equilibrium strategy is characterized by (P (x), qs(x)), with price offerP (x) : R 7→ [0, P ] and quitting decision qs(x) : R 7→ {0, 1}. Buyer’s equilibrium strategyis characterized by (a(x, P ), qb(x)), with buying decision a(x, P ) : R2 7→ {0, 1} and quit-ting decision qb(x) : R 7→ {0, 1}. Since everything depends on list price P , which is fixedthroughout the game, I do not include P from notations.

Buyer incurs continuation cost of cb ∗∆t and Seller incurs continuation cost cs ∗∆t at theend of each period if a(xt) = 0 and qs(xt) = qb(xt) = 0. If a(xt) = 1, Buyer receives utilityxt − Pt and Seller receives Pt. If either player quits, the game ends and players get outsideoptions of πb and πs, which are normalized to 0.

Given a strategy profile φ, let u(x, φ) denote the seller’s expected utility in state x, andv(x, φ) denote buyer’s expected utility in state x. Let V1(x) denote Buyer’s value function inequilibrium, and U1(x) denote Seller’s equilibrium value function. Let V2(x) denote Buyer’sexpected value from continuing to the next period. That is, V2(x) represents Buyer’s expectedutility of rejecting the trade and if both players choose not to quit this period.

42

We can write V2(x) recursively as:

V2(x) = −cb∆t+1

2V1(x+ σ

√∆t) +

1

2V1(x− σ

√∆t) (33)

When players trade, Seller must offer the highest price that Buyer is willing to accept.If players quit at x, then Seller can charge P (x) = max{x, P}. If players don’t quit at x,and if V2(x) ≥ 0, Seller can charge up to x−V2(x). If players don’t quit at x and V2(x) < 0,Seller can charge up to x. Also Seller cannot charge above the list price. This implies:

V1(x) =

max{0, x− P} ∀x s.t. q(x) = 1

max{0, V2(x)} ∀x ≤ P s.t. q(x) = 0

max{V2(x), x− P} ∀x > P s.t. q(x) = 0

(34)

Before we proceed, note that there is an infinite number of trivial equilibria in whichplayers quit simultaneously. If the opponent is quitting in a given state, then the player isindifferent between quitting and not quitting, thus quitting is always optimal. By the samereasoning, the opponent’s quitting decision is optimal too. As a result, we can constructequilibria with arbitrary quitting rules, as long as both players quit at the same time. Theseequilibria exist due to the simultaneity of players’ quitting decisions, and do not offer anyinsight. One extreme example is that qb(x) = qs(x) = 1 ∀x, P (x) = max{0, min{x, P} },and a(x, P , P ) = I{x ≥ 0} constitutes an equilibrium. In this equilibrium, players alwaysquit no matter what the state x is; Seller charges a price equal to the value of the product(bounded between 0 and P ); Buyer buys immediately if product value is positive because he’sindifferent between buying and quitting. In this equilibrium, the game ends immediately, nomatter what the initial position is.

To remove these simultaneous quitting equilibria, I introduce the following refinement,which is equivalent to applying trembling hand only to players’ quitting strategies.

Refinement. Player quits if and only if quitting is strictly preferred.

Under this refinement, players cannot quit simultaneously. If the opponent quits, thenthe player is indifferent and will choose to stay. This refinement is not restrictive. Supposein an equilibrium, Seller quits for x ≤ xs and Buyer quits for x ≤ xb. Now keep P , P (·), anda(·) as before, and see whether xs − ε and xb constitute a equilibrium, and whether xb − εand xs constitute a equilibrium. If neither is an equilibrium, that means that players quitonly because the other player is quitting. If either perturbation on xs or on xb constitutesan equilibrium, then the new equilibrium has the same equilibrium outcome as before, andthe payoffs are not affected by the restriction that xs 6= xb.

Now we are ready to prove the following result. Let V∆t denote an index family ofBuyer’s equilibrium value functions for the index ∆t. Let U∆t denote an index family ofSeller’s equilibrium value functions for the index ∆t. Let q(x) = max{qb(x), qs(x)} andx = sup{x|q(s) = 1}. Let U(x) = lim∆t→0 U∆t and V (x) = lim∆t→0 V∆t denote the limit ofvalue functions as the game approaches continuous time.

Theorem 4. As ∆t→ 0, V∆t → V (x, x) from equation (21). Condition 1 is satisfied in thelimit.

43

Step 1. There exists an x such that a(x) = 1 and P (x) = P iff x ≥ x.

Here we claim that there exists a threshold such that, when x is above the threshold,Seller charges P and Buyer buys immediately.

First we see that, if P (x) = P and a(x) = 1, then x ≥ P . If x ≥ P , then V1(x) =max{V2(x), x− P} if q(x) = 0 and V1(x) = x− P if q(x) = 1. Seller must charge P (x) = P

if V2(x) < x− P . Let P (x) denote the highest price that Buyer is willing to pay at x > P .

Buyer buys at P if and only if P = P .Now we prove that P must be non-decreasing for x > P . Suppose P decreases for

some x < P . Then there exists x′ > P such that P (x′ − σ√

∆t) > P (x′). This implies

that P (x′) < P , so players do not quit at x. Then P (x′) = x′ − V2(x′) = 12(wideP (x′ −

σ√

∆t) + P (x′+ σ√

∆t) + cb∆t. By rearranging the terms, we get: P (x′+ σ√

∆t)− P (x′) =

P (x′) − P (x′ − σ√

∆t) − 2cb∆t < 0. Thus P (x′ + σ√

∆t) < P (x′) − cb∆t. By induction,

P → −∞ as x→∞. When P < 0, there cannot be trade since Seller can’t charge a negativeprice. Players must not quit if P < P . Thus neither trading or quitting can occur after xpasses some threshold. However, this implies that players’ value functions must approach−∞, so players should quit, which is a contradiction.

Given that P must be non-decreasing, we need to show that P must reach P at somepoint. Suppose P (x) never reaches P . Previously we showed that if x > P and P (x) < P ,

then P (x+σ√

∆t)− P (x) = P (x)− P (x−σ√

∆t)−2cb∆t < 0. Because P is non-decreasing,

P (x)− P (x− σ√

∆t)− 2cb∆t > 0, thus P (x)− P (x− σ√

∆t) > 2cb∆t for all x > P . Then

P must reach P for some large x. This concludes the proof. We can find x by taking themin{x | P = P}.

Step 2. For ∆t small enough, there exists an x < P such that q(x) = 1 iff x ≤ x.

There exists a threshold x such that, players quit if and only if the current state is belowthe threshold. First, we can see that Buyer must quit for some states that are low enough.When x is negative, there cannot be trade, and players pay costs to continue the game. Asx approaches −∞, continuation value also goes −∞ and so players must quit.

Next, for x ≤ P , we can show that if q(x) = 1, then q(x− σ√

∆t) = 1. If q(x) = 1, thenwe must have V1(x) = 0. Suppose that qb(x− σ

√∆t) = 0; this implies that

V2(x− σ√

∆t) = −cb∆t+1

2V1(x) +

1

2V1(x− 2σ

√∆t) ≥ 0 (35)

which means that V1(x − 2σ√

∆t) > V2(x − σ√

∆t ≥ 0. By induction, we have V1(x′) > 0for all x′ < x− 2σ

√∆t. Thus players never quits for x′ < x, which is a contradiction.

Lastly, I show that, for ∆t small enough, we must have q(x) = 0 if x > P . Supposenot, then we have some x′ > P such that q(x′) = 1. Then we must have P (x′) = P anda(x′) = 1, which implies that x′ ≥ x. Then the continuation value for Buyer V2(x′) =−cb∆t+ 1

2V1(x′−σ

√∆t)+ 1

2V1(x′+σ

√∆t) ≥ −cb∆t+ 1

2(x′+σ

√∆t−P ) > −cb∆t+ 1

2σ√

∆t,which is strictly positive for ∆t small enough, so Buyer does not quit. Seller’s continuationvalue U2(x′) = −cs∆t+ 1

2U1(x′− σ

√∆t) + 1

2U1(x′+ σ

√∆t) ≥ −cs∆t+ 1

2P , which is strictly

positive for ∆t small enough, so Seller does not quit. Thus q(x′) = 0, a contradiction. Thisconcludes the proof.

44

Step 3. Let {V∆t(x) : X(∆t) 7→ R}∆t be an index family of Buyer’s value functions. ThenV∆t(x)→ V (x) as ∆t→ 0.

Given the existence of x and x, we can now specify V1(x) in three cases:

V1(x) =

0, ∀x ≤ x

−cb∆t+ 12V1(x+ σ

√∆t) + 1

2V1(x− σ

√∆t), ∀x < x < x

x− P , ∀x ≥ x

(36)

Rearranging the terms for x < x < x, we get

2c

σ2=

V1(x+σ√

∆t)−V1(x)

σ√

∆t+ V1(x)−V1(x−σ

√∆t)

σ√

∆t

σ√

∆tfor x < x < x

Also, at x, we must have:

V1(x) ≥ −cb∆t+1

2V1(x+ σ

√∆t) +

1

2V1(x− σ

√∆t)

which rearranges to:V1(x)− V1(x− σ

√∆t)

σ√

∆t≥ 1− 2c

σ

√∆t (37)

Let V (x) denote the limit of V1(x)→ V (x) as ∆t→ 0. In the limit, the above 5 conditionsconverge to:

V (x) = 0 ∀x ≤ x

V ′′(x) = 2cσ2 for x < x < x

V (x) = x− P ∀x ≥ x

V ′−(x) ≥ 1

Fixing an x, one can solve for V (x) using the above 4 conditions. The solution is:

V (x|P ) =

0, x ≤ xcbσ2 (x+ η − P )2 + 1

2(x+ η − P ) + σ2

16cb, x ∈

(x, x)

x− P , x ≥ x

(38)

where η = 14σ2

cb− (x − P ) −

√−σ2

cb(x− P ) and x = P + 1 +

√1− (x− P )2 − σ2

cb(x− P ).

This is V (x, x) from equation (21) for x < P .

Step 4. Either U(x) = U ′(x) = 0, or V (x) = V ′(x) = 0.

Our refinement implies that only one player can quit at a given state. So either qb(x) = 1or qs(x) = 1.

Suppose qb(x) = 1. That means the continuation value must be less than the outside

45

option for Buyer:

0 ≤ −cb∆t+1

2V1(x− σ

√∆t) +

1

2V1(x+ σ

√∆t)

Thus V1(x+σ√

∆t)−V1(x−σ√

∆t)

σ√

∆t≤ cb√

∆t.

Similarly, suppose qs(x) = 1. That means the continuation value must be less than theoutside option for Seller:

0 ≤ −cs∆t+1

2U1(x− σ

√∆t) +

1

2U1(x+ σ

√∆t)

Thus U1(x+σ√

∆t)−U1(x−σ√

∆t)

σ√

∆t≤ cs√

∆t.So for any ∆t, we must have:

min{V1(x+ σ

√∆t)− V1(x− σ

√∆t)

σ√

∆t−cb√

∆t,U1(x+ σ

√∆t)− U1(x− σ

√∆t)

σ√

∆t−cs√

∆t}≤ 0

Thus in the limit as ∆t→ 0, we have min{U ′(x), V ′(x)} ≤ 0.Suppose V ′(x) ≤ 0. Since we know V (x) = 0, and equilibrium value function cannot be

negative, then V ′(x) = 0. Suppose U ′(x) ≤ 0. If x > 0, then U(x) = x for x ≤ x becauseBuyer is wiling to accept price up to x. This implies U ′−(x) = 1, a contradiction. Thusx ≤ 0, so we must have U(x) = 0.

This concludes the proof. In the limit, either U(x) = U ′(x) = 0 or V (x) = V ′(x) = 0must hold.

46

2. Alternative Equilibrium Concept

Standard equilibrium notions such as subgame-perfection may not work well incontinuous-time games. A strategy profile can produce no well-defined outcome in con-tinuous time, or produce equilibrium outcomes that do not correspond to anything in thediscrete-time game, because SPE does not impose the same restrictions on strategies in con-tinuous time as in discrete time. See Simon and Stinchcombe (1989) for more discussions. Inthe main text of the paper, I constrain players to strategies such that the resulting stoppingtime τθ is a measurable function. One undesirable feature from doing so is that each player’savailable deviation strategies now depend on the opponent’s strategy. In this section, I de-scribe an alternative equilibrium concept of the game without such joint restriction on thestrategy space. I first describe each component of the equilibrium concept, before formallydefining it at the end.

Stationarity I focus on stationary behavior with pure strategies. Players’ equilibriumstrategies depend on state x but not on time t. Seller’s actions in equilibrium can becharacterized by the list price P ∈ R+ ∪ +∞, her price offering in each state P (x, P ) :R × R+ 7→ (−∞, P ), and her quitting decision qs(x, P ) : R × R+ 7→ {0, 1}, where qs = 1means quitting. Buyer’s actions can be characterized by his buying decision a(x, P, P ) :R2×R+ 7→ {0, 1} where a = 1 means accepting the offer, and his quitting decision qb(x, P ) :R × R+ 7→ {0, 1}. Since everything depends on the list price P , which is fixed throughoutthe game, I will drop P from notations going forward. We will treat the list price as anexogenous parameter, and solve the equilibrium for any arbitrary list price, then let Sellerchooses the list price that maximizes her ex-ante utility from bargaining under such list price.

Buyer’s Problem Given Seller’s strategy, Buyer has to decide between three actionsfor each product value x. Buyer can accept Seller’s offer P (x), which gives utility x− P (x);he can reject the offer and quit the game, which gives utility πb; or he can reject the offerand continue to wait. This is an optimal stopping problem, with stopping value Wb =max{x − P (x), πb}, subjects to states in which Seller leaves. Buyer chooses the strategythat maximizes his expected payoff, supτ E[−c ∗ τ + Wb(xτ )], where τ is the stopping time.Buyer’s equilibrium strategy a(x, p) and qb(x) has to be a solution to this problem.

Highest Acceptable Offer If Seller makes an offer that Buyer accepts, Seller shouldmake the highest offer that Buyer is willing to accept. Otherwise, Seller can profitablydeviate by charging a slightly higher price. Thus in equilibrium, if a(x, P (x)) = 1, then weshould have P (x) = sup{P |a(x, P ) = 1}.

Seller’s Problem Given this, Seller also decides between three actions at each moment.Seller can make a sale, which gives utility P (x) = sup{P |a(x, P ) = 1}; she can quit the game,which gives utility of πs; or she can continue to wait (by making an unacceptable and do notquit). This is an optimal stopping problem, with stopping value Ws = max{sup{P |a(x, P ) =1, πs}, subject to states in which Buyer leaves. Buyer chooses the strategy that maximizes hisexpected payoff, supτ E[−c ∗ τ +Wb(xτ )], where τ is the stopping time. Seller’s equilibriumstrategy P , P (x), and qb(x) has to be a solution to this problem.

Outcome Given a strategy profile described above, the games ends at time τ =inf{t | a(xt, P (xt) = 1 or qs(x) + qb(x) > 0}, which is the earliest time by which an offer isaccepted or one of the players quits. Given a starting value x0, the equilibrium outcome canbe described by a quadruple {A,Q,U(x), V (x)}, where A = {x|a(x, P (x)) = 1} is the set

47

of states such that players reach agreement, Q = {x|qs(x) + qb(x) > 1} is the set of statessuch that some player quits, U(x) is Seller’s equilibrium value function, and V (x) is Buyer’sequilibrium value function.

Combining these elements, we get the following definition of equilibrium:

Definition 1. A strategy profile is an equilibrium if it can be describe by {P (x), qs(x),a(x, P ), qb(x)} such that:

1. (Buyer’s Problem) Given {P (x), qs(x)}, {a(x, P ), qb(x)} solves supτ E[−c∗τ+Wb(xτ )].

2. (Highest Acceptable Price) P (x) = sup{P |a(x, P ) = 1} if a(x, P (x)) = 1.

3. (Seller’s Problem) Given {a(x, P ), qb(x)}, {P (x), qs(x)} solves supτ E[−c∗τ+Ws(xτ )].

4. (Trivial Multiplicity) A and Q are closed. 18

18The game can have trivial equilibrium outcomes that differ only on sets of measure 0 with no relevanceto payoffs. I restrict A and Q to be closed sets to rule out this type of multiplicity. See Ortner 2017 for asimilar restriction.

48

3. Time Discounting

The paper assumes that players incur flow costs but do no have time discounting. Theexistence of flow costs creates the hold-up problem, because Buyer quits if he does not expecthis payoff from future bargaining to compensate for his costly discovery effort today. In thissection, I show that the modelling choice between flow cost and time discounting does notaffect the results of the paper as long as players have positive outside options so that theyhave incentives to quit.

Let r denote the common discount rate for both players. For non-terminating states, theplayers’ value functions are of the form:

V (x) = Abe

√2rσ2x

+Bbe−√

2rσ2x

U(x) = Ase

√2rσ2x

+Bse−√

2rσ2x

(39)

for some constants Ab, As, Bb, and Bs.For simplicity, assume that πb > 0 and πs = 0. This mimics the baseline model in Section

3, where only Buyer has an incentive to quit. Given a list price P , one can calculate thelower bound on Buyer’s utility, V (x|P ), by solving for Buyer’s optimal stopping problemfacing a fixed price of P . The value-matching and smooth-pasting boundary conditions are:Abe

√2rσ2x

+Bbe−√

2rσ2x

= πb and Abe

√2rσ2x −Bbe

−√

2rσ2x

= 0

Abe

√2rσ2x

+Bbe−√

2rσ2x

= x− P and Abe

√2rσ2x −Bbe

−√

2rσ2x

=√

σ2

2r

(40)

Solving this system of equations, one can find that

V (x|P ) =1

2(

√σ2

2r+ πb2+

√σ2

2r)e

√2rσ2

(x−P−√σ2

2r+πb2)

+1

2(

√σ2

2r+ πb2−

√σ2

2r)e−√

2rσ2

(x−P−√σ2

2r+πb2)

(41)

Buyer’s buying threshold is x = P +√

σ2

2r+ πb2, and Buyer’s quitting threshold is x =

P +√

σ2

2r+ πb2 −

√σ2

2rlog(

√σ2

2rπb2+√

σ2

2rπb2+ 1).

The equilibrium with V (x) = V (x) can be constructed as before. Seller offers pricesP (x) = x − V (x) in states where Seller wants to trade, and offers higher prices in otherstates. The problem transforms into Seller’s optimal stopping problem with stopping valueP (x) = x − V (x), subject to Buyer’s quitting threshold x = x. The solution of Seller’sstopping problem is also an equilibrium outcome, because Buyer’s optimal strategy is tocomply with Seller’s trading threshold. Buyer receives V (x) in this equilibrium.

The qualitative results are the same as with flow costs. One can see that x increaseswith P , and P (x) decreases with P , thus the optimal list price reduces Buyer’s hold-upconcern at the cost of Seller’s bargaining power. Also, the derivative of P (x) approaches 0as x approaches x, which implies that Seller’s optimal stopping point must be lower than x.Thus, players always trade at a price below the list price, regardless of the level of the listprice.

If Seller’s outside option, πs, is also positive, then Seller may choose to quit when the

49

state is bad enough. One can again prove that, on the equilibrium path, Buyer must quitbefore Seller does. The intuition is the same as in Section 4. If Seller’s quitting thresholdis higher than Buyer’s, then the hold-up problem is not binding; thus, raising the list priceincreases Seller’s stopping payoff x − V (x) for all x. This is strictly beneficial to Seller.Furthermore, if Seller commits to a fixed price, then when πs

πbis above a certain threshold,

the game must end at T = 0. If Seller allows bargaining, then the players must stay anddiscover beyond t = 0. Thus, bargaining is necessary for “conversation” when selling is morecostly than buying; here the cost comes from delaying the realization of the outside option.

50

4. Non-retractable Offers

In the baseline model, an offer expires instantly if Buyer does not accept it. Seller makesno promise regarding future prices beside an upper bound. As a result, Seller can actuallyraise price over time. However, one may imagine circumstances in which social norms preventSeller from doing so. In this extension, I study what happens if Seller cannot retract pastoffers, so the standing price has to be decreasing over time.

If Buyer can recall all previous price offers, then each new lowest offer becomes thenew list price, and this list price can only decrease over time. Let U(x|P ) denote Seller’sexpected utility at x with list price P . Let U(x) denote Seller’s expected utility with a listprice optimally chosen for x. As before, let x denote the trading threshold. Let x denotethe point at which Seller lowers list price to avoid Buyer leaving. Then

U(x|P ) =x− xx− x

U(x) +x− xx− x

P (42)

The function is strictly increasing in x for U(x) > x− 14σ2

cb, which must hold, otherwise Seller

should sell at x with price x− 14σ2

cb. This implies that U(x|P ) is strictly increasing in x. The

idea is that Seller prefers to maintain the highest price possible. So the only time that Sellerdecreases list price is when Buyer is about to leave as x hits x = P − 1

4σ2

cb. Figure 11 gives

an example of the equilibrium path.We can then calculate Seller’s ex-ante utility with non-retractable offers, U(x0). We have

a boundary condition that U(−14σ2

cb) = 0, because Buyer quits at −1

4σ2

cbwhen list price is

0. The differential equation that pins down U(x0) is U ′(x0) = P−U(x0)x−x , which is derived by

taking derivative of equation (42) with P = x0 + 14σ2

cb. We obtain:

U(x0) =1

2

σ2

cbe−

12−2x0cb/σ

2 − 1

4

σ2

cb+ x0 (43)

Does Seller prefer to make non-retractable offers? We can compare Seller’s utility fromequation (43) to Seller’s utility from equation (9). There exists a threshold, x = [−1

2log(1

2)−

14]σ

2

cb, such that for x0 below, Seller prefers to use non-retractable offers, and for x0 above,

Seller prefers the offers to be retractable. The intuition is straightforward. Using non-retractable offers allow Sellers to lower list price over time to prevent Buyer from quitting.This is particularly beneficial when ex-ante product value is low. The disadvantage of non-retractable offers is that it takes away Seller’s ability to close sale of discount. Any newdiscount lowers the list price, which increases Buyer’s continuation value. This limits Seller’sability to extract utility when product value is high. Thus Seller prefers to not commit onoffers when x0 is high.

This extension and the baseline model explore the two extreme cases of offer expiration:an offer either expires instantly or never expires. In practice, sellers can choose in between.A seller may make an offer with a future expiration date, and make a new offer when theprevious one expires. Such mechanism are difficult to study for two reasons. First, theproblem becomes non-stationary. Players’ strategy must depend on time. Second, there aremultiple state dimensions. In the baseline model, the only state variable relevant to strategy

51

Figure 11: Non-retractable Offers

and utility is product value x. In a model with endogenous expiration date, both time tillexpiration and the standing offer (which can take any value) enter into players’ continuationvalues. The outcome will also be path dependent. The same strategy profile can lead todifferent trading prices. Such model is very complex. However, the two extreme cases canprovide some insights into the problem. A long expiration date is valuable when Buyer wantto leave. On the other hand, a short expiration date is better at value capturing. So if Sellerhas the freedom to endogenously set an expiration date on each offer, we can expect Seller tooffer longer expiration when the match value with Buyer is low, and offer shorter expirationwhen the math value with Buyer is high.

52

5. Buyer Offers and Price Floor

Because the prevalence of bargaining in B2B markets, one may wonder what is the effectof allowing Buyer to make offers. It is difficult to extend the baseline model to allow foralternating offers due to the inherent asymmetry that list price imposes. When Seller isproposing, list price puts a fixed upper bound on the proposer’s utility, P . But when Buyeris proposing, the same list price puts a linear lower bound on the proposer’s utility, xt − P .As a result, equilibrium must be asymmetric. Such a model is beyond the scope of thispaper, but remain an important question for future research.

Instead, to see how the problem depends on who has the pricing power, this sectiondiscusses what happens if Buyer makes all the offers. If Buyer makes all the offers, listprice becomes useless. In a subgame-perfect equilibrium, Buyer never offers any price above0 because Buyer controls all future prices. A price ceiling in the form of list price cannotprevent Seller from being held-up, and if Seller has any cost, Seller would quit immediately.Instead, Seller may benefit from a price floor that bound offers from below. Such a price flooris common in practice. B2B firms often set a “lowest allowed price” or a “deepest discount”for their salespersons. These policies bound transaction prices from below. A survey byHansen et al. (2008) shows that 61% of firms set restrictions to how much their salespersonscan discount, while only 11% of firms give their salespersons full pricing authority.

The model is modified as follows: (1) Buyer makes offers, and Seller decides whether toaccept or reject; (2) at time 0, Seller chooses a price floor, P , such that all future offers belowP are automatically rejected. Note that Buyer does not have make an offer at a give t, soSeller cannot force a sale at P .

Because Buyer makes all future offers, Seller knows that Buyer never offers above P . SoSeller will accept as soon as Buyer offers P . Thus Buyer’s problem is to decide when it iswilling to pay P for the product. This is equivalent to having a fixed price. The optimallower bound is P = σ2

8cb. Buyer’s offers P when xt reaches P + σ2

4cband quits when xt reaches

P − σ2

4cb.

We can allow Seller to set both list price and price floor, whether Buyer or Seller makesoffers. However, when Buyer makes offers, it is always optimal to set list price to infinity,and when Seller makes offers, setting price floor to 0 is always optimal. So only one of twoconstraints is meaningful.

One can compare the equilibrium outcome with Buyer offers to the equilibrium outcomewith Seller offers. This is equivalent to a comparison between fixed price and the baselineequilibrium, which is presented in Corollary 1 of Section 3. Transaction price is actuallyhigher when Buyer makes offers. The sales process is expected to be longer when Buyermakes offers. Utilities move in expected directions. Buyer’s utility increases and Seller’sutility decreases when Buyer makes offers. However, social welfare is lower when Buyermakes offers.

What happens if there is uncertainty regarding who makes offers? Suppose with someprobability φ, the incoming Buyer is more dominant and makes all the offers, otherwiseSeller makes all the offers. Seller has to set P and P before observing Buyer. Note that theoptimal price floor if Seller know that Buyer makes offers, σ2

8cb, is higher than the equilibrium

final price when Seller makes offers, σ2

9cb. This means that price floor becomes a binding

constraint on Seller’s offers. When facing the probability of a more dominant Buyer, Seller

53

has incentive to set a higher price floor to avoid Buyer low-balling. But this price floor limitsSeller’s ability to discount when facing a less dominant Buyer. The optimal P has to balancethese two effects. Note that with a binding P , final transaction prices are the same whetherBuyer or Seller makes offers. But pricing power still affects utilities as well as length of thesales process.

54

6. Finite Mass of Attributes

The baseline model assumes an infinite mass of attributes in order to study stationaryequilibria. In reality, a product may have only a limited number of features, or consumersmay care about only a finite subset of a product’s features. As a result, players exhaustall the information they can learn from the other party if the sales process continues longenough. Suppose that the product has a finite mass of attributes, T . The product valuext follows a Brownian motion between time 0 and time T . Equilibrium can no longer bestationary. Players’ strategies and value functions now depend on both state x and time t.Equilibrium value functions satisfy the following partial differential equations:

Vxx =2

σ2(cb − Vt)

Uxx =2

σ2(cb − Ut)

(44)

where Vxx and Uxx are second-order derivatives w.r.t x and Vt and Ut are first-order deriva-tives w.r.t t.

The analysis is similar to Section 3. One first solves for the lower bound on Buyer’sequilibrium value function, V (x, t|P ), by solving for Buyer’s optimal stopping problem facinga fixed price of P . Buyer’s stopping thresholds, x(t) and x(t), now change with time t. The

optimal x(t) and x(t) must satisfy the value-matching and smooth-pasting conditions:{V (x(t), t) = x(t)− P and V (x(t), t) = 0

Vx(x(t), t) = 1 and Vx(x(t), t) = 0(45)

The solution to these boundary conditions can be solved numerically. By the same logicas the proof of Lemma 1, one can construct an equilibrium where Buyer gets this lowerbound as his utility, V (x, t) = V (x, t). We can then construct the equilibrium outcomeby solving Seller’s stopping problem. At each moment, Seller decides whether to stop byoffering P (x, t) = xt − V (x, t) to Buyer, or to continue the game. Seller’s choice of tradingthreshold x(t) must satisfy: {

U(x(t), t) = x(t)− V (x(t), t)

Ux(x(t), t) = 1− V x(x(t), t)(46)

Figure 12 presents the solution for the case of T = 1, cb = 0.25, σ2 = 1, and x0 = 0.As time goes, the remaining problem shrinks in scale. Thus both the trading threshold

and trading price decline over time towards 0, and Buyer’s quitting threshold increases overtime towards 0. Also, players never wait until they exhaust all attributes. Buyer’s quittingthreshold must approach P as time approaches T . The quitting threshold crosses x = 0strictly before time T . If the quitting threshold is above 0, then instead of letting Buyerwith product value x is between 0 and the quitting threshold quit, Seller wants to sell atthe price of P (x, t) = x. As a result, the agreement threshold and the quitting thresholdcollapse at 0. So the sales process must end before Buyer’s quitting threshold crosses 0,

55

Figure 12

which happens strictly before players exhaust all attributes at time T .

56

7. Bayesian Learning

Instead of matching on attribute, an alternative way to model the discovery process is tohave players observe signals on the true value of the product and update their beliefs usingBayes’ rule. If the true value of the product does not change over time, then players get“closer and closer” to the truth by receiving more signals. The game becomes non-stationary.

Suppose that, before the negotiation begins, Buyer and Seller have a common prior onthe true value x∗ that follows a normal distribution with mean ν and variance e2. Eachmoment during the game, players observe a common signal that equals true value plus anormal error term that is independent across time. Players update their beliefs about theproduct value as bargaining continues.

Specifically, let St denote the cumulative signal up to time t. The signal is assumed tofollow the process dSt = x∗dt+ ηdW . An incremental signal of size dt is thus the true valuex∗dt plus a normal error with variance η2dt. Bayesian updating on the normal prior impliesthat the expected value of the product after observing t signals, xt, can be written as

xt =ν/e2 + St/η

2

1/e2 + t/η2(47)

The change in expected product value xt can be decomposed as

dxt =x∗ − xt

η2(1/e2 + t/η2)dt+

1

η(1/e2 + t/η2)dW (48)

The expected change in the expected product value, conditional on observing St, E[dxt|St],must be 0, because E[x∗|St] = xt by definition of xt. The variance of the change in expectedproduct value with dt signals is σ2

t dt = 1η2(1/e2+t/η2)2

dt. Thus, the process xt can be describedas a Brownian motion with a variance that decreases with time t.

The rest of the analysis is carried out similarly to earlier sections. Equilibrium valuefunctions satisfy the following partial differential equations:

Vxx =2

σ2t

(cb − Vt)

Uxx =2

σ2t

(cb − Ut)(49)

One first solves for Buyer’s optimal stopping problem facing a fixed price of P . This givesthe lower bound on Buyer’s equilibrium value function, V (x, t|P ), and Buyer’s quittingthreshold x(t). One can then construct an equilibrium where Buyer gets V (x, t) = V (x, t) ashis utility. At each moment, Seller decides whether to stop and receive P (x, t) = xt−V (x, t),or to continue. Buyer optimally chooses to comply with Seller’s strategy. Trade happenswhen product value reaches Seller’s stopping threshold x(t), and Buyer quits when productvalue reaches x(t) = x(t). Solving Seller’s optimal stopping problem gives the equilibriumoutcome. Figure 12 illustrate the case of cb = 0.25, ν = 0, e2 = 1, η2 = 1, at a list price of0.5.

Similar to the previous extension to finite attributes, one can show that both the trad-

57

Figure 13

ing threshold and trading price decline towards 0, and Buyer’s quitting threshold increasestowards 0. Furthermore, there exists a time by which the players must trade or quit, eventhough the game does not impose an explicit deadline. This implicit deadline happens whenBuyer’s quitting threshold hits 0. Similar to the finite attributes case, the game must endbefore Buyer’s quitting threshold and agreement threshold collapse at 0.

Note that the posterior belief xt becomes stationary if the true value of x∗t evolves with arandom walk of variance ρ2, due to Buyer’s evolving preference. Then the posterior variancefollows dσt

dt= −σ2

t /η2 + ρ2. If σ0 = ρη, then σt = σ0 remains constant for all t, otherwise σt

approaches ρη asymptotically over time. Thus the baseline model can be interpreted as aBayesian learning model with an evolving preference.

Declining Importance of Attributes

In the baseline model, all attributes are assumed to be equally important. In reality, someattributes may be more important for a product’s value, and the two parties can match onthese attributes first. As players discover, later attributes have less impact on the expectedvalue of the product. This differs from the base model because the discovery process is nowrepresented by dxt = σtdWt, where σt decreases in t. Note that this is closely related to theBayesian updating extension above, where σ2

t dt = 1η2(1/e2+t/η2)2

dt. Once σt is given, the restof the analysis can be carried out in the same way. In the equilibrium, both the tradingthreshold and trading price decline over time and Buyer’s quitting threshold increases overtime. The game must end before Buyer’s quitting threshold hits 0, which happens in finitetime.

58