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

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

• 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

Actual average payoffs per game

games V(1,2) V(1,3) V(2,3)ActualPayoff 1

ActualPayoff 2

ActualPayoff 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

Shapley value Nucleolus Quotas

Aumann-MaschlerBargaining set(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

games V(1,2) V(1,3) V(2,3)ActualPayoff 1

ActualPayoff 2

ActualPayoff 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

Game 10: Experimental results, its agency model solution, and these compared with other theoretical values for the game

Game 10 V(1,2) = 70V(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)

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

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

Phase 1 Phase 2

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

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

.3.4

.5.6

.7.8

Mea

n co

unt

1 2 3 4 5 6 7 8 9 10Game

Meanequal UpperMeanequal/LowerMeanequal

Equal split

Relative frequency of equal split, pooled over all periodsper game

002

02

04

04

06

06

000

20

20

40

40

60

60

002

02

04

04

06

06

0

-2-2 00 22 44 -2-2 00 22 44

-2-2 00 22 44 -2-2 00 22 44

11 22 33 44

55 66 77 88

99 1010

x-axis: (-1=no coalition, hardly ever) (1=player 1 representative, 2= player 2 representative, 3=player 3 representative)

Relative frequency of representative in each game

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 implic-ation of the results derived directly from the experiments.