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

1. Towards Automated Bargaining in Electronic Markets: a Partially Two-Sided Competition Model N. Gatti, A. Lazaric, M. Restelli {ngatti, lazaric, restelli}@elet.polimi.it DEI, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, 20133, Italy

2. 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 • Outline • We discuss the bargaining problem in electronic markets • We propose a protocol that extends the alternating-offers protocol to electronic markets • We study agents’ equilibrium strategies with complete information • We provide a solving algorithm and we experimentally evaluate it

3. 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]

4. 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

5. Alternating-Offers with Deadlines (formal) • Players • Player function • Actions • Preferences

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

7. 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 nodeadline • 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

8. Our Original Contributions • 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 protocol • Given the negotiation model, we study agents’ equilibrium strategies when information is complete

9. 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 dbi – and by sj the j-th seller agent – her parameters will be RPsj, Tsj, and dsj • 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

10. The Proposed Protocol (2) • 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)

11. The Proposed Protocol (3)

12. An Example • 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 • At t = 0 • b2 and s1 match • they start to negotiate at t = 1 • the platform selects b2 to open the negotiation • 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 t = 0 t = 1 t = 2 t = 3 … time

13. 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

14. 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

15. The Solving Algorithm The algorithm develops in three steps • 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 • 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 • 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)

16. Experimental Evaluation • 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: di  (0, 1), Ti {2, 100}, RPbi = 1, RPsi = 0 • Experimental results

17. 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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