The Agencies Method for Coalition Formation in Experimental Games John Nash (University of Princeton, USA) Rosemarie Nagel (Universitat Pompeu Fabra, Spain) Axel Ockenfels (University of Köln, Germany) Reinhard Selten (University of Bonn, Germany) Stony Brook Experimental Economics Workshop July 2007

Motivation Comparison of some solution concepts with actual behavior Shapley Value Nucleolus Agency model simulations Bargaining set Equal division payoff bounds …

Bargaining Procedure Phase I Phase II No coalition Two person coalition Grand Coalition Phase III

Experimental design Characteristic function games 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 Game 1 - 4: no core

Actual average payoffs per game games V(1,2) V(1,3) V(2,3) Actual Payoff 1 Payoff 2 Payoff 3 Efficiency 1 120 100 90 43.69 36.15 37.9 .98 2 70 44.28 41.95 31.42 3 50 45.42 37.94 30.72 .95 4 30 44.46 35.88 32.99 .94 5 41.86 38.88 37.13 6 42.01 41.99 31.90 .97 7 37.95 39.33 40.03 8 40.51 37.65 38.02 9 39.75 38.40 36.67 .96 10 40.84 37.69 35.72

games V(1,2) V(1,3) V(2,3) Actual Payoff 1 Payoff 2 Payoff 3 1 120 100 90 43.69 36.15 37.9 2 70 44.28 41.95 31.42 3 50 45.42 37.94 30.72 4 30 44.46 35.88 32.99 5 41.86 38.88 37.13 6 42.01 41.99 31.90 7 37.95 39.33 40.03 8 40.51 37.65 38.02 9 39.75 38.40 36.67 10 40.84 37.69 35.72 shapley value Quotas Nucleolus Game 1 2 3 46.67 41.67 31.67 65 55 35 53.33 43.33 23.33 38.33 28.33 75 45 25 66.67 36.67 16.67 60 85 15 80 30 10 4 21.67 95 5 93.33 3.33 48.33 33.33 56.67 26.67 6 70 20 7 61.67 83.33 13.33 8 50 40 9 72.50 32.50 15.00 57.50 37.50 25.00

Aumann-Maschler (min requirement) games V(1,2) V(1,3) V(2,3) Actual Payoff 1 Payoff 2 Payoff 3 1 120 100 90 43.69 36.15 37.9 2 70 44.28 41.95 31.42 3 50 45.42 37.94 30.72 4 30 44.46 35.88 32.99 5 41.86 38.88 37.13 6 42.01 41.99 31.90 7 37.95 39.33 40.03 8 40.51 37.65 38.02 9 39.75 38.40 36.67 10 40.84 37.69 35.72 shapley value Nucleolous Quotas Aumann-Maschler (min requirement) Selten: equal Division. payoff bounds (min requirement game 1 2 3 46.67 41.67 31.67 53.33 43.33 23.33 65 55 35 47.50 37.50 17.50 60 45 15 38.33 28.33 66.67 36.67 16.67 75 25 62.50 32.50 12.50 10 80 30 85 78 28 8 4 21.67 93.33 3.33 95 5 92.50 22.50 2.50 48.33 33.33 56.67 26.67 40 50 20 6 70 6.67 7 61.67 83.33 13.33 9 72.50 15.00 57.50 25.00

Games 9 and 10, its solutions and actual data V(1,2) = 90 V(1,3) = 70 V(2,3) = 30 Player 1 2 3 Actual payoffs 39.75 38.40 36.67 Agency method 43.81 40.51 34.08 Shapley value 56.67 26.67 Nucleolus 72.50 32.50 15.00 Quotas 65 25 5 Aumann Maschler Selten 45 16.67 10 Efficiency (.96) Game 10 V(1,2) = 70 V(1,3) = 50 V(2,3) = 30 Player 1 2 3 Actual payoffs 40.84 37.69 35.72 Agency method 40.71 39.73 37.52 Shapley value 50 40 30 Nucleolus 57.5 37.50 25.00 Quotas 45 25 5 Aumann Maschler Selten 23.33 16.67 Efficiency (.95)

Game 9 V(1,2) = 90 V(1,3) = 70 V(2,3) = 30

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

Game 9, combinations of payoffs of player 1, player 2 1 Payoff player 2 Payoff Player 1

Game 10, combinations of payoffs of player 1, player 2 Payoff player 2 Payoff Player 1

Phase 1 Phase 2 Relative frequencies of random rule in phase 1 and phase 2, per game

Relative frequency of equal split, pooled over all periods per game

Relative frequency of representative in each game x-axis: (-1=no coalition, hardly ever) (1=player 1 representative, 2= player 2 representative, 3=player 3 representative)

Some thoughts Combination: Innovative theoretic model to approach three person coalition formation, simulation, experiment