1. Ethical Norms Realizing Pareto-Efficiency in Two-Person Interactions:
3rd-Joint-Conference (2005) June at Sapporo
Ethical Norms Realizing Pareto-Efficiency in Two-Person Interactions:
Game Theoretic Analysis with Social Motives
Good morning everyone. I’m Masayoshi MUTO, a graduate student of Tokyo Institute of Technology.
I study “Ways of Caring about Others in Interactions” by using Game theory.
In this presentation I report a result of my research.
In this title” Ethical Norms” means “Ways of Caring about Others“.
I show What “Ways of Caring about Others“ realize Pareto-Efficiency.
Masayoshi MUTO Tokyo Institute of Technology

2. 1 INTRODUCTION. 2 OR-UTILITY FUNCTION. 3 GAME-TRANSFORMATION
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 Ann’s payoff into account.
I take Bob’s 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-Efficient unanimously better
4, 4
1, 5
5, 1
2, 2
Pareto-Inefficient unanimously worse
to make sure

6. 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
Social Psychologists and Economists introduced “Other-Regarding Utility Function” (OR-Utility Function) for explaining game-experiments data of few games. Prisoners’ Dilemmas, Ultimatum Game, Coordination Games...
But we don’t know what game is played in daily-life.
We have to need theory not depended particular game.
So I make a 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
1 INTRODUCTION 2 OR-UTILITY FUNCTION 3 GAME-TRANSFORMATION 4 CONCLUSION
Let me go to the formulation of Other-Regarding utility function.

9. Other-Regarding Utility Function 1
v(x ; y) = (1－p)x + py　　
MacClintock 1972
x my payoff
y the other’s payoff
p my WEIGHT for the other
v my subjective payoff
But NOT expressing EGALITARIANISM !
objective
subjective
EGALITARIANISM are observed in many experiments.
(Why can’t (*) represent egalitarianism.
Because (*) can’t express |x-y|: the absolute value of the deference of players’ payoffs.
See 11 page, please. )

10. Other-Regarding Utility Function 2
Schulz&May 1989, Fehr&Schmidt 1999
p if my payoff is BETTER than the other’s
q if my payoff is WORSE than the other’s
There are two cases.
One is that my payoff is better than the other’s.
The Other is that my payoff is worse than the other’s.
If my my payoff is better than the other’s, the weight for the other is p.
If my my payoff is worse than the other’s, the weight for the other is q.
－∞＜p＜+∞, －∞＜q＜+∞

11. ＊Egalitarianism：(p－q ) is large
p>0, q<0 is sufficient for weak Egalitarianism
much heavier →　Egalitarianism
This formulation can express Egalitarianism.
When this weght( (p-q)/2 ) is much heavier, this formulation express Egalitarianism.
The three term, 1.my payoff, 2.other’s payoff, and 3.the difference of my and other’s payoffs , are weighed respectively.
And weighting the only difference of means Egalitarianism.
p>0, q<0 is sufficient for weak Egalitarianism.

12. Family of OR-Utility Functionsq
ANTI-EGL.
SACRIFICE
0.5 1 ∞ -∞ O
p＋q = 1
Family of OR-Utility Functions
MAXMAX
ALTRUISM
altruistic
JOINT
egalitarian
p = q
EGOISM
MAXMIN p
This is a graph of p, q.
An each point (p, q) represents an OR-Utility Functions.
Competition, egoism, joint altruism and sacrifice, this is more altruistic in this order.
But they don’t weigh (regard) equity.
And Maxmin is less egalitarian than Egalitarianism.
COMPETITION
EGALITARIANISM

13. 1 INTRODUCTION. 2 OR-UTILITY FUNCTION 3 GAME-TRANSFORMATION
1 INTRODUCTION 2 OR-UTILITY FUNCTION 3 GAME-TRANSFORMATION CONCLUSION
Let me go to payoff-structure transformation by using these utility function

14. Payoff Transform row-player’s subjective payoff obj. C D 1, 1 0, 6
6, 0
2, 2
subj. C D
(1-p)+p
0(1-q)+6q
6(1-p)+0p
2(1-p)+2p
Now this is payoff transformation for one player.
Red circle means Nash eq.

15. Payoff Transform row-player’s subjective payoff obj. C D 1, 1 0, 6
6, 0
2, 2
subj. C D
(1-p)+p
0(1-q)+6q
6(1-p)+0p
2(1-p)+2p
calculate
This is result of calculation.
subj. C D
1, 1
6q, 6－6p
6－6p, 6q
2, 2

16. Payoff Transform row-player’s subjective payoff obj. C D 1, 1 0, 6
6, 0
2, 2
subj. C D
(1-p)+p
0(1-q)+6q
6(1-p)+0p
2(1-p)+2p
for both players
Now this is both player’s payoff transformation.
subj. C D
1, 1
6q, 6－6p
6－6p, 6q
2, 2

17. Payoff Transform row-player’s subjective payoff obj. C D 1, 1 0, 6
6, 0
2, 2
subj. C D
(1-p)+p
0(1-q)+6q
6(1-p)+0p
2(1-p)+2p
p =1, q =0
:MAXMIN
For example, when p=1, q=0 (that is maxmin) like this.
This is coordination game.
subj. C D
1, 1
6q, 6－6p
6－6p, 6q
2, 2
subj. C D
1, 1
0, 0
2, 2
ex.

18. Payoff Transform by Some OR-Utility Functionsq 1 O
0.5
This is summary of last slide’s transform process.
The objective payoff matrix is only the example.
The payoff matrix can be chicken or coordination or 3by3 or 4by4 and so on.
MAXMIN (1, 0)
1, 1
0, 6
6, 0
2, 2
1, 1
0, 0
2, 2 p
example
0.5 1

19. Payoff Transform by Some OR-Utility Functionsq 1 O
ALTRUISM (1, 1)
1, 1
6, 0
0, 6
2, 2
0.5
Altruism is represented by (1, 1) that is p=1, q=1
Now by using Altruism we transform this matrix.
Altruism exchange my payoff for other’s payoff.
MAXMIN (1, 0)
1, 1
0, 6
6, 0
2, 2
1, 1
0, 0
2, 2 p
example
0.5 1

20. Problem in ALTRUISM p =1，q =1 1, 1 0, 6 6, 0 2, 2 1, 1 6, 0 0, 6 2, 2
“The Gift of the Magi”
Della＼Jim
present
not
1, 1
0, 6
6, 0
2, 2
subjective
Delｌa＼Jim
present
not
1, 1
6, 0
0, 6
2, 2
Then Altruism have a problem.
We assume that subjective Eq realize as constant state.
But objectively this is inefficient! (in this case)
In “The Gift of the Magi,” Altruism causes Pareto inefficiency.
INEFFICIENT!

21. Problem in EGALITARIANISM
p→∞，q→－∞
“Leader Game”
follow
lead
3, 3
4, 7
7, 4
1, 1
subjective
follow
lead
0, 0
-2, -2
INEFFICIENT!
Similarly, there is a problem in Egalitarianism.
In “Leader Game”, Altruism causes Pareto inefficiency.

22. Theorem WAYS of Other-Regarding ALTRUISTIC p,q≧0
existing Social States
which are
Pareto EFFICIENT in objective level
and　　
Pure Nash EQUILIBRIA in subjective level
for any two-person games
ALTRUISTIC p,q≧0
and　　
IMPARTIAL p +q =1 ＝
Then, what ways of other-regarding can realize Pareto-efficiency in any games?
That’s like this.

23. IMPARTIAL Ways q ∞ IMPARTIAL p＋q = 1 1 0.5 p -∞ anti-egl sacrificeO
IMPARTIAL
p＋q = 1
maxmax
altruism
IMPARTIAL Ways
joint
egoism
maxmin p
This line means impartial.
competition
egalitarian

24. ALTRUISTIC and IMPARTIAL Waysq
anti-egl
ALTRUISTIC p, q≧0
0.5 1 ∞ -∞ O
ALTRUISTIC and IMPARTIAL Ways
MAXMAX
altruism
including mixture
JOINT
egoism
MAXMIN p
This area means altruistic.
Altruistic and Impartial way are this segment, typically, maxmax, joint, and maxmin.
Altruistic and Impartial way include mixture of Maxmax and Maxmin.
competition
egalitarian

25. ALTRUISTIC and IMPARTIAL Ways: Payoff Transform exampleq
0.5 1 O
MAXMAX
1, 1
6, 6
2, 2
ALTRUISTIC and IMPARTIAL Ways: Payoff Transform example
JOINT
1, 1
3, 3
2, 2
JOINT
MAXMIN
1, 1
0, 6
6, 0
2, 2
1, 1
0, 0
2, 2
egoism p
Typical altruistic and impartial ways are Maxmin, Joint, and Maxmax.
On the impartial way, in each sell, both players’ subjective payoffs are the same.
Objective LV

26. 1 INTRODUCTION. 2 OR-UTILITY FUNCTION
1 INTRODUCTION 2 OR-UTILITY FUNCTION 3 GAME-TRANSFORMATION 4 CONCLUSION
Let me go to the conclusion

27. 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 equally If my payoff is worse than the other’s, regard a little
But not impartial (p+q = 0.8＜1)
→ Theorem requires a strict ethic

28. Implication 1
→ Only altruistic and impartial ways of other regarding can realize Pareto efficiency in ANY two-person games

29. Implication 2 Extreme-Egalitarianism isn’t good
　　weight for difference of payoffs (|e| ) ≦ weight for sum of payoffs (1/2)
Other-Regarding utility function satisfying to be altruistic and impartial can be said in other words, like this.
The absolute value of e is smaller than half.
egalitarianism
utilitarianism
⇒ 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 Efficiency in any two-person interactions.
Let me make a summary.
不偏性のような超越的な規範概念が無数の相互行為からボトムアップに導出される

32. Bibliography
Shulz, U and T. May “The Recording of Social Orientations with Ranking and Pair Comparison Procedures.” European Journal of Social Psychology 19:41-59
MacClintock, C. G “Social Motivation: A set of propositions.” Behavioral Science　17:
Fehr, E. and K. M. Schmidt “A Theory of Fairness, Competition, and Cooperation.” Quarterly Journal of Economics 114(3):

33. Defection through Egoism
p =0，q =0
“Prisoners’ Dilemma”
stay silent
confess
4, 4
0, 6
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 m×n ASYMMETRIC game
efficient action profiles in objective level of game g
existing
{v | ∀g　Eff(g)∩NE(vg)≠φ}
= {v | p +q =1, p≧0, q≧0 }