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. Gatti, Lazaric, RestelliAMEC IX 2008 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 1.We discuss the bargaining problem in electronic markets 2.We propose a protocol that extends the alternating-offers protocol to electronic markets 3.We study agents’ equilibrium strategies with complete information 4.We provide a solving algorithm and we experimentally evaluate it

3. Gatti, Lazaric, RestelliAMEC IX 2008 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. Gatti, Lazaric, RestelliAMEC IX 2008 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. Gatti, Lazaric, RestelliAMEC IX 2008 5 Alternating-Offers with Deadlines (formal) Players Player function Actions Preferences

6. Gatti, Lazaric, RestelliAMEC IX 2008 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 narrow uncertainty settings, e.g. [Rubinstein 1985; Cramton et al. 2004; Gatti et al. 2008]

7. Gatti, Lazaric, RestelliAMEC IX 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 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

8. Gatti, Lazaric, RestelliAMEC IX 2008 8 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 protocol 2.Given the negotiation model, we study agents’ equilibrium strategies when information is complete

9. Gatti, Lazaric, RestelliAMEC IX 2008 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 b i the i-th buyer agent – her parameters will be RP b i, T b i, and  b i – and by s j the j-th seller agent – her parameters will be RP s j, T s j, and  s j Each agent, both b i and s j, will be characterized by a time point denoted by A b i and A s j, respectively, where she enters the market Matching mechanism (At each time point) At first, each b i announces the seller with which she wants to be matched Then, each s j chooses the buyer to match among the ones that have announced s j 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. Gatti, Lazaric, RestelliAMEC IX 2008 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(s k )) 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(b i ) ), 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. Gatti, Lazaric, RestelliAMEC IX 2008 11 The Proposed Protocol (3) statestageagenttime pointsavailable actions b i (s j ) is not matched 1bibi any nonmatchable, matchable(s k ) if s k is not matched 2 bibi any wait, exit if b i has made nonmatchable sjsj any wait, match(b i ) if b i has made matchable(s j ), exit b i and s j are negotiating 2 bibi alternatelyoffer, accept, exit sjsj alternatelyoffer, accept, exit

12. Gatti, Lazaric, RestelliAMEC IX 2008 12 At t = 1 b 3 and s 2 match they start to negotiate at t = 2 the platform selects s 2 to open the negotiation b 1 leaves the market An Example time A simple setting Three buyers: b 1, b 2, b 3 Two sellers: s 1, s 2 All the agents are present in the market from time t = 0 The value of d is set to d = 1 t = 0t = 1t = 2t = 3… b1b1 b2b2 b3b3 s1s1 s2s2 stage 1nonmatchablematchable(s 1 ) stage 2wait match(b 2 )wait At t = 0 b 2 and s 1 match they start to negotiate at t = 1 the platform selects b 2 to open the negotiation b1b1 b2b2 b3b3 s1s1 s2s2 stage 1nonmatchablematchable(s 2 ) stage 2exitoffer(…)waitmatch(b 3 )

13. Gatti, Lazaric, RestelliAMEC IX 2008 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. Gatti, Lazaric, RestelliAMEC IX 2008 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. Gatti, Lazaric, RestelliAMEC IX 2008 15 The Solving Algorithm The algorithm develops in three steps 1.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: b i : s 1 s 3 s 4 s 5 s 8 … s n it requires m linear searches among n elements and n linear searches among m elements and 3.Iteratively, each pair (b i, s j ) such that s j is the first choice for b i and b i is the first choice for s j is removed from the problem The proof can be easily produced The asymptotically computational complexity is O(m·n)

16. Gatti, Lazaric, RestelliAMEC IX 2008 16 Experimental Evaluation min{m,n}successmin{m,n}successmin{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 10 5 different settings In each settings agents’ parameters are chosen with uniform probability distribution from the following ranges:  i  (0, 1), T i  {2, 100}, RP b i = 1, RP s i = 0 Experimental results

17. Gatti, Lazaric, RestelliAMEC IX 2008 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

18. Gatti, Lazaric, RestelliAMEC IX 2008 18 Thank for your attention