Towards Automated Bargaining in Electronic Markets: a Partially Two-Sided Competition Model

Towards Automated Bargaining in Electronic Markets: a Partially Two-Sided Competition ModelN. Gatti, A. Lazaric, M. Restelli {ngatti, lazaric, restelli}@elet.polimi.it

DEI, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, 20133, Italy

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Gatti, Lazaric, RestelliAMEC IX 2008

Aim and Outline

• Aim We aim at providing a satisfactory extension of the

alternating-offers protocol to electronic markets and at game theoretically analyzing it

• Outline1. We discuss the bargaining problem in electronic

markets2. We propose a protocol that extends the alternating-

offers protocol to electronic markets3. We study agents’ equilibrium strategies with complete

information4. We provide a solving algorithm and we experimentally

evaluate it

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Introduction to the Bargaining Problem

• A bargaining situation involves two parties, which can cooperate towards the creation of a commonly desirable surplus, over whose distribution both parties are in conflict [Serrano 2008]

• Bargaining is the most common form of negotiation and plays a crucial role in automated negotiations

• Bargaining is studied in depth both as cooperative problem [Nash 1953] and non-cooperative problem [Rubinstein 1982]

• The alternating-offers protocol [Rubinstein 1982] is considered the principal protocol for bilateral negotiations and it has received a lot of attention • In economics, to analyze human transactions [Osborne

and Rubinstein 1990]• In computer science, to automate electronic

transactions [Kraus 2001]

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Alternarting-Offers with Deadlines (informal)• It is an extensive-form game in which agents alternately act,

e.g. s acts at t=0, b acts at t=1, s acts at t=2, and so on • The model studied in computer science [Fatima 2002] is an

extension of [Stahl 1972] and [Rubinstein 1982]• Game mechanism:

• The agent that acts at t=0 is a parameter of the protocol• Agents’ allowed actions are:

• Offer a value x• Accept the last opponent’s offer• Exit the negotiation

• Agents’ preferences:• Agents have opposite preferences and temporal

discounting factors• Agents have reservation values, e.g.

• Buyer’s reservation value expresses the maximum price at which she would buy the item

• Seller’s reservation value expresses the minimum price at which she would sell the item

• Agents have deadlines and after these they strictly prefer not to reach any agreement rather than to reach

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Alternating-Offers with Deadlines (formal)

• Players

• Player function

• Actions

• Preferences

)()(

sellersbuyerb

)1()()0(

tti

exitaccept

xoffer )(

s

sti

sss

b

bti

bbb

sb

TtTtRVx

txU

TtTtxRV

txU

tNoAgreemenUtNoAgreemenU

1)()(

),(

1)()(

),(

0)()(

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Known Results

• One-issue complete information settings• Agents’ equilibrium strategies can be simply inferred by

backward induction similarly to [Stahl 1972]• Multiple-issue complete information settings

• Several procedures can be followed to negotiate different issues (e.g., price and quality), the most efficient is the in-bundle: all the issues are negotiated together

• The problem of finding agents’ equilibrium strategies with multiple-issues can be reduced (in linear time in the number of the issues) to the problem of negotiating one issue [Di Giunta and Gatti 2006; Fatima et al. 2006]

• Uncertain information settings• Several partial results have been provided in narrow

uncertainty settings, e.g. [Rubinstein 1985; Cramton et al. 2004; Gatti et al. 2008]

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Bargaining and Markets

• Within a market of bargaining agents two aspects coexist: • The matching of two opponents (a buyer and a seller) • The negotiation between two matched opponents

• Classic models from economic literature do not effectively capture the negotiation between autonomous agents in electronic markets [Rubinstein and Wolinsky 1985; Binmore et al. 1989]:• They assume the matching between two agents to be

random• They assume all the buyers (sellers) to be the same (agents

have the same parameter values) and agents have no deadline

• In electronic markets, we expect that:• Agents can choose the opponent with which to negotiate• Agents can be different, having different values for the

utility parameters

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Our Original Contributions

1. We provide a satisfactory model for capturing bargaining in markets• Our model rules both the matching between agents

and the negotiation• Our model extends the alternating-offers protocol, i.e.

in presence of one buyer and one seller, agents’ equilibrium strategies are those in the original

protocol2. Given the negotiation model, we study agents’

equilibrium strategies when information is complete

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The Proposed Protocol (1)

• We consider the following situation:• The items sold by the sellers are equal• All the sellers have exactly one item to sell• All the buyers are interested in buying exactly one item

• Agent characterization• We denote by bi the i-th buyer agent – her parameters will be RPbi, Tbi, and

bi – and by sj the j-th seller agent – her parameters will be RPsj, Tsj, and sj • Each agent, both bi and sj, will be characterized by a time point denoted by

Abi and Asj , respectively, where she enters the market• Matching mechanism

(At each time point)• At first, each bi announces the seller with which she wants to be matched• Then, each sj chooses the buyer to match among the ones that have

announced sj• Negotiation mechanism

• Once two opponents matched at time t, they start to negotiate at time t+d (the value of d is set by the negotiation platform)

• The agent that starts the negotiation is selected by the negotiation platform at random with probability 0.5

• During the negotiation agents can make the classic actions available in the alternating-offers protocol

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• Protocol extension:• Each time point is divided in two stages• In the first stage, each non-matched buyer announces the

seller with which she wants to be matched (matchable(sk)) or announce that she wants not to be matched (nonmatchable)

• In the second stage:• Each non-matched buyer can wait for a time point (wait) or

leave the market (exit)• Each non-matched seller can wait for a time point (wait), match

a buyer that has announced her in the first stage (match(bi)), or leave the market (exit)

• Each matched buyer and each matched seller negotiate alternately as prescribed by the classic protocol

• Action redefinition:• Action exit imposes agents to leave the market (in addition

to the negotiation)

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state stage agent time points available actions

bi (sj) is not matched

1 bi anynonmatchable, matchable(sk)

if sk is not matched

2

bi anywait, exit if bi has

made nonmatchable

sj anywait, match(bi) if bi has

made matchable(sj), exit

bi and sj are negotiating 2

bi alternately offer, accept, exit

sj alternately offer, accept, exit

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At t = 1• b3 and s2 match

• they start to negotiate at t = 2

• the platform selects s2 to open the negotiation

• b1 leaves the market

An Example

time

• A simple setting• Three buyers: b1, b2, b3

• Two sellers: s1, s2

• All the agents are present in the market from time t = 0• The value of d is set to d = 1

t = 0 t = 1 t = 2 t = 3 …

b1 b2 b3 s1 s2

stage 1 nonmatchable

matchable(s1)

matchable(s1)

stage 2 wait wait wait match(b2) wait

At t = 0• b2 and s1 match

• they start to negotiate at t = 1

• the platform selects b2 to open the negotiation

b1 b2 b3 s1 s2

stage 1 nonmatchable

matchable(s2)

stage 2 exit offer(…) wait match(b3)

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Main Results (1)

• In presence of one buyer and one seller• Agents prefer to match themselves immediately rather

than to wait for one or more time points and subsequently match themselves

• Once two agents were matched, their equilibrium strategies are exactly those in the classic alternating-offers protocol

• If action exit does not impose agents to leave the market• Once two agents were matched, their equilibrium

strategies are different from those in the classic alternating-offers protocol

• The buyer or the seller can exploit the action exit to leave the negotiation and subsequently start a new negotiation with the same opponent

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Main Results (2)

• In presence of more buyers and more sellers• The problem is essentially a matching problem, since,

once two opponents matched, they negotiate as is prescribed by classic alternating-offers protocol

• We provide an algorithm that produces the equilibrium matching for a large range of the parameters and we experimentally evaluate it

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The Solving Algorithm

The algorithm develops in three steps1. The outcomes of all the possible negotiations are calculated:

they are 2·m·n, where m is the number of buyer and n is the number of sellers

2. The utility expected by each agent from matching each possible opponent is computed and agents’ preferences over the opponents to match are found:

bi: s1 s3 s4 s5 s8 … sn

it requires m linear searches among n elements and n linear searches among m elements and

3. Iteratively, each pair (bi, sj) such that sj is the first choice for bi and bi is the first choice for sj is removed from the problem

• The proof can be easily produced• The asymptotically computational complexity is O(m·n)

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Experimental Evaluation

min{m,n} success min{m,n} success min{m,n} success

2 ~99.7% 6 ~90.1% 10 ~74.2%

3 ~98.2% 7 ~86.1% 15 ~55.5%

4 ~96.2% 8 ~82.4% 20 ~37.0%

5 ~93.4% 9 ~78.1% 25 ~24.7%

• We evaluate the success of the proposed algorithm • Experimental setting

• For each value of min{m, n}{1, . . . , 25} we have considered 105 different settings

• In each settings agents’ parameters are chosen with uniform probability distribution from the following ranges: i (0, 1), Ti {2, 100}, RPbi = 1, RPsi = 0

• Experimental results

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Conclusions and Future Works

• Conclusions• The problem of bargaining in electronic markets is of

extraordinary importance• Literature lacks of satisfactory model• We provide a satisfactory bargaining model that extends the

classic alternating-offers protocol in electronic markets• We analyze agents’ equilibrium strategies with complete

information• The computational complexity of the proposed algorithm is

O(m·n)• Future works

• We will complete our solving algorithm by resorting to Gale-Shapley stable marriage algorithm

• We will enrich the bargaining model by introducing the outside option (i.e., the possibility of leaving a negotiation to start a new negotiation)

• We will study agents’ equilibrium strategies in presence of uncertainty

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Thank for your attention

Towards Automated Bargaining in Electronic Markets: a Partially Two-Sided Competition Model

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