Static versus Dynamic InformationalRobustness in Intertemporal Pricing

J������� L������� ��� X��������M�

Department of Economics, Harvard University

February 18th, 2017

A�������. We study intertemporal price discrimination in a se�ing with endogenousinformation. A seller commits to a price path, and a buyer observes signals ofher value according to some information structure. �e seller does not know theinformation structure and thus chooses prices to maximize the worst-case pro�t.We show that the optimal price path depends on the timing of information arrival.Speci�cally, if information can arrive arbitrarily over time, only the one-period pro�tcan be guaranteed. But if information arrives only in the �rst period, declining pricescan improve the seller’s pro�t.

Suppose a monopolist has developed a completely new durable product and is deciding how toset prices to maximize pro�t. Consulting the literature on intertemporal pricing, the monopolistmay at �rst think that if the pool of potential consumers does not change over time, pro�t wouldbe maximized by charging the optimal one-period price in each period.1 However, if the productis completely new, then the monopolist should consider the possibility that consumers will learnsomething about their value a�er pricing decisions have been made. For example, when the AppleWatch, Amazon Echo, and Google Glass were released, consumers with li�le prior experiencewould rely on journalists or product reviewers to inform their willingness-to-pay. But how muchand what information is provided about the product may depend on the monopolist’s pricing

C������. jlibgober@fas.harvard.edu, xiaoshengmu@fas.harvard.edu. We are particularlyindebted to Drew Fudenberg for guidance and encouragement. We also thank Gabriel Carroll, Ed Glaeser, Ben Golub,Johannes Horner, Michihiro Kandori, Sco� Kominers, Eric Maskin, Tomasz Strzalecki, Juuso Toikka, and our seminaraudience at Harvard for comments. Errors are ours.

1See Stokey (1979), Bulow (1981), Conlisk, Gerstner and Sobel (1984), among others. �ese papers show that a sellerwith commitment does not bene�t from choosing lower prices in later periods.

2

decision. In general, this possibility of information arrival can make the monopolist’s problemquite complicated.

�is paper develops an intertemporal pricing model where a buyer observes signals of hervalue, possibly over time. We assume that the seller does not know the information structure thatinforms the buyer of her value, and that he chooses prices as if the information structure were theworst possible given the prices posted. One standard justi�cation for this worst-case analysis isthat the seller may want to guarantee a good outcome, no ma�er what the information structureactually is.2 For our application, another justi�cation is that a competitor may be interested inminimizing the seller’s pro�t (even without a directly competing product). Our framework wouldbe appropriate if other �rms were able to release information on the product in a way that isoutside of the seller’s control.

We show that if the buyer can receive multiple signals of her value over time, then the sellerachieves the optimal pro�t with a constant price path. One explanation for this result is as follows:in each period, the adversary could release information that minimizes the pro�t in that period.Doing so would make the seller’s problem separable across time, eliminating potential gains fromdecreasing prices. �is intuition is incomplete, because the worst-case information structuresfor di�erent periods need not be consistent, in the sense that past information may prevent theadversary from minimizing pro�ts in the future. A key step of our argument is to show that theworst-case information structure in any period takes a partitional form, so that the seller’s pro�tcan be minimized period-by-period.

In contrast, Proposition 3 shows that if the buyer receives all information in the �rst period,then the seller increases his pro�t by charging di�erent prices in di�erent periods. By doing so,the seller ensures that any adversary who provides information only once cannot minimize pro�tsin all periods simultaneously. �is reasoning provides a novel explanation for intertemporal pricediscrimination, as far as we are aware.

In the applications listed above, our results imply that when information is concentratedaround the release date—say, through the Consumer Reports review—then intertemporal pricediscrimination may be optimal. But if information can arrive at later dates—say, through Amazonratings or even word of mouth—then the optimality of constant prices is restored.

We are not the �rst to consider the application of dynamic pricing for new products, and infact there is by now a sizable literature that uses maxmin objectives in these se�ings. However, theprevious literature considers ambiguity over the value distribution itself, and not over informationstructures.3 Our paper relates optimal pricing to learning and information arrival. Information

2A more complete discussion of this justi�cation can be found in the robust mechanism design literature, in particular:Chung and Ely (2007), Frankel (2014), Yamashita (2015), Bergemann, Brooks and Morris (2016), Carroll (2015, 2017).

3While Carrasco et. al. (2015) and Handel and Misra (2014) study the case where the buyer can purchase multiple

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I������������ R��������� �� I������������ P������ 3

arrival places restrictions on how the value evolves, and rules out the main cases considered inthe literature.4

Outside of the intertemporal pricing literature, ambiguity over information has been studied inmechanism design: Bergemann, Brooks and Morris (2016) examines a static auction environmentwith common values. �ough their se�ing is quite di�erent from ours, we share similar generalmotivation for considering informational robustness. We contribute to this literature by showinghow the possibility of dynamic information arrival in�uences the designer’s optimal policy.5

We proceed as follows. �e next section sets up the model, de�nes di�erent informationalrobustness concepts and discusses our assumptions. In Section 2 we consider the one-periodbenchmark, and in Section 3 we show that when information arrival is unrestricted, a longerselling horizon does not help the seller. Section 4 shows the optimality of intertemporal price dis-crimination when information only arrives in the �rst period. We consider two natural extensionsin Section 5: arriving buyers and seller information disclosure. Section 6 concludes. All omi�edproofs and additional results can be found in the Appendix.

1. MODEL

A seller (he) sells a durable good to a buyer (she)6 at time t = 1, 2, . . . , T , where T 1. Both theseller and the buyer have discount factor �. �e product is costless for the seller to produce, whilethe buyer has unit demand and obtains discounted lifetime utility from purchasing the objectequal to v. �e value v has distribution F supported on R+, with 0 < E[v] < 1. We let v denotethe minimum value in the support of F . �e distribution F is �xed and common knowledge, andthe buyer’s value for the object does not change over time.

However, the buyer does not directly know v; instead, she observes signals about v accordingto some information structure. We de�ne a static information structure to be a set S and a functionI : R+ ! �(S). �e interpretation is that the buyer observes a signal s 2 S at time 1, prior tomaking a purchase, and the conditional distribution of s is I(v) given the true value v. A dynamic

units (whereas our buyer purchases at most a single unit), the case of durable goods has been studied by Caldentey,Liu, Lobel (2016), Liu (2016) and Chen and Farias (2016).

4Several Bayesian se�ings allow for the value to change over time without explicitly modeling information arrival.Stokey (1979) assumes the value changes deterministically given the initial type. Deb (2014) and Garre� (2016)allow for stochastically changing values, but in these papers the value changes in ways that violate the martingalecondition for expectations. Deb (2014) assumes the value is independently redrawn upon Poisson shocks. For Garre�(2016), the value follows a two-type Markov-switching process.5�e implication of dynamic information arrival has been considered in the related literature of information design,see Ely (2017).

6Our analysis is unchanged in the case of a continuum of buyers who do not know their true values, as the worstcase would simply involve each buyer having the same information. �e case with arriving buyers is considered inSection 5.1.

3

4

information structure is a sequence of sets (St

)

T

t=1 and functions It : R+ ⇥S1 ⇥S2 ⇥ · · ·⇥St�1 !

�(St

). �is means that the buyer observes st

2 St

at time t, whose conditional distribution isIt

(v, s1, s2, . . . , st�1) given the true value v and the history of realized past signals.�e timing of the model is as follows. At time 0, the seller commits to a price path (p

t

)

T

t=1,where we allow p

t

= 1 to mean that the seller refuses to sell in period t. �en nature chooses aninformation structure I 2 �, where � denotes a set of information structures deemed possibleby the seller. In our main analysis, � represents either the set of static information structures orthe (larger) set of dynamic information structures. Having observed both the price path and theinformation structure, the buyer chooses a strategy to maximize expected discounted value lessprice:

⌧ ⇤ 2 argmax

⌧

E⇥�⌧�1

(E[v|s1, . . . , s⌧ ]� p⌧

)

⇤.

Here ⌧ represents a stopping time adapted to the information arrival process under I . �is meansthat the buyer purchases the good whenever ⌧ dictates her to stop, given the signals observed sofar. We allow ⌧ to take any positive integer value T , or ⌧ = 1 to mean the buyer never buys.

�e seller evaluates payo�s as if the information structure chosen were the worst possible,given his own choice of the price path (p

t

)

T

t=1 and buyer’s optimizing behavior. Hence the seller’spayo� is:

sup

(pt)Tt=1

inf

I2�,⌧

⇤E[�⌧⇤�1p

⌧

⇤] s.t. ⌧ ⇤ is optimal given (p

t

)

T

t=1 and I.

Note that when the buyer faces indi�erence, ties are broken against the seller. Breaking indi�erencein favor of the seller would add cumbersome details without changing the results.7

1.1. Discussion of Assumptions

An important assumption of the model is that the monopolist has commi�ed to a price pathbefore the information structure is chosen. One may wonder whether a more realistic assumptionmight involve a seller who commits to prices a�er the information structure has been picked. �eone-period case of such a model has been considered by Roesler and Szentes (2017). However, ifwe insist that nature cannot condition the information structure on prices in the future, then theseprices should also be hidden from the buyer. �is would bring us to a di�erent model withoutseller commitment, where the seller might be worse o�.8 Achieving the worst-case guaranteethen requires the seller to commit to the price path.7When ties are broken against the seller, it follows from our analysis that the sup inf is achieved asmaxmin. �iswould not be true if ties were broken in favor of the seller.8More precisely, the Coase conjecture predicts vanishing pro�ts for a seller without commitment when the buyerknows her value and is su�ciently patient. See Fudenberg, Levine and Tirole (1985), Gul, Sonnenschein and Wilson(1986).

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Another important assumption of the model is that the monopolist is restricted to choosingprice paths. In particular, we are ruling out mechanisms which utilize random prices. Bergemannand Schlag (2011) and Carrasco et. al. (2015) show that with one period, these random mechanismsmay improve beyond a single posted price. �e contrast with these papers arises because in ourmodel, nature can condition the worst-case information structure on the realized prices, elimi-nating possible gains from randomization. Whether the restriction to price paths is appropriatedepends very much on the technology of the environment in practice. Our approach follows theintertemporal pricing literature, where posted price mechanisms are typically assumed.

2. ONE-PERIOD BENCHMARK

We start with the case where the seller only has one period to sell the object. To solve this problem,we de�ne a transformed distribution of F . For expositional simplicity, the following de�nitionassumes F is continuous. Our results apply equally to the discrete case, though the generalde�nition requires additional care and is relegated to Appendix A.

De�nition 1. Given a continuous distribution F , the transformed distribution G is de�ned asfollows. For y 2 R+, let L(y) denote the conditional expectation of v ⇠ F given v y. �en G is thedistribution of L(y) when y is drawn according to F .

�e distribution G is useful because for any price p, nature can only ensure that the objectremains unsold with probability G(p). �is holds since the worst-case information structure hasthe property that a buyer who does not buy has expected value exactly p. Otherwise nature could�nd an information structure that makes a non-purchasing buyer slightly more optimistic abouther value and decreases the probability of sale. It follows that the worst-case information structureinvolves telling the buyer whether her value is above or below F�1

(G(p)), making 1�G(p) theprobability of sale.

�is discussion gives us the following proposition.

Proposition 1. In the one-period model, the maxmin optimal price p⇤ solves the following maxi-mization problem:

p⇤ 2 argmax

p

p(1�G(p)). (1)

For future reference, denote by ⇧

⇤= max

p

p(1�G(p)) the one-period maxmin pro�t.

It is worth comparing the optimization problem (1) to the standard model without ambiguity.If the buyer knew her value, the seller would maximize p(1� F (p)). In our se�ing, the di�erenceis that the transformed distribution G takes the place of F , which will be useful for the analysis inlater sections. �e following example illustrates:

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6

Example 1. Let v ⇠ Uniform[0,1], so that G(p) = 2p. �en p⇤ = 14 and ⇧

⇤=

18 . �e information

structure chosen by nature tells the buyer whether or not v > 1/2. Relative to the case where thebuyer knows her value, the seller charges a lower price and obtains a lower pro�t under informationalambiguity. In Appendix B, we show that this comparative static holds generally.

3. MULTIPLE PERIODS WITH DYNAMIC INFORMATION STRUCTURES

In this section we show that having multiple periods to sell does not improve the seller’s pro�twhen nature can choose amongst dynamic information structures. Since the seller can alwayssell exclusively in the �rst period, the one-period pro�t ⇧⇤ forms a lower bound for the seller’smaxmin pro�t.

To show that ⇧⇤ is also an upper bound, we take advantage of the partitional form of worst-case information structures derived in the previous section. �is property allows us to explicitlyconstruct an information structure for any given price path, such that the seller’s pro�t againstthis information structure decomposes into a convex combination of one-period pro�ts.

Proposition 2. For any price path (pt

)

T

t=1, there is a dynamic information structure (and a cor-responding optimal stopping time) that yields expected pro�t no more than ⇧

⇤. Hence the seller’smaxmin pro�t against all dynamic information structures is ⇧⇤, irrespective of the time horizon T

and the discount factor �.

To prove this proposition, we �rst review the sorting argument when the buyer knows hervalue. In this case, given a price path (p

t

)

T

t=1, we can �nd time periods 1 t1 < t2 < · · · T andvalue cuto�s v

t0 = 1 > vt1 > v

t2 > · · · � 0, such that a buyer with v 2 [vtj , vtj�1 ] optimally buys

in period tj

(as in Stokey (1979)). �is implies that in period tj

, the object is sold with probabilityF (v

tj�1)� F (vtj).

Inspired by the one-period problem, we construct an information structure under which inperiod t

j

, the object is sold with probability G(vtj�1)�G(v

tj) (that is, where G replaces F ). �efollowing information structure I satis�es this condition:

• In each period tj

, a buyer is told whether or not her value is in the lowestG(vtj)-percentile.9

• In all other periods, no information is revealed.

We describe the buyer’s optimal stopping behavior under I in the following lemma (with theproof in Appendix A):

9If G(vtj ) = 1, or equivalently vtj � E[v], then no information is revealed.

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I������������ R��������� �� I������������ P������ 7

Lemma 1. An optimal stopping time ⌧ ⇤ under I involves the buyer buying in the �rst period tj

when she is told her value is not in the lowest G(vtj)-percentile.

Using this lemma, we can now prove the proposition by computing the seller’s pro�t underthe information structure I and the stopping time ⌧ ⇤:

Proof of Proposition 2. Since a buyer with value v in the percentile range (G(vtj), G(v

tj�1)] buysin period t

j

, the seller’s discounted pro�t is given by:

⇧ =

X

j�1

�tj�1ptj ·

�G(v

tj�1)�G(vtj)

�

=

X

j�1

(�tj�1ptj � �tj+1�1p

tj+1) · (1�G(vtj))

=

X

j�1

(�tj�1 � �tj+1�1)v

tj · (1�G(vtj))

�t1�1 · ⇧⇤,

(2)

where the second line is by Abel summation,10 the third line is by vtj ’s indi�erence between

buying in period tj

or tj+1, and the last inequality uses v

tj(1�G(vtj)) ⇧

⇤, 8j. ⌅

One may wonder which price paths allow the seller to achieve ⇧⇤ as worst-case pro�t. �eanswer is given by the following corollary:11

Corollary 1. When the one-period maxmin optimal price p⇤ is unique, the optimal pricing strategiesof the seller are those with p⇤ = p1 p

t

, 8t.

4. MULTIPLE PERIODS WITH STATIC INFORMATION STRUCTURES

We now show that the seller can improve upon the one-period pro�t if nature is restricted tochoosing amongst static information structures. Since the seller can set p

t

= p2 for t > 2, itsu�ces to show this result for two periods, which we assume throughout this section. Staticinformation structures are much simpler to represent than dynamic information structures, sinceit is equivalent to work with the distribution of the buyer’s posterior expected value. LetD denote

10Abel summation says thatP

j�1 ajbj =

Pj�1

⇣(aj � aj+1)

Pji=1 bi

⌘for any two sequences {aj}1j=1, {bj}1j=1

such that aj ! 0 andPj

i=1 bi is bounded. We take aj = �tj�1ptj and bj = G(vtj�1)�G(vtj ).11Above we showed that the worst-case pro�t can be achieved by se�ing p1 = p⇤ and pt = 1 for t � 2. In contrastto the known value case, it is not obvious that other price paths (such as constant prices of p⇤) do as well. �is isformally proved as Lemma 3 in Section 5.1.

7

8

this posterior distribution under nature’s choice of information structure. �e seller’s pro�t isgiven by:

⇧ = p1(1�D(v1)) + �p2(D(v1)�D(v2))

= (1� �)v1(1�D(v1)) + �v2(1�D(v2)),(3)

where v1 = p1��p2

1��

and v2 = p2 are the threshold values for buying in period one and period tworespectively, assuming p1 � p2. We will �nd v1 and v2 such that ⇧ > ⇧

⇤+ " for any distribution

D that can arise under some information structure.To do this, we introduce a result of Rothschild and Stiglitz (1970) (elaborated on by Gentzkow

and Kamenica (2016)), which yields a tractable representation of D:

Lemma 2 (Rothschild and Stiglitz (1970), Gentzkow and Kamenica (2016)). If D is a distributionof the posterior expected value, then

Zx

0

D(s)ds Z

x

0

F (s)ds, 80 x 1. (4)

�is result is useful because it allows us to give a joint upper bound of D(v1) and D(v2). Ouranalysis from the one-period benchmark shows that D(v

i

) G(vi

) (because G(vi

) maximizesthe probability of no sale). By using (4), we can further show that for an appropriate choice of(v1, v2), the inequality D(v

i

) G(vi

) must be su�ciently slack at some vi

.12 Together with (3),we will be able to conclude ⇧ > ⇧

⇤+ ".

Proposition 3. Suppose T = 2 and p⇤ > v.13 �en for all � 2 (0, 1), there exist prices p1, p2 andsome ✏ > 0, such that the seller’s pro�t against any static information structure is at least ⇧⇤

+ ✏.

Proof of Proposition 3. We take v2 = p⇤, with v1 > p⇤ to be determined. From (4), for all x > v1 >

p⇤ it holds thatZ

x

0

F (s)ds �Z

x

0

D(s)ds � (v1 � p⇤)D(p⇤) + (x� v1)D(v1), (5)

where the second inequality follows from the monotonicity of D.

12Intuitively, if this were not the case,D would have to arise from a bipartite information structure with two di�erentthresholds (corresponding to v1 and v2).

13�e assumption p⇤ > v is not dispensable. For example, suppose F ⇠ Uniform[0.5, 1]. �en Stokey (1979) impliesthat in the known value case, the seller can do no be�er than se�ing p1 = p2 = 0.5. Since nature can alwaysinform the buyer perfectly in the �rst period, ⇧⇤

= 0.5 is also the seller’s maxmin pro�t against static informationstructures.

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�0

xF(s) ds = (x- p*)G(p*)

�* x �[v]v1

∫���(�)��

∫���(�)��

�(�*)(�-�*)

(�- [�])+

v

Figure 1: Graphical illustration of the choice of v1

Let us choose x = F�1(G(p⇤)). Recall from the one-period problem that the lowest G(p⇤)-

percentile of the distribution F has expected value p⇤, so that (by F (x) = G(p⇤))

p⇤ =1

F (x)

Zx

0

sdF (s). (6)

Since p⇤ > v, we have G(p⇤) > 0 and x > p⇤. Using integration by parts as well as theprevious equation, we obtain

Zx

0

F (s)ds = xF (x)�Z

x

0

sdF (s) = xF (x)� p⇤F (x) = (x� p⇤)G(p⇤). (7)

�is crucial step of our argument is illustrated in the following �gure, for the distribution F (s) =

s2.

Combining (5) and (7), we deduce the following inequality:

D(v1)�G(p⇤) v1 � p⇤

x� v1· (G(p⇤)�D(p⇤)). (8)

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10

Plugging this last inequality into the objective function (3), we conclude that

⇧ = (1� �)v1(1�D(v1)) + �p⇤(1�D(p⇤))

> (1� �)p⇤(1�D(v1)) + �p⇤(1�D(p⇤)) + ✏

= p⇤ [(1�G(p⇤) + �(G(p⇤)�D(p⇤))� (1� �)(D(v1)�G(p⇤))] + ✏

� p⇤(1�G(p⇤)) + ✏

= ⇧

⇤+ ✏.

(9)

We note that the inequality in the second line above holds for D(v1) < 1 and ✏ su�ciently small.�is is guaranteed by (8) and G(p⇤) < 1, so long as we choose v1 larger than but su�ciently closeto p⇤. �e inequality in the penultimate line holds by (8) whenever �

1��

� v1�p

⇤

x�v1, which is again

because v1 can be chosen close to p⇤. ⌅Since the seller can at most obtain ⇧

⇤ by choosing p2 � p1, we obtain the following simplecorollary:

Corollary 2. Suppose T = 2 and p⇤ > v. �en against static information structures, any optimalpricing strategy of the seller consists of strictly decreasing prices.

5. EXTENSIONS

5.1. Arriving Buyers

Our analysis extends without change to a continuum of buyers (as argued in Footnote 6). Onemay wonder if the same conclusions would hold if buyers were to arrive over time, as in Conlisk,Gerstner and Sobel (1984), Sobel (1991), Board (2008) and Garre� (2016). In contrast to the singlebuyer case, selling only once is no longer optimal as the monopolist may want to capture buyerswho only arrive in later periods. On the other hand, low prices in the future could allow nature tochoose an information structure that delays purchase, which may be costly for the seller.

To comment on this possibility, we modify the model by assuming that in each period t, a newbuyer arrives and decides when to buy the object. To be precise, our timing is as follows:

• At time 0, the seller chooses a price path (pt

)

T

t=1.

• At time t 2 {1, . . . , T}, a buyer arrives with value v(t) drawn from F , independently fromprevious buyers.

• Once a buyer arrives at time t, nature chooses an information structure I(t) according towhich this buyer learns her value.

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I������������ R��������� �� I������������ P������ 11

• Given the price path and the information structure I(t), the buyer arriving at time t choosesan optimal stopping time ⌧ (t) to purchase the object.

If every buyer learns only when she arrives, our analysis in the previous section directly impliesthat intertemporal price discrimination still has value. Hence we focus on dynamic informationstructures. If the seller were able to set personalized prices, Proposition 2 would imply that hismaxmin pro�t is 1��

T

1��

⇧

⇤. We will show that the seller can achieve the same pro�t by a constantprice path of p⇤, without conditioning prices on the arrival time.

Under known values, any arriving buyer facing a constant price path would buy immediately(if she were to buy at all), due to impatience. However, the promise of future information mayinduce delay. In the following lemma, we show that the seller can eliminate the potential damageof delayed purchases by commi�ing to never lower the price.

Lemma 3. In the multi-period model with dynamic information structures (and one buyer), the sellercan guarantee ⇧⇤ with any price path (p

t

)

T

t=1 satisfying p⇤ = p1 pt

, 8t.We present the intuition here and leave the formal proof to Appendix A. Let us �x a non-

decreasing price path. For any dynamic information structure nature can choose, we consideran alternative static information structure that simply informs the buyer (in the �rst period)of her stopping time. �is replacement is in the spirit of the revelation principle; however, itdi�ers due to the fact that we push nature’s recommendation to time 1.14 �e proof shows thatfor non-decreasing prices, we can �nd a replacement such that the buyer still follows nature’srecommendation of whether or not to buy. �is replacement has the property that the seller’spro�t is decreased. Since the seller receives at least ⇧⇤ against any static information structure,we obtain the lemma. We comment that more generally, for an arbitrary price path, the sameworst-case pro�t is obtained if nature only reveals information a�er a price drop.

Armed with this lemma, we can show the following:

Proposition 4. In the multi-period model with dynamic information structures and arriving buyers,the seller can guarantee 1��

T

1��

⇧

⇤ with a constant price path charging p⇤ in every period. No otherprice path generates a higher worst-case pro�t.

5.2. Seller Information Disclosure

So far we have assumed that the seller has no control over the information the buyer receives.In practice, however, the seller may disclose information himself, for example via an advertising14�is contrasts with the obfuscation principle of Ely (2017), which implies that it is without loss to recommend anaction at every time.

11

12

campaign (Johnson and Mya� (2006)). In this section we show that this possibility does not changeour conclusions.

We modify the model in Section 1 by assuming that in addition, the buyer observes somesignal s0 2 S0 at time 0. �e signal space S0 as well as the conditional distributions H(· | v)are common knowledge between the buyer and the seller, and we denote this initial informationstructure by H.15 We allow nature’s choice of information structure to depend on s0 but keep allother aspects of the model identical.

A signal s0 induces a posterior distribution on the buyer’s value, which we denote by Fs0 . We

de�ne Gs0 to be the transformed distribution of F

s0 , following De�nition 1. �e same analysis asin Section 2 yields the following result:

PROPOSITION 1’: In the one-period model where the buyer observes initial information structureH,the seller’s maxmin optimal price p⇤H is given by:

p⇤H 2 argmax

p

p(1� Es0 [Gs0(p)]). (10)

We denote the maxmin pro�t in this case by ⇧

⇤H.

�e expression (10) is familiar in two extreme cases: ifH is perfectly informative, then Fs0 is

the point-mass distribution on s0. �is means Gs0(p) is the indicator function for p � s0, so that

Es0 [Gs0(p)] = F (p). In contrast, ifH is completely uninformative, we return to Equation (1).

We now turn to the multi-period problem. Against dynamic information structures, ourprevious proof carries over and shows that the seller does not bene�t from a longer sellinghorizon:

PROPOSITION 2’: In the multi-period model where the buyer observes initial information structureH, the seller’s maxmin pro�t against all dynamic information structures is ⇧⇤

H, irrespective of thetime horizon T and the discount factor �.

For the static information structure case, the analysis extends if the support of Fs0 is indepen-

dent of s0. But in general, we need additional assumptions to ensure that the seller can improveupon the one-period pro�t. �e di�culty is related to the hypothesis p⇤ > v in the statement ofProposition 3. We present a version of such a result, with further discussion in Appendix B.

PROPOSITION 3’: De�ne vs0to be the minimum value in the support of F

s0 . Suppose that the initialinformation structureH has �nitely many signals, and that the one-period maxmin optimal price

15In principle, the seller may choose whether or not the buyer has access toH. In Appendix B, we show that the selleralways weakly prefers to release this information. Intuitively, releasing information limits the set of informationstructures with which nature can damage pro�ts.

12

I������������ R��������� �� I������������ P������ 13

p⇤H is distinct from all vs0. �en for some ✏ > 0, there exists a decreasing price path giving the seller

pro�t ⇧⇤H + ✏ against all static information structures.

6. CONCLUSION

In this paper, we have studied optimal monopoly pricing in relation to information availability,utilizing a robustness approach to provide a sharp answer on how they interact. A monopolistwho has developed a completely new durable product should consider whether buyers can learnmore about the product a�er the initial release date. In our model, if buyer learning is restrictedto the �rst period, then intertemporal price discrimination can improve pro�ts. Without anyrestrictions, the one-period pro�t achieved by a constant price path is the best the monopolist cando.

�e main lesson of this paper—that static and dynamic informational robustness may lead toqualitatively di�erent optimal policies—is one that we hope will be scrutinized in other contextsand under other modeling assumptions. In particular, many of the models inspired by Stokey (1979)could be reevaluated using our techniques. �e case of limited commitment seems compelling inour opinion, though there are technical di�culties associated with formalizing (seller) learningunder ambiguity (see Epstein and Schneider (2007)). One could also ask these same questions inmore general dynamic mechanism design se�ings, without restricting to posted prices.

�is paper contributes to a growing literature which employs the maxmin approach in analyz-ing the optimal design of mechanisms. In our se�ing, the maxmin approach allows us to focuson particular information structures—the partitional ones—with clear economic interpretations.We therefore avoid the di�culty in working with the entire space of information structures andstopping times. While it is certainly worthwhile to analyze the Bayesian case, doing so would �rstrequire a similarly tractable restriction as the one we have provided here.

A. APPENDIX A: PROOFS OMITTED FROMMAIN TEXT

We �rst de�ne the transformed distribution G in cases where F need not be continuous.

DEFINITION 1’: Given a percentile ↵ 2 (0, 1], de�ne g(↵) to be the expected value of the lowest↵-percentile of the distribution F . In case F is a continuous distribution, g(↵) = 1

↵

RF

�1(↵)

0 vdF (v).But in general, g is continuous and weakly increasing.

Let v be the minimum value in the support of F . For � 2 (v,E[v]], de�ne G(�) = sup{↵ :

g(↵) �}. We extend the domain of this inverse function to R+ by se�ing G(�) = 0 for � v andG(�) = 1 for � > E[v].16

16If F does not have a mass point at v, g(↵) is strictly increasing and G(�) is its inverse function which increases

13

14

We now provide proofs of the results in the main text, in the order in which they appeared.

Proof of Proposition 1. Given a price p set by the seller, the worst-case pro�t occurs when thereis maximal probability of signals that lead the buyer to have posterior expected value p. Firstconsider the information structure I that tells the buyer whether her value is above or belowF�1

(G(p)). By De�nition 1, the buyer’s expected value is exactly p upon learning the la�er. �isshows that, under I , the buyer’s expected value p with probability G(p).

Now we show that G(p) cannot be improved upon. To see this, note that it is without loss ofgenerality to consider information structures which recommend that the buyer either “buy” or“not buy”. Nature chooses an information structure to minimize the probability that the buyerbuys. By Lemma 1 in Kolotilin (2015), this minimum is achieved by a partitional informationstructure, namely by recommending “buy” for v > ↵ and “not buy” for v ↵. From this, we seethat the particular information structure I above is the worst case.

�us, for �xed price p, the seller’s minimum pro�t is p(1 � G(p)). �e proposition followsfrom the seller optimizing over p. ⌅

Proof of Lemma 1. De�ne “LOW” to be the signal where a buyer is told her value is below theG(v

tj)-percentile, and “HIGH” to be the opposite signal. Fix some period t, we will show thefollowing facts that together imply the lemma:Fact 1. A buyer who has only received “LOW” signals in the past �nds it optimal to delay herpurchase.Fact 2. A buyer receiving a “HIGH” signal �nds it optimal to buy immediately.

To prove Fact 1, suppose t < t1. Since no information has been revealed, the buyer’s expectedvalue is E[v], which clearly belongs to some interval [v

tj , vtj�1 ]. Hence if no more informationwere to be revealed, such a buyer would buy in some later period t

j

> t (or never). Since moreinformation could only improve the buyer’s payo�, in our se�ing the buyer still prefers to notbuy at time t.

Suppose instead that the current period t satis�es tk

t < tk+1. By assumption the buyer

has received “LOW” signals in periods t1, t2, . . . , tk. According to I , this means her value is inthe lowest G(v

tk)-percentile, so her expected value is simply min{E[v], v

tk}. Again, such a buyer

would �nd it optimal to buy in some later period if no more information were to be revealed.Hence she also does not buy at time t.

To prove Fact 2, suppose t = tj

and the buyer receives her �rst “HIGH” signal. �is means v isnot in the lowest G(v

tj)-percentile. By de�nition, the lowest G(vtj)-percentile has expected value

continuously. If instead F (v) = m > 0, then g(↵) = v for ↵ m and it is strictly increasing for ↵ > m. In thatcase G(�) = 0 for � v, a�er which it jumps to m and increases continuously to 1.

14

I������������ R��������� �� I������������ P������ 15

vtj , so the buyer knows with certainty that her value v is at least v

tj . Since under known values,any buyer with value v � v

tj prefers to buy in period tj

than at any later time, this also holdswhen the buyer only knows that v belongs to the percentile range (G(v

tj), G(vtj�1)]. ⌅

Proof of Corollary 1. Su�ciency follows from Lemma 3. To prove necessity, note that the lastinequality in (2) holds equal only when t1 = 1 and every v

tj is a one-period maxmin optimal price.�us when p⇤ is unique, there is no other t

j

except t1 = 1. By de�nition, this means under knownvalues, a buyer with v > p⇤ optimally buys in the �rst period; otherwise he never buys. �usp1 = p⇤ and p

t

� p⇤. ⌅

Proof of Lemma 3. Fix a dynamic information structure (St

, It

)

T

t=1 and an optimal stopping time ⌧of the buyer. As discussed in the main text, we will construct a static information structure (S, I)which weakly reduces the seller’s pro�t. We take S = {s, s}, corresponding to the recommen-dation to buy and not to buy, respectively. To specify the distribution of s conditional on v, lets1, s2, . . . be drawn according to the original dynamic information structure (and conditional on v).If ⌧ < 1, let the buyer receive the signal s with probability �⌧�1 at time 1. With complementaryprobability and when ⌧ = 1, let the buyer receive the opposite signal s.

We claim that under this static information structure, a buyer receiving the signal s hasexpected value at most p1. We actually show something stronger, namely that the buyer hasexpected value at most p1 conditional on the signal s and any realized s1.17 To prove this, notethat since stopping at time ⌧ is weakly be�er than stopping at time 1, we have

E[v|s1]� p1 Es2,··· ,sT ⇥�⌧�1

(E[v|s1, s2, · · · , s⌧ ]� p⌧

)

⇤. (11)

Since p⌧

� p1, we obtain by simple algebra

E[v|s1] Es2,··· ,sT ⇥�⌧�1E[v|s1, s2, · · · , s⌧ ] + (1� �⌧�1

)p1⇤. (12)

By noting that E[v|s1] = Es2,··· ,sT[E[v|s1, s2, · · · , s⌧ ]] from Doob’s Optional Sampling�eorem,

we derive the following inequality:

p1 � Es2,··· ,sT[(1� �⌧�1

)E[v|s1, s2, · · · , s⌧ ]]Es2,··· ,sT

[1� �⌧�1]

. (13)

17Technically we will only consider those s1 such that s occurs with positive probability given s1.

15

16

On the other hand, we can compute that

E[v|s1, s] = Es2,··· ,sT[(1� �⌧�1

)E[v|s1, s2, · · · , sT , s]]Es2,··· ,sT

[1� �⌧�1]

=

Es2,··· ,sT[(1� �⌧�1

)E[v|s1, s2, · · · , s⌧ ]]Es2,··· ,sT

[1� �⌧�1]

.

(14)

�e �rst equality follows directly from the de�nition of conditional expectation, as 1� �⌧�1 is theconditional probability of s given s1, s2, · · · , sT . �e second equality holds because �rst, s doesnot provide more information about v beyond s1, s2, . . . , sT , and second,

Es2,··· ,sT[(1� �⌧�1

)E[v|s1, s2, · · · , sT ]] = Es2,··· ,sT[(1� �⌧�1

)E[v|s1, s2, · · · , s⌧ ]]

because ⌧ is a stopping time.Comparing (13) and (14), we see that E[v|s1, s] p1 as claimed. �is implies that under

the static information structure constructed above, it is optimal for the buyer to only buy uponreceiving the signal s. Moreover, all sale happens in the �rst period as p

t

� p1, 8t. �us theprobability of sale is at most E[�⌧�1

], and the seller’s pro�t is at most E[�⌧�1] · p1. �is is no more

than E[�⌧�1 · p⌧

], the discounted pro�t under the original dynamic information structure. We havethus proved that with a non-decreasing price path, the seller’s pro�t is at least what he wouldobtain by selling only once at the price p1. Taking p1 = p⇤ shows the lemma. ⌅

Proof of Proposition 4. By the previous lemma, a constant price path of p⇤ delivers expected pro�t⇧

⇤ whenever a buyer arrives. However, Proposition 2 shows that the worst-case pro�t obtainedfrom this buyer is ⇧⇤. �e result follows from computing the discounted pro�t. ⌅

Proof of Proposition 1’ and 2’. �e proofs are direct adaptations of those presented in the maintext, for the model without an initial information structure. ⌅

Proof of Proposition 3’. De�ne xs0 = F�1

s0(G(p⇤H)). �is is the analogue of x from the proof of

Proposition 3 in the main text. As with the proof there, we take T = 2 and consider a seller whochooses v2 = p2 = p⇤H and v1 =

p1��p2

1��

slightly larger than p⇤H.Fix any realized initial signal s0. �ere are three possibilities, and in each case we will show

that the seller prefers the above price path to selling only once:

1. p⇤H � E[v|s0]. By charging p⇤H, �e seller obtains zero pro�t from such a buyer in theone-period problem. �e price path p1 > p2 = p⇤H also yields zero pro�t.

16

I������������ R��������� �� I������������ P������ 17

2. vs

< p⇤H < E[v|s0]. In this case we can follow the argument in the proof of Proposition 3to show that as long as (v1 � p⇤H) is su�ciently small compared to (x

s0 � p⇤H), the sellerdoes strictly be�er with two periods to sell. We can satisfy this condition for every s0, sincethere are �nitely many of them.

3. p⇤H < vs

. Suppose we have chosen v1 su�ciently close to p⇤H, so that v1 < vs

. From thede�nition of v1, we deduce that a buyer receiving the signal s0 always prefers to buy in the�rst period, regardless of what additional information she might receive from nature. Sincep1 > p⇤H, the seller again does strictly be�er.

Since the seller’s one-period maxmin pro�t is strictly positive, either of the la�er two possibil-ities above occurs with positive probability. Hence the seller achieves pro�t strictly greater than⇧

⇤H using a decreasing price path. ⌅

B. APPENDIX B: ADDITIONAL RESULTS

B.1. Ambiguity Leads to Lower Price

In Example 1, we mentioned that ambiguity over the information structure leads the seller tochoose a lower price than if the buyer knew her value. �e following proposition generalizes thisobservation to any continuous distribution.

Proposition 5. For any continuous distribution F , let p be an optimal price under known values:

p 2 argmax

p

p(1� F (p)). (15)

�en any maxmin optimal price p⇤ satis�es p⇤ p. Equality holds only if p⇤ = p = v.

Proof of Proposition 5. It su�ces to show that the function p(1�G(p)) strictly decreases whenp > p, until it reaches zero. By taking derivatives, we need to show G(p) + pG0

(p) > 1 for p > p

and G(p) < 1.From de�nition, the lowest G(p)-percentile of the distribution F has expected value p. �at is,

pG(p) =

ZF

�1(G(p))

0

vdF (v), 8p 2 [v,E[v]]. (16)

17

18

Di�erentiating both sides with respect to p, we obtain

G(p) + pG0(p) =

@

@p(F�1

(G(p))) · F�1(G(p)) · F 0

(F�1(G(p)))

= G0(p) · F�1

(G(p)).

(17)

�is enables us to write G0(p) in terms of G(p) as follows:

G0(p) =

G(p)

F�1(G(p))� p

. (18)

�us,

G(p) + pG0(p) =

G(p) · F�1(G(p))

F�1(G(p))� p

. (19)

We need to show that the RHS above is greater than 1, or that F�1(G(p)) < p

1�G(p) wheneverp > p and G(p) < 1. �is is equivalent to G(p) < F (

p

1�G(p)), which in turn is equivalent to

p

1�G(p)·✓1� F

✓p

1�G(p)

◆◆< p. (20)

From the de�nition of p, we see that the LHS above is at most p(1� F (p)) p < p, as we claimto show. Moreover, when p > v, the last inequality p(1� F (p)) < p is strict. Tracing back theprevious arguments, we see that G(p) + pG0

(p) > 1 holds even at p = p. In that case we wouldhave the strict inequality p⇤ < p as desired. ⌅

B.2. More on Seller Information Disclosure

One way to interpret the release of information by the seller is to consider it as a restrictionon the space of information structures that nature can minimize over. �e se�ing in Section5.2 is then one where the buyer has access to some information structureH (which is commonknowledge) and potentially more information chosen by nature (which the seller is uncertain about).By Blackwell (1951), we can equivalently consider nature choosing an information structure thatis more information thanH.18

As the seller releases more informative signals, nature is minimizing over a smaller set ofinformation structures. �us the worse case pro�t of the seller can only go up. �at is,

Proposition 6. SupposeH is Blackwell more informative thanH0. �en ⇧

⇤H � ⇧

⇤H0 .

18See also Bergemann and Morris (2016) for an extension of Blackwell’s partial ordering to multi-agent se�ings.

18

I������������ R��������� �� I������������ P������ 19

We conclude by studying an example in detail to illustrate the limits of Proposition 5 andProposition 3’.

Example 2. In contrast to the conclusion of Proposition 5, if an additional signal is released bythe seller, then the maxmin optimal price need not be lower than the known value price. Supposev ⇠ Uniform[0, 1], and the initial information structure H tells the buyer whether v � 3/5 or not.Selling only to buyers who are told v � 3/5 results in pro�t equal to 6/25, which turns out to be themaxmin pro�t. Note that p⇤H = 3/5 > 1/2 = p.

We show that the seller cannot improve upon 6/25 with multiple periods to sell, even if naturecan only choose amongst static information structures. Suppose nature chooses the following staticinformation structure (which is Blackwell more informative thanH). �ere are 3 signals s1, s2, s3,corresponding to v 2 [0, .1) [ [.3, .4), v 2 [.1, .2) [ [.4, .5) and v 2 [.2, .3) [ [.5, .6), respectively;and if v � .6, the value is revealed perfectly.

Given nature’s choice, the buyer’s distribution of posterior expected value has a 15 mass at each of

v = .2, .3, .4, while the remaining 25 mass is uniformly distributed on [.6, 1]. By Stokey (1979), the

seller’s maximum pro�t against this �xed distribution of values is achieved by selling only once. Inour case, simple computation shows that p = .3, .4, .6 are all optimal prices that yield a pro�t of 6/25,which is thus the seller’s maxmin pro�t. �is suggests that the assumptions stated in Proposition 3’are to some extent indispensable for the conclusion to hold.

19

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