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

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v(23)v(13)v(12)games

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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.