The Agencies Method for Coalition Formation in Experimental Games John Nash (University of Princeton) Rosemarie Nagel (Universitat Pompeu Fabra, ICREA, BGSE) Axel Ockenfels (University of Cologne) Reinhard Selten (University of Bonn) Stony Brook 2013 Motivation • How to reach cooperation in a world of unequal bargaining circumstances (based on Nash JF (2008) The agencies method for modeling coalitions and cooperation in games Int Game Theory Rev 10(4):539–564) – Repeated interaction and acceptance of agencies through a voting mechanism • Combination of non-cooperative and cooperative game theory – Coalition formation, selection of agencies through non-cooperative rules – Multiplicity of non-cooperative equilibria • A way out of multiplicity: structuring the strategy space through cooperative solution concepts (e.g. Shapley value, nucleolous) and equal split • Run experiments letting behavior determine the outcome Experimental bargaining procedure • In a two step procedure an active player decides whether or not to accept another player as his agent. The final agent divides the coalition value. – If there are ties between accepted agents then a random draw decides who becomes the (final) agent. – If nobody accepts another agent then the procedure is repeated or a random stopping rule terminates the round with zero payoffs or two person coalition payoff Bargaining Procedure Start Phase I 1 Every player accepts at most one other player. 2 3 No 4 Yes Is there an eligible pair? No Stop? Yes with prob. 1/100 Random selection of an eligible pair (X,Y) 7 X and Z do or do not accept the other active player Z or X 8 Yes No 10 No Stop? Yes with prob. 1/100 9 Is (X,Z) or (Z,X) an eligible pair? Yes Yes Random selection of an eligible pair (U,V) of X and Z Two person coalition No coalition Final payoffs zero: pA= pB= pC= 0 5 X chooses final payoff division (pX, pY) of v(X,Y) pZ =0 Phase II 13 Grand Coalition 11 U chooses final payoff division (pA,pB,pC) of v(ABC) 6 12 15 End End End 14 Phase III Experimental design • In every period an agency is voted for (who divides the coalition value) • The grand coalition always has value 120. • 3 subjects per group • 10 independent groups per game • 40 periods • Maintain same player role in same group and same game • All periods are paid Characteristic function games games 1 2 3 4 5 6 7 8 9 10 v(12) 120 120 120 120 100 100 100 90 90 70 v(13) 100 100 100 100 90 90 90 70 70 50 v(23) 90 70 50 30 70 50 30 50 30 30 Game 1 - 5: no core Theoretical solutions • Non cooperative solutions – One shot game: any coalition can be an equilibrium outcome with any final agent demanding coalition value for himself In supergame any payoff division can be equilibrium division • Cooperative solution concepts – We discuss Shapley value and Nucleolous • Equal split as a good descriptive theory Average results and cooperative solution concepts Example GAME 6 Average group results (“+” = one group) 3 Equal split Shapley value 90 50 ++ Nucleolus 1 100 Game 6 2 Single group results over time Payoffs over time for all three players for each group, game 10 2 3 4 5 6 7 8 0 40 80 120 0 40 80 120 1 0 20 30 40 0 10 20 30 40 10 80 120 9 10 0 40 Game 10 V(1,2) = 70 0 10 20 30 40 0 10 20 30 40 time Payoffs 1 Graphs by Group Payoffs 2 Payoffs 3 V(1,3) = 50 V(2,3) = 30 number of observations (out of 100 groups) 35 30 25 Number of times of equal split in each group, e.g. 30% of all groups divide fairly in 36- 40 rounds => High heterogeneity 20 15 10 5 0 0-3 4-7 8-11 12-15 16-19 20-23 24-27 28-31 32-35 36-40 number of periods (out of 40) with equal split divisions Average vector of Strong player division in game 6 Average payoff vectors across all 3 periods in game 6 C Many near equal split 90 1 50 100 Game 6 90 Many near Shapl. Value, nucleolous 50 2 A 100 Game 6 B Why is there equalization of payoffs over time, given that the strong player demands on average very much for himself within a single period? Equalization through reciprocity and balancing of power through voting mechanism Payoff offers between A&B or A&C or B&C What final agents offer to each other: Rank correlation significantly positive: If you offer “high” to me I offer “high” to you and similar with “low” offers => Equalization across periods THROUGH RECIPROCITY Number of times being agent (out of 40) and own payoff demand If you demand too much for yourself, less likely to be voted as final agent Equalization across periods THROUGH balance of power Conclusion • A theoretical model to reach cooperation in three person coalition formation using – a non-cooperative model of interacting players – implement experiments • Both the Shapley value and the nucleolus (cooperative concepts) seem to give comparatively more payoff advantage to player 1 than would appear to be the implication of the average results across periods derived directly from the experiments. • Equalization of payoffs through reciprocity and balance of power.