The Logic of Bargaining

Transcription

1 The Logic of Bargaining Dongmo Zhang Intelligent Systems Lab University of Western Sydney Australia Thematic Trimester on Game IRIT, France 6 July 2015

2 Bargaining Theory A two-person bargaining situation involves two individuals who have the opportunity to collaborate for mutual benefit in more than one way. [John Nash, 1950] Under such a definition, nearly all human interaction can be seen as bargaining of one form or another. [Ken Binmore et al. 1992]

3 Multi-agent systems A multi-agent system (MAS) is a system composed of multiple interacting intelligent agents, each of which is: Autonomous: acts on its own. Self-interested: directs its activity towards achieving its goals. Decentralized: no designated controlling agent. Research in multi-agent systems is concerned with the study, behaviour, and construction of a collection of possibly pre-existing autonomous agents that interact with each other and their environments. [Katia P. Sycara, 1998]

4 Outline 1 Motivation 2 Game-theoretic solutions 3 Logical model 4 Solution construction 5 Axiomatic system 6 Continuous domain 7 Conclusion

5 Bargaining problems Bargaining problem: Pie Devision. Devision range: x [0, 1], y = 1 x Utility of player 1: u 1 (x) Utility of player 2: u 2 (y) Bargaining game: (S, d), where S R 2 & d S Bargaining solution: f (S, d) S

6 Game-theoretic bargaining solutions Nash s bargaining solution (NBS): the maximizer of the product of utilities. Kalai-Smorodinsky s Solution (KSS): the maximizer of the points in S on the segment connecting d and a(s,d).

7 Axiomatization of Bargaining Solutions A bargaining solution is the NBS iff it satisfies: Pareto-optimality Symmetry Scale invariance Independence of irrelevant alternatives. A bargaining solution is the KSS iff it satisfies: Pareto-optimality Symmetry Scale invariance Restricted Monotonicity

8 Split a pie Example Two players bargain over the split of a pie. u 1 (x) = x u 2 (y) = y 2 S = {(x, y 2 ) : 0 x 1 and y = 1 x}, d = (0, 0). Nash s prediction: (33.3, 66.7) Kalai-Smorodinsky s prediction: (38.2, 61.8)

9 The numbers illusion Can this prediction be tested as in the sciences? The use of numbers, even if analytically convenient, obscures the meaning of the model and creates the illusion that it can produce quantitative results. I am not convinced that Nash s theory has done more than clarify the logic of one consideration which influences bargaining outcomes. I can not see how this consideration will comprehensively explain real-life bargaining results. Were game theorists to use a more natural language to specify the model, the solution would become clearer and more meaningful. [Ariel Rubinstein, 2000]

10 Example: Political negotiation Two political parties in the Parliament bargain over a government rescue plan in response to the 2008 financial crisis. Proposals from the parties: Inject funds into struggling financial institutions Rescue car makers Relieve homeowners of heavy house mortgage Sponsor job training and job creation. Increase taxes Obviously each party has their benefits from different industries therefore has preference on different rescue plans. However, representing the preference for each item in numbers can be a hard job for each party.

11 Logical solution: a possibility Represent bargaining terms in logic: bank: fund financial institutions; car: rescue car makers; house: help house mortgagors; training: create training opportunities; inctax: increase taxes. Represent constraints in logic: (bank house): mortgagees and mortgagors shouldn t be both funded. (car bank) inctax: it is impossible to rescue both car industry and financial institutions without increasing taxes.

12 Representation of bargaining problems: demands Bargainers demands: Party A wants to inject almost all funds into the major banks but a small amount for job training. Tax increase is never a policy of party A. Party B insists on funding car makers and individual homeowners. Both parties know that there is no need to support both sides of house mortgage. Also the government budget does not allow to rescue both car industry and financial institutions unless increase taxes. A: {bank, training, inctax, (bank house), (car bank) inctax} B: {car, house, (bank house), (car bank) inctax}

13 Representation of bargaining problems: conflicts A: {bank, training, inctax, (bank house), (car bank) inctax} B: {car, house, (bank house), (car bank) inctax} Conflicts between the negotiation parties: Conflict 1 {bank, house, (bank house)} Conflict 2 {bank, car, inctax, (car bank) inctax}

14 Representation of Preferences Each party ranks their bargaining items in total pre-order, representing the firmness the party insists the items (the higher the firmer). Party A (bank house) (car bank) inctax intax bank training Party B (bank house) (car bank) inctax car house

15 Logical model of bargaining Bargaining game: G = ((X 1, 1 ),, (X n, n )). Bargaining solution: f (G) = (C 1,, C n ), where C i X i. Agreement: A(G) = f i (B) i N

16 Solution construction: the simultaneous concession solution (SCS) (bank house) (car bank) inctax intax bank training (bank house) (car bank) inctax car house The solution is { (bank house), (car bank) inctax, intax}, meaning that nothing is agreed.

17 Dummy demands Party A (bank house) (car bank) inctax intax bank training Party B (bank house) (car bank) inctax car house coffee The solution is { (bank house), (car bank) inctax, intax,car}, meaning that rescuing car makers has been agreed. Key idea We can use dummy demands to represent intentional delays, which plays a similar role as non-linearity of utility functions.

18 Axiomatic system The logic solution is characterized by the following axioms: 1 Consistency: f i (G) is consistent. i N 2 Comprehensiveness: For each i, f i (G) is comprehensive. 3 Collective Rationality: If G is non-conflictive, then f i (G) = X i for all i. 4 Disagreement: If G represents a disagreement situation, then f i (G) = for all i. 5 Contraction independence: If G max G, then f (G) = f (G ) unless G is non-conflictive.

19 Solution characterisation Theorem A logical bargaining solution is the simultaneous concession solution if and only if it satisfies Consistency Comprehensiveness Collective Rationality Disagreement Contraction independence

20 Continuous domain Pie devision problem Devision range: x 1 [0, 1], x 2 = 1 x 1 Utility of player 1: u 1 (x 1 ) Utility of player 2: u 2 (x 2 ) For each i = 1, 2, let P i (l) represent the following proposition: x i u 1 i ((1 l L i )u i (0) + l L i u i (1)) (1) i.e., evenly divide player i s utility into L i pieces. It means that each time, a player gives up equally amount of utility. Let C represent the constraints that rule out invalid pie division. X 1 = C {P 1 (1),, P 1 (L 1 )} X 2 = C {P 2 (1),, P 2 (L 1 )}

21 Continuous domain Round Player A s demand Player B s demand Agreement 0 x x x 1 90 x x 1 80 x x 1 70 x x 1 60 x x 1 50 x x 1 40 x x 1 30 x where L 1 = L 2 = 10. Both the Nash solution (33.3, 66.7) and the Kalai-Smorodinsky solution (38.2, 61.8) belong to the range. When L A = L B = 100, the solution ranges become x 1 [38, 38.4] and x 2 [61.6, 62], which means that it approaches to Kalai-Smorondinsky solution.

22 Conclusion A logical theory for multi-issue, n-person, raw domain bargaining. An elegant axiomatic system: simple and intuitive. The proposed solution approaches to Kalai-Smorondinsky solution for continuous domain. Bargaining reasoning: Identify conflicts. Modeling risk. Representation of bargaining problems Logical language + ordering > utility function The logical theory of bargaining is not a rival but a complementary of game-theoretic bargaining theory.