ethical norms realizing pareto-efficiency in two-person interactions:
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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
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