Altruism and Resultant Mixed Nash Equilibrium
This chapter is mainly technical, and requires a little bit of mathematics. Those who find it difficult can skip it over, but a rough browse would still be helpful.
In the game of Angel and Devil, usual pursuit of self interest of both parties leads to the nash equilibrium of both playing Devils. On the other hand, unconventional "forced" pursuit of opponents interest by both players would lead to a modified Nash of two Angels, which is certainly preferable to both players. Moreover, there seems to be even better way of 'forcing" both players to mix self interest and opponents's interest that results in even more preferable results. We shall make this story line of ours more robust and rigorous in this chapter.
Imagin that the game is played many times over, and half of the time, at random, the players are cheated by a third party (the referee) and given the table containing reversed entries for row and column. Because of the randomness, both players have no way of knowing what table would be given beforehand, so will resort to roll dice and change hands with a probability. We call this mixed strategies. In this case, players would have to assume that the table used in calculating the payoff is, on average, given by a table given by the sum devided by two of two tables. namely, both players will use
| your score |
opp. Devil |
opp. Angel |
| you Devil |
0
|
3/2
|
| you Angel |
3/2
|
１
|
|
|
| opp score |
opp. Devil |
opp. Angel |
| you Devil |
0
|
3/2
|
| you Angel |
3/2
|
１
|
|
Supose you go Angel by probaility s, and go Devil by (1-s). Your opponent plays Angel and Devil by probabilities x, and (1-x) respectively. Using the next table
|
your score
|
1-x
|
x
|
|
1-s
|
0
|
3/2
|
|
s
|
3/2
|
１
|
|
|
|
opp. score
|
1-x
|
x
|
|
1-s
|
0
|
3/2
|
|
s
|
3/2
|
１
|
|
we calculate your expected payoff Pyou(s, x) and opponet's expected payoff Popp(x, s) as
Pyou(s, x) = (1-s) { (1-x)*0 + x * 3/2 } + s { (1-x)*3/2 + x*1 }
Popp(s, x) = (1-x) { (1-s)*0 + s * 3/2 } + x { (1-s)*3/2 + s*1 } .
(They are ientical in this case.) You control your strategy by tweaking s to maximize Pyou(s, x) , and your opponent does the same for Popp(s, x) with x. Namely, we have
dPyou(s, x) /ds = 0 , dPopp(s, x) /ds = 0 .
Both are linear equations whose solutions are easily obtained as
s = x = 3/4
Pyou(3/4, 3/4) = Popp(3/4, 3/4) = 9/8 .
This type of outcome based on the probablistic (mixed) strategy is called mixed nash equilibrium.
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