Ethical Norms Realizing Pareto-Efficiency

in Two-Person Interactions:

Masayoshi MUTOTokyo Institute of Technology

3rd-Joint-Conference (2005) June at Sapporo

Game Theoretic Analysis with Social Motives

1 INTRODUCTION1 INTRODUCTION2 OR-UTILITY FUNCTION3 GAME-TRANSFORMATION4 CONCLUSIONS

Motivation of Research In everyday life, people interact

TAKING EACH OTHER INTO ACCOUNT But we have few such theories in

Game Theory

Ann Bob

I take Bob’s

payoff into account.

I take Ann’s payoff into account.

Overview QUESTIONHow should we take others into

account to realize PARETO-EFFICIENCY?

ANSWERIn two-person interactions we should be ALTRUISTIC and IMPARTIAL

＊ Pareto Efficiency

4, 4 1, 55, 1 2, 2

Pareto-Efficientunanimously better

Pareto-Inefficientunanimously worse

Existing Research “Other-Regarding Utility Function”

（ =OR-Utility Function ） for explaining experiments data of few games Prisoners’ Dilemma, Ultimatum Game...

But we don’t know what game is played in daily-life

↓General Theory about Ways of Other-Regarding in Many Situations

Scope Conditions

Situations: Any TWO-person games

Both players share AN Other-Regarding Utility Function ex. altruism, egalitarianism, competition

1 INTRODUCTION2 OR-UTILITY FUNCTION2 OR-UTILITY FUNCTION

3 GAME-TRANSFORMATION4 CONCLUSION

Other-Regarding Utility Function 1

x my payoff y the other’s payoff p my WEIGHT for the other v my subjective payoff

But NOT expressing EGALITARIANISM !

MacClintock 1972

v(x ; y) = (1 － p)x + py　　

objective

subjective

Other-Regarding Utility Function 2

p if my payoff is BETTER than the other’s q if my payoff is WORSE than the other’s

)()1(

)()1();(

yxqyxq

yxpyxpyxv

Schulz&May 1989, Fehr&Schmidt 1999

－∞＜ p ＜ +∞, －∞＜ q ＜+∞

＊ Egalitarianism ： (p－ q ) is large

Large if :0)(2

22)

21(

)()1(

)()1();(

qpyxqp

yxqp

yqp

xqp

yxqyxq

yxpyxpyxv

≒

p>0, q<0 is sufficient for weak Egalitarianism

much heavier → 　Egalitarianism

0.5

1

∞

∞1　0.5-∞　 O

-∞

EGOISM

ALTRUISM

p

q

COMPETITION

JOINT

EGALITARIANISM

MAXMIN

MAXMAX

SACRIFICEANTI-EGL.

Family ofOR-Utility Functions

)()1(

)()1();(

yxqyxq

yxpyxpyxv

p＋ q = 1

p = q

altru

istic

egalitarian

1 INTRODUCTION2 OR-UTILITY FUNCTION

3 GAME-3 GAME-TRANSFORMATION TRANSFORMATION

4 CONCLUSION

Payoff Transform

obj. C D

C 1, 1 0, 6

D 6, 0 2, 2

subj. C D

C (1-p)+p 0(1-q)+6q

D 6(1-p)+0p

2(1-p)+2p

row-player’s subjective payoff

)()1(

)()1();(

yxqyxq

yxpyxpyxv

Payoff Transform

subj. C D

C 1, 1 6q, 6 －6p

D 6 － 6p, 6q

2, 2

calculate

obj. C D

C 1, 1 0, 6

D 6, 0 2, 2

subj. C D

C (1-p)+p 0(1-q)+6q

D 6(1-p)+0p

2(1-p)+2p

row-player’s subjective payoff

Payoff Transform

subj. C D

C 1, 1 6q, 6 －6p

D 6 － 6p, 6q

2, 2

for both players

obj. C D

C 1, 1 0, 6

D 6, 0 2, 2

subj. C D

C (1-p)+p 0(1-q)+6q

D 6(1-p)+0p

2(1-p)+2p

row-player’s subjective payoff

Payoff Transform

subj. C D

C 1, 1 6q, 6 －6p

D 6 － 6p, 6q

2, 2

subj. C D

C 1, 1 0, 0

D 0, 0 2, 2

p =1, q =0 :MAXMIN

obj. C D

C 1, 1 0, 6

D 6, 0 2, 2

subj. C D

C (1-p)+p 0(1-q)+6q

D 6(1-p)+0p

2(1-p)+2p

row-player’s subjective payoff

ex.

1

1　　 Op

)()1(

)()1();(

yxqyxq

yxpyxpyxv

1, 1 0, 6

6, 0 2, 2

1, 1 0, 0

0, 0 2, 2

MAXMIN (1, 0)MAXMIN (1, 0)

example

Payoff Transform by Some OR-Utility

Functionsq

0.5

0.5 1

1

1　　 Op

)()1(

)()1();(

yxqyxq

yxpyxpyxv

1, 1 0, 6

6, 0 2, 2

1, 1 0, 0

0, 0 2, 2

MAXMIN (1, 0)MAXMIN (1, 0)

1, 1 6, 0

0, 6 2, 2

ALTRUISM (1, 1)ALTRUISM (1, 1)

example

Payoff Transform by Some OR-Utility

Functionsq

0.5

0.5 1

subjectiveDel ｌ a ＼

Jimpresen

tnot

present 1, 1 6, 0not 0, 6 2, 2

Problem in ALTRUISM

“The Gift of the Magi”Della ＼

Jim present not

present 1, 1 0, 6not 6, 0 2, 2

p =1 ， q =1

INEFFICIENT!

subjective

follow lead

follow 0, 0 -2, -2lead -2, -2 0, 0

Problem in EGALITARIANISM

“Leader Game”

follow lead

follow 3, 3 4, 7lead 7, 4 1, 1

p→∞ ， q→ －∞

INEFFICIENT!

INEFFICIENT!

TheoremWAYS of Other-

Regarding existing Social States which are Pareto EFFICIENT

in objective level and 　　

Pure Nash EQUILIBRIAin subjective level

for any two-person games

ALTRUISTICp,q≧0and 　　

IMPARTIAL p +q =1

＝

)()1(

)()1(

);(

yxqyxq

yxpyxp

yxv

0.5

1

∞

∞1　0.5-∞　 O

-∞

egoism

altruism

p

q

competition

joint

egalitarian

maxmin

maxmax

sacrificeanti-egl

IMPARTIAL Ways

IMPARTIAL

p＋ q = 1

0.5

1

∞

∞1　0.5-∞　 O

-∞

egoism

altruism

p

q

competition

JOINT

egalitarian

MAXMIN

MAXMAX

anti-egl

ALTRUISTIC and

IMPARTIALWays

ALTRUISTIC

p, q≧0

ALTRUISTIC

p, q≧0

including

mixture

0.5

1

1　0.5　 Oegoism p

q

JOINT

ALTRUISTIC and

IMPARTIALWays:Payoff

Transform example

1, 1 0, 6

6, 0 2, 2

Objective LV

1, 1 0, 0

0, 0 2, 2

MAXMIN

1, 1 6, 6

6, 6 2, 2

MAXMAX

1, 1 3, 3

3, 3 2, 2

JOINT

)()1(

)()1();(

yxqyxq

yxpyxpyxv

1 INTRODUCTION2 OR-UTILITY FUNCTION 3 GAME-TRANSFORMATION

4 CONCLUSION4 CONCLUSION

Implication 1

p = 0.5, q =0.3 　 appears to be good for Pareto efficiency

： If my payoff is better than the other’s, regard equallyIf my payoff is worse than the other’s, regard a little

But not impartial (p+q = 0.8 ＜ 1)

→ Theorem requires a strict ethic

Implication 1

→ Only altruistic and impartial ways of other regardingcan realize Pareto efficiency in ANY two-person games

Implication 2 Extreme-Egalitarianism isn’t good

21

with)(21

1,0,satisfy

≦

≧

eyxyxev

qpqpv

　　weight for difference of payoffs (|e| ) ≦ weight for sum of payoffs (1/2) ⇒ e =1/2 means “MAXMIN”

Implication 2

↓MAXMIN is the “Maximum Egalitarianism with Pareto-Efficiency” in any two-person games

Summary Altruistic and Impartial

Ways of Other-Regarding(that is “from Maxmin to Maxmax”)are justified as the only ways realizing Pareto Efficiencyin any two-person interactions.

Bibliography Shulz, U and T. May. 1989. “The Recording o

f Social Orientations with Ranking and Pair Comparison Procedures.” European Journal of Social Psychology 19:41-59

MacClintock, C. G. 1972. “Social Motivation: A set of propositions.” Behavioral Science 　17:438-454.

Fehr, E. and K. M. Schmidt. 1999. “A Theory of Fairness, Competition, and Cooperation.” Quarterly Journal of Economics 114(3):817-868.

Defection through Egoism

In “Prisoners’ Dilemma”, Egoism causes Pareto non-efficiency.

“Prisoners’ Dilemma”stay silent confess

stay silent 4, 4 0, 6confess 6, 0 2, 2

p =0 ， q =0

＊ Mathematical Expression of Theorem

The following v expresses possible “ways of other-regarding” to realize Pareto-Efficiency in any two-person interaction.

{v | ∀g 　 Eff(g)∩NE(vg)≠φ}= {v | p +q =1, p≧0, q≧0 }

two-person finite game including m×nASYMMETRIC game

efficient action profiles in objective level of game g

equilibrium action profiles in subjective level of game g existing

)()1(

)()1();(

yxqyxq

yxQyxQyxv