If both confess, then they each get 8 years. So if one confesses and the other doesn't, the confessor only gets 3 months in prison, while the partner serves 10 years. They are each offered a deal: if they confess and rat on their partner, they will receive a reduced sentence. Two partners in crime are arrested for burglary and sent to separate rooms. In order to better understand non-zero-sum games we look at two classic games. What are some possible strategies for each player? Might some strategies depend on what a player knows about her opponent?Ĭan you see that some of the analysis might be better understood with psychology than with mathematics? OK, so now, how do we analyze these games? Exercise 4.2.6. Each player only cares about his or her own payoff, not the payoff of the other player. Similarly, we don't want to apply the expected value solution since Player 1 does not care if Player 2's expected values are equal. Notice, that now the expected value for Player 1 is 5, which is better than 10/3! Again, since Player 2 is not trying to keep Player 1 from gaining, there is no reason to apply the maximin strategy to non-zero-sum games. But if Player 2 plays only C, then Player 1 should abandon her (1/3, 2/3) strategy and just play B! This results in the payoff vector (5, 10). Then the expected payoff to Player 2 for playing pure strategy C, \(E_2(C)\text\) is 5/3. Now suppose, Player 1 plays the (1/3, 2/3) strategy. But are our players motivated to play as defensively in a non-zero-sum game? Not necessarily! It is no longer true that Player 2 needs to keep Player 1 from gaining! Recall we developed this strategy as a “super defensive” strategy. Similarly we can determine that Player 2 should play a (2/3, 1/3) mixed strategy for an expected payoff of 10/3. If we apply the graphical method for Player 1 to the above game, we get that Player 1 should play a (1/3, 2/3) mixed strategy for an expected payoff of 10/3.
We saw in Section 4.1, that our methods for analyzing zero-sum games do not work very well on non-zero-sum games. If no communication is allowed, we say it is non-cooperative.
If communication is allowed in the game, then we say the non-zero-sum game is cooperative. For example, if Player 1 says that she will choose A no matter what, then it is in Player 2's best interest to choose D. Since it is now possible for BOTH players to benefit at the same time, it might be a good idea for players to communicate with each other. Does Player 1 prefer one of the equilibria from Exercise 4.2.4 over the other? As we saw in Section 4.1, the equilibrium points in non-zero-sum games need not have the same values.
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