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 Two person coalitionGrand Coalition Phase I Phase II Phase III No coalition

Game 1 - 5: no core Characteristic function games In every period an agency is voted for (who divides the coalition value) The grand coalition always has value subjects per group 10 independent groups per game 40 periods Maintain same player role in same group and same game All periods are paid Experimental design

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) Equal split Shapley value Nucleolus ++

Actual average payoffs per game games V(1,2)V(1,3) V(2,3) Actual Payoff 1 Actual Payoff 2 Actual Payoff 3Efficiency Strong player typically gets highest payoff; average payoff vector closest to equal split.

Shapley valueNucleolusQuotas Aumann-Maschler Bargaining set (min requirement) Selten: equal Division. payoff bounds (min requirement game games V(1,2)V(1,3) V(2,3) Actual Payoff 1 Actual Payoff 2 Actual Payoff Game 1 - 5: no core

Game 10: Experimental results, its agency model solution, and these compared with other theoretical values for the game Game 10V(1,2) = 70 V(1,3) = 50 V(2,3) = 30 Player123 Experimental results Agency method Shapley value Nucleolus Quotas45255 Aumann Maschler45255 Selten Efficiency (.95)

Single group results over time

Game 10 V(1,2) = 70 V(1,3) = 50 V(2,3) = 30

Average payoff vectors across all periods in game 6 Average vector of Strong player division in game 6 Number of times of equal split in each group, e.g. 30% of all groups divide fairly in rounds => High heterogeneity Many near equal split Many near Shapl. Value, nucleolous

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

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 Payoff offers between A&B or A&C or B&C 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.