1. 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

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

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

4. 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

5. 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

6. 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

7. 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

8. 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)

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

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

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

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

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

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

15. 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)

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