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Respond to the question: Reductio Ad Absurdum of PD?

 08/28/2003 01:12 AM by David May; risk of being cheated vs. potential gain of co-operation
To make things easyer, I neglect a strategy where one prisoner hopes on cheating the other prisoner and I neglect the difference between 2-days and 0-days. As the difference is so small, especially compared to the worst-cases of 18-years or 20-years, there is no real incentive to attempt to cheat

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.

If
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.

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 08/16/2003 12:43 AM by Sunsawed; Reductio Ad Absurdum of PD
Lemme get this straight... Let's say if both prisoners stay silent they get two days more in jail (with intense questioning). If one squeals and the other doesn't, the one that squeals if free to go and the other gets 20 years. But [View full text and thread]