Economic and Game Theory
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I assume that they have two strategies: (1) best-worst-case i.e. choose the option where the worst result is not as bad as with the other option and (2) best-sum choose the strategy were the added gain for both players is best. Best-worst-case leads to an 18-years/18-years game, best-sum leads to a 2-days/2-days game. However the prisoners have to take into account that they might be cheated in their attempt to go for the best-sum, thus it is normally assumed that the best-worst-case strategy is dominant.
In this game, the risk of beeing cheated is 2-years and the potential gain by going for the best-sum is 18-years. Furthermore the incentive for the other prisoner to cheat is minimal. So the risk of being cheated is rather small compared to the potential gain of co-operation - even in a single game.
Compare this to Axelrods reward matrix (rewards _not_ years in prison):
best-sum 3/3 - mismatch 4/0 or 0/4 - best-worst-case 1/1
Here the risk of being cheated is 1 and the potential gain only 2, while the incentive to cheat is 1. Here - especially when we are looking at a single game - the risk of being being cheated in combination with the opponents incentve to cheat is rather big compared to the potential gain of cooperation.
I would suggest the following formula for a single game (I am no economist, mathematician or the like):
potential gain of co-operation = best-sum - best-worst-case
risk of being cheated = best-worst-case - cheated
The logic of this is that instead of going for best-worst-case I offer best-sum, but get cheated.
oponents incentive to cheat = opponent's cheating - oponent's best-sum
The logic of this is that the opponent counts on you going for best-sum, but she/he wants to cheat.
potential gain of mutual co-operation
------------------------------------------------------------ > 1
A * risk of being cheated + B * opponents incentive to cheat
then go for mutual co-operation.
A is a factor acounting for the player's willingness to take risks and B is a factor for the player's assumption about the other player's reasoning. A and B are situtaional and personal factors.
For Axelrod's reward-matrix this formula gives only 1 even without taking the factors A and B into account.