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, ICREA,
  BGSE)
• Axel Ockenfels (University of Cologne)
• Reinhard Selten (University of Bonn)
• Stony Brook 2013

2
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)539564)
• 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

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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

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Bargaining Procedure
Phase I
Phase II
Two person coalition
Grand Coalition
Phase III
No coalition
5
Characteristic function games
Experimental design

• 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

Game 1 - 5 no core
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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

7
Average resultsand cooperative solution
concepts
8
Example GAME 6 Average group results ( one
group)
Equal split
Shapley value

Nucleolus
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Actual average payoffs per game
games V(1,2) V(1,3) V(2,3) Actual Payoff 1 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
Strong player typically gets highest payoff
average payoff vector closest to equal split.
10
games V(1,2) V(1,3) V(2,3) Actual Payoff 1 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
5 100 90 70 41.86 38.88 37.13
6 100 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
Game 1 - 5 no core
Shapley value Shapley value Nucleolus Nucleolus Quotas Quotas Aumann-Maschler Bargaining set (min requirement) Aumann-Maschler Bargaining set (min requirement) Aumann-Maschler Bargaining set (min requirement) Selten equal Division. payoff bounds (min requirement Selten equal Division. payoff bounds (min requirement Selten equal Division. payoff bounds (min requirement
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
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Game 10 Experimental results, its agency model
solution, and these compared with other
theoretical values for the game
Game 10 V(1,2) 70 V(1,3) 50 V(2,3) 30
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) Efficiency (.95) Efficiency (.95) Efficiency (.95)
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Single group resultsover time
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Game 10 V(1,2) 70 V(1,3) 50 V(2,3) 30
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Number of times of equal split in each group,
e.g. 30 of all groups divide fairly in 36- 40
rounds gt High heterogeneity
Average vector of Strong player division in game
6
Average payoff vectors across all periods in
game 6
Many near equal split
Many near Shapl. Value, nucleolous
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• 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

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Number of times being agent (out of 40) and own
payoff demand
Payoff offers between AB or AC or BC
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 gt Equalization across periods
THROUGH RECIPROCITY

• If you demand too much for
• yourself, less likely to be voted as
• final agent
• Equalization across periods
• THROUGH balance of power

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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.