Evolutionary Ultimatum Game with Responder's Dissatisfaction

Yougui WangDepartment of Systems Science, School of Management,

Beijing Normal University, Beijing, P. R. China

2008-06-30

Courses

Introduction Standard Game Game with Responder’s Dissatisfaction Discussion and Conclusion

The Ultimatum Game Two people divide a pie or a sum of money:

total sum: m

proposer’s offer: p

responder: accept/reject

proposer responder

accept

reject

(m-p, p)

(0,0)

p

Game Theory:

Responder accept any offer p>0

Proposer offer the smallest share ε

Sub-game perfect equilibrium: (ε, accept)

Experimental Results:

Proposers’ offer : 40%-50% of the “pie”.

When offer was less than 30%, half of the responders reject it.

Explanation of Experimental Results Utility functions

: monetary payoff : other aspects players care about (fairness, envy, altruism, etc) Weakness of this explanation Bounded rationality Ignorance of learning effect Individual purpose

))(,( 1 nxxfUU

ix

Evolutionary Ultimatum Game Repeated games Players interact with each other more than once.

Adaptive Strategy

Players change their strategies

Successful strategy spread

Collective Dynamics

Replicator equation

Evolutionary Ultimatum Game (Nowak, 2000; Page, 2000, 2002, Hardling, 2007).

1. Roles of the players were decided randomly

2. Only monetary payoff was considered

Our work

1.The role of player is fixed. (Heterogeneous)

2. Responder’s dissatisfaction is taken into account

The standard game

The reward of money is standardized to 1 H=“high offer” h, L=“low offer” l. (h>l>0) A=“accept”, R=“reject”.

proposer

H

L

responder

responder

(1-h, h)

A

R

(1-l, l)

(0, 0)

A

Proportion of players with each strategy

Responder Proposer

“accept”: x high offer H: y

“reject”: 1-x low offer L: 1-y

Unsymmetrical replicator dynamics

)( rrxx

)( ppyy

Players change their strategies according to the payoffs of their “parents” or how much they received last time.

Equilibrium of Standard game Differential equations

)]1()1)[(1(

)1)(1(

lxhyyy

lyxxx

The evolutionary stable equilibrium:

Meaning: All proposers willing to offer the low share l; all responders accept it.

The low offer is chosen freely below the high one, so the equilibrium will lead to sub-game perfect: (ε, accept)

)0,1( yx

Responder’s Dissatisfaction When offered l, responder was dissatisfied

proposer

H

L

responder

responder

(1-h, h)

A

R

(1-l, l-c)

(0, 0)

A

The dissatisfaction c will be incorporated into the responder’s choice when she chooses strategies and is assumed the same among all responders.

Analyzing the differential equations

The case of

The evolutionary stable equilibrium:

Meaning: All proposers will offer the high share h and all the responders plan to choose “reject” if they receive the low offer l

l is not enough to compensate dissatisfaction

)]1()1)[(1(

)1)(1(

lxhyyy

xlcxxx

lc )1,0( yx

The case of

The planar steady state solutions:

The line and respectively orientate the evolution of y and x.

Define

lc

)]1()1)[(1(

)1)(1(

lxhyyy

xlcxxx

)0,0( yx )0,/)(( ylclx ]1,0[1 xy

)1/()1( lhx lclx /)(

lcllhk /)()1/()1(

(1) k=0

x

1

y

10L

H

L

CLx

1

1

“origin”

indifferent equilibrium

No Stable Equilibrium

(2) k>0

x

1

y

10L

Hx

1

1

“origin”

“saddle”

L

CLx

indifferent equilibrium

Evolutionary stable Equilibrium: )1,(L

CL

(3) k<0

x

1

y

10L

Hx

1

1

“origin”

“convergence”

L

CLx

indifferent equilibrium

Evolutionary stable Equilibrium: )0,(L

CL

Discussion

Dissatisfaction parameter c is assumed as a constant, high offer is normalized to 0.5

Define

Calculate the identification term

clchllkllf )()1()( 2

)())(( ksignlfsign

4/134)2/1( 22 cccc

Function with different levels of C)(lf

Conlusions The proposer may be more selfish when the

dissatisfaction is small, but the strategy of too small offer can not prevail. Responder will reject the unfair offer with a remarkable proportion.

When the dissatisfaction is large enough, proposer will be afraid of rejection for bringing such feeling to responder.

People would maintain their rights and interests

well if they had strong feelings of unfairness.

Thanks!