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
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)
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.
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
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= p B = p C = 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
•
•
•
•
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
Actual average payoffs per game
Actual
V(1,3) V(2,3) Payoff 1
games V(1,2)
Actual
Payoff 2
Actual
Payoff 3
Efficiency
1
120
100
90
43.69
36.15
37.9
.98
2
120
100
70
44.28
41.95
31.42
.98
3
120
100
50
45.42
37.94
30.72
.95
4
120
100
30
44.46
35.88
32.99
.94
5
100
90
70
41.86
38.88
37.13
.98
6
100
90
50
42.01
41.99
31.90
.97
7
100
90
30
37.95
39.33
40.03
.98
8
90
70
50
40.51
37.65
38.02
.97
9
90
70
30
39.75
38.40
36.67
.96
10
70
50
30
40.84
37.69
35.72
.95
games
V(1,2)
Actual
Payoff 1
V(2,3)
V(1,3)
Actual
Payoff 2
Actual
Payoff 3
1
120
100
90
43.69
36.15
37.9
2
120
100
70
44.28
41.95
31.42
3
120
100
50
45.42
37.94
30.72
4
120
100
30
44.46
35.88
32.99
5
6
100
90
70
41.86
38.88
37.13
100
90
50
42.01
41.99
31.90
7
100
90
30
37.95
39.33
40.03
8
90
70
50
40.51
37.65
38.02
9
90
70
30
39.75
38.40
36.67
10
70
50
30
40.84
37.69
35.72
Shapley
value
Nucleolus
Selten: equal
Division. payoff
bounds (min
requirement
Aumann-Maschler
Bargaining set
(min requirement)
Quotas
Game 1 - 5:
no core
game
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
46.67
41.67
31.67
53.33
43.33
23.33
65
55
35
47.50
37.50
17.50
60
45
15
2
53.33
38.33
28.33
66.67
36.67
16.67
75
45
25
62.50
32.50
12.50
60
35
10
3
60
35
25
80
30
10
85
35
15
78
28
8
60
25
0
4
66.67
31.67
21.67
93.33
23.33
3.33
95
25
5
92.50
22.50
2.50
60
15
0
5
48.33
38.33
33.33
56.67
36.67
26.67
60
40
30
55
35
25
50
35
20
6
55
35
30
70
30
20
70
30
20
70
30
20
50
25
6.67
7
61.67
31.67
26.67
83.33
23.33
13.33
80
20
10
80
20
10
50
15
6.67
8
50
40
30
60
40
20
55
35
15
55
35
15
45
25
10
9
56.67
36.67
26.67
72.50
32.50
15.00
65
25
5
65
25
5
45
16.67
10
10
50
40
30
57.50
37.50
25.00
45
25
5
45
25
5
40
23.33
16.67
Game 10: Experimental results, its agency model solution, and
these compared with other theoretical values for the game
V(2,3) =
30
Game 10
V(1,2) = 70
V(1,3) =
50
Player
1
2
3
Experimental results
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
45
25
5
Selten
40
23.33
16.67
Efficiency (.95)
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
Phase 1
Phase 2
games
1
2
3
4
v(12)
120
120
120
120
v(13)
100
100
100
100
v(23)
90
70
50
30
5
6
7
8
9
10
100
100
100
90
90
70
90
90
90
70
70
50
70
50
30
50
30
30
Relative frequencies of random rule in phase 1 and phase 2, per game
.6
.5
.4
.3
Mean count
.7
.8
Equal split
1
2
3
4
5
6
7
8
9
10
Game
Meanequal
UpperMeanequal/LowerMeanequal
Relative frequency of equal split, pooled over all periods
per game
Relative frequency of representative in each game
2
3
4
5
6
7
8
0
20
40
60
0
20
40
60
1
-2
2
4
-2
0
2
4
10
0
20
40
60
9
0
-2
0
2
4
-2
0
2
4
x-axis: (-1=no coalition, hardly ever) (1=player 1 representative,
2= player 2 representative, 3=player 3 representative)
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 implication of the results derived directly from the
experiments.