The Agencies Method for Coalition Formation in Experimental Games

1. The Agencies Method for Coalition Formation in Experimental Games John Nash (University of Princeton) Rosemarie Nagel (Universitat Pompeu Fabra) Axel Ockenfels (University of Cologne) Reinhard Selten (University of Bonn) LEEX-UPF-COLOGNE Workshop Experimental Economics across the Fields Nov. 2007

2. Introduction • Some results from an experiment, based on 3-person games defined by characteristic function descriptions, in which coalition formation and cooperation must be achieved through actions and of surrender and acceptance by the individual players • This can be called the “method of agencies” (Nash 1996) • Comparison of some solution concepts with actual human behavior • Shapley Value (Shapley, 1953) • Nucleolus (Schmeidler, 1969) • Bargaining set (Aumann, Maschler 1968) • Equal division payoff bounds (Selten, 1984) • Agency model simulations (Nash, 2002)

3. Characteristic function of our partially symmetric 3 person games v(1)=v(2)=v(3)=0, v(1,3) = v(2,3) = bz = 0, v(1,2) = b3, v(1,2,3) = 1, v(i, j) is the value of the coalition of players i and j The imbalance of the payoffs to the different players resulting from the calculations based on our agency model can be well measured by comparing p1+p2 with 2*p3 because in the calculations that were made for the graphs the games were such that players P1 and P2 were symmetrically situated.

4. Agency method model results compared with Shapley value and nucleolus for games with v(1,3) = v(2,3) = bz = 0, v(1,2) = b3 (see x-axis) b3

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

6. games games v(12) v(12) v(13) v(13) v(23) v(23) 1 1 120 120 100 100 90 90 2 2 120 120 100 100 70 70 3 3 120 120 100 100 50 50 4 4 120 120 100 100 30 30 5 5 100 100 90 90 70 70 6 6 100 100 90 90 50 50 7 7 100 100 90 90 30 30 8 8 90 90 70 70 50 50 9 9 90 90 70 70 30 30 10 10 70 70 50 50 30 30 Characteristic function games Experimental design • 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 - 5: no core

7. Actual average payoffs per game

8. Game 1 - 5: no core

9. Game 10: Experimental results, its agency model solution, and these compared with other theoretical values for the game

11. games v(12) v(12) v(13) v(13) v(23) v(23) 1 1 120 120 100 100 90 90 2 2 120 120 100 100 70 70 3 3 120 120 100 100 50 50 4 4 120 120 100 100 30 30 5 5 100 100 90 90 70 70 6 6 100 100 90 90 50 50 7 7 100 100 90 90 30 30 8 8 90 90 70 70 50 50 9 9 90 90 70 70 30 30 10 10 70 70 50 50 30 30 Phase 1 Phase 2 Relative frequencies of random rule in phase 1 and phase 2, per game

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

13. 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)

14. Conclusion • A theoretical model to approach three person coalition formation: • a model of interacting players • relating experiments • Both the Shapley value and the nucleolus seem to give comparatively more payoff advantage to player 1 than would appear to be the implic-ation of the results derived directly from the experiments.