Ethical Norms Realizing Pareto-Efficiency in Two-Person Interactions:

1
Ethical Norms Realizing Pareto-Efficiency in
Two-Person Interactions
3rd-Joint-Conference (2005) June at Sapporo
Game Theoretic Analysis with Social Motives

• Masayoshi MUTOTokyo Institute of Technology

2
?1 INTRODUCTION 2 OR-UTILITY FUNCTION 3
GAME-TRANSFORMATION 4 CONCLUSIONS
3
Motivation of Research

• In everyday life, people interact TAKING EACH
  OTHER INTO ACCOUNT
• But we have few such theories in Game Theory

I take Anns payoff into account.
I take Bobs payoff into account.
Ann
Bob
4
Overview

• QUESTION
• How should we take others into account to
  realize PARETO-EFFICIENCY?
• ANSWER
• In two-person interactions we should be
  ALTRUISTIC and IMPARTIAL

5
Pareto Efficiency
Pareto-Efficientunanimously better
4, 4 1, 5
5, 1 2, 2
Pareto-Inefficientunanimously worse
6
Existing Research

• Other-Regarding Utility Function
  (OR-Utility Function) for explaining experiments
  data of few games
• Prisoners Dilemma, Ultimatum Game...
• But we dont know what game is played in
  daily-life
• ?
• General Theory about Ways of Other-Regarding in
  Many Situations

7
Scope Conditions

• Situations Any TWO-person games
• Both players share AN Other-Regarding Utility
  Function
• ex. altruism, egalitarianism, competition

8
1 INTRODUCTION?2 OR-UTILITY FUNCTION 3
GAME-TRANSFORMATION 4 CONCLUSION
9
Other-Regarding Utility Function 1
v(x y) (1-p)x py
MacClintock 1972

• x my payoff
• y the others payoff
• p my WEIGHT for the other
• v my subjective payoff
• But NOT expressing EGALITARIANISM !

10
Other-Regarding Utility Function 2
SchulzMay 1989, FehrSchmidt 1999

• p if my payoff is BETTER than the others
• q if my payoff is WORSE than the others

-8ltplt8, -8ltqlt8
11
Egalitarianism(p-q ) is large
12
Family ofOR-Utility Functions
q
ANTI-EGL.
SACRIFICE
pq 1
ALTRUISM
MAXMAX
altruistic
JOINT
egalitarian
p q
EGOISM
MAXMIN
p
COMPETITION
EGALITARIANISM
13
1 INTRODUCTION 2 OR-UTILITY FUNCTION ?3
GAME-TRANSFORMATION 4 CONCLUSION
14
Payoff Transform
row-players subjective payoff
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
15
Payoff Transform
row-players subjective payoff
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
calculate
subj. C D
C 1, 1 6q, 6-6p
D 6-6p, 6q 2, 2
16
Payoff Transform
row-players subjective payoff
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
for both players
subj. C D
C 1, 1 6q, 6-6p
D 6-6p, 6q 2, 2
17
Payoff Transform
row-players subjective payoff
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
p 1, q 0
MAXMIN
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
ex.
18
Payoff Transform by Some OR-Utility Functions
q
0.5
MAXMIN (1, 0)
1, 1 0, 6
6, 0 2, 2
1, 1 0, 0
0, 0 2, 2
p
example
0.5
1
19
Payoff Transform by Some OR-Utility Functions
q
ALTRUISM (1, 1)
1, 1 6, 0
0, 6 2, 2
0.5
MAXMIN (1, 0)
1, 1 0, 6
6, 0 2, 2
1, 1 0, 0
0, 0 2, 2
p
example
0.5
1
20
Problem in ALTRUISM
p 1,q 1
The Gift of the Magi The Gift of the Magi The Gift of the Magi
Della\Jim present not
present 1, 1 0, 6
not 6, 0 2, 2
subjective subjective subjective
Della\Jim present not
present 1, 1 6, 0
not 0, 6 2, 2
INEFFICIENT!
21
Problem in EGALITARIANISM
p?8,q?-8
Leader Game Leader Game Leader Game
follow lead
follow 3, 3 4, 7
lead 7, 4 1, 1
subjective subjective subjective
follow lead
follow 0, 0 -2, -2
lead -2, -2 0, 0
22
Theorem

• WAYS 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?0
• and
• IMPARTIAL p q 1

23
IMPARTIAL Ways
q
anti-egl
sacrifice
IMPARTIAL pq 1
maxmax
altruism
joint
maxmin
egoism
p
egalitarian
competition
24
ALTRUISTIC and IMPARTIALWays
q
ALTRUISTICp, q?0
anti-egl
MAXMAX
altruism
includingmixture
JOINT
MAXMIN
egoism
p
egalitarian
competition
25
ALTRUISTIC and IMPARTIALWaysPayoff Transform
example
q
MAXMAX
1, 1 6, 6
6, 6 2, 2
JOINT
1, 1 3, 3
3, 3 2, 2
JOINT
MAXMIN
1, 1 0, 6
6, 0 2, 2
1, 1 0, 0
0, 0 2, 2
egoism
p
Objective LV
26
1 INTRODUCTION 2 OR-UTILITY FUNCTION 3
GAME-TRANSFORMATION?4 CONCLUSION
27
Implication 1

• p 0.5, q 0.3 appears to be good for Pareto
  efficiency
• If my payoff is better than the others,
  regard equallyIf my payoff is worse than the
  others, regard a little
• But not impartial (pq 0.8lt1)
• ? Theorem requires a strict ethic

28
Implication 1

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

29
Implication 2

• Extreme-Egalitarianism isnt good

weight for difference of payoffs (e ) ?
weight for sum of payoffs (1/2)
? e 1/2 means MAXMIN
30
Implication 2
? MAXMIN is the Maximum Egalitarianism with
Pareto-Efficiency in any two-person games
31
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.

32
Bibliography

• Shulz, U and T. May. 1989. The Recording of
  Social Orientations with Ranking and Pair
  Comparison Procedures. European Journal of
  Social Psychology 1941-59
• MacClintock, C. G. 1972. Social Motivation A
  set of propositions. Behavioral Science
  17438-454.
• Fehr, E. and K. M. Schmidt. 1999. A Theory of
  Fairness, Competition, and Cooperation.
  Quarterly Journal of Economics 114(3)817-868.

33
Defection through Egoism
p 0,q 0
Prisoners Dilemma Prisoners Dilemma Prisoners Dilemma
stay silent confess
stay silent 4, 4 0, 6
confess 6, 0 2, 2

• In Prisoners Dilemma, Egoism causes Pareto
  non-efficiency.

34
Mathematical Expression of Theorem

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

equilibrium action profiles in subjective level
of game g
two-person finite game including mnASYMMETRIC
game
efficient action profiles in objective level of
game g
existing
v ?g Eff(g)nNE(vg)?f v p q 1, p?0, q?0