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
Start 1
Every player accepts at most one other player.
2 Is there an eligible pair?
3
Stop? Yes with prob.
1/100
4
No No
Yes
Random selection of an eligible pair (X,Y)
7
Yes
X and Z do or do not accept the other active player Z or X
8
Is (X,Z) or (Z,X) an
eligible pair?
9 Stop? Yes with prob.
1/100
10
No
X chooses final payoff division (pX, pY) of v(X,Y) pZ =0
11 Final payoffs zero: pA= pB= pC= 0
5
End 6
Yes
End 12
Random selection of an eligible pair (U,V) of X and Z
13
U chooses final payoff division (pA,pB,pC) of v(ABC)
14
No
End 15
Yes
Bargaining Procedure
Two person coalition Grand Coalition
Phase I
Phase II
Phase IIINo 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 120.
• 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
Experimental design
30507010
3070909
5070908
30901007
50901006
70901005
301001204
501001203
701001202
901001201
v(23)v(13)v(12)games
30507010
3070909
5070908
30901007
50901006
70901005
301001204
501001203
701001202
901001201
v(23)v(13)v(12)games
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 resultsand
cooperative solution concepts
1 2
3
100
90 50
Game 6
Example GAME 6 Average group results (“+” = one group)
Equal split
Shapley value
Nucleolus
++
Single group resultsover time
040
8012
00
4080
120
040
8012
0
0 10 20 30 40 0 10 20 30 40
0 10 20 30 40 0 10 20 30 40
1 2 3 4
5 6 7 8
9 10
Payoffs 1 Payoffs 2 Payoffs 3
payo
ffs p
laye
rs 1
, 2, 3
time
Graphs by Group
Payoffs over time for all three players for each group, game 10
Game 10V(1,2) = 70
V(1,3) = 50
V(2,3) = 30
0
5
10
15
20
25
30
35
0-3 4-7 8-11 12-15 16-19 20-23 24-27 28-31 32-35 36-40
number of periods (out of 40) with equal split divisions
nu
mb
er
of
ob
serv
ati
on
s
(o
ut
of
100 g
rou
ps)
1 2
3
100
90 50
Game 6A B
C
100
90 50
Game 6
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 36- 40 rounds=> High heterogeneity
Many near equal splitMany 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 periodsTHROUGH 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.