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The Prisoner's Dilemma
Of course, it is possible to construct payoff grids in many different ways, and they do not always have to be zero-sum. In fact, constructing game theory problems that are intentionally less symmetrical than Mixed Pennies and Cake Division can lead to some very perplexing "games." One famous game theory problem is called the Prisoner's Dilemma. It is from this problem that William Poundstone takes the title of his book. He describes the "story" behind this game as follows: Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police admit they don't This table shows the outcomes for Player 1, the player that wins if the pennies match. Because this is another zero-sum game, the utility for Player 2 is the inverse of Player 1's payoff. What is the proper way to play this game? What strategy should a rational player choose: heads or tails? There does not seem to be a single best answer to the question. If one player decided to pick heads or tails as a permanent strategy, the other player could take advantage of this strategy and win every time. But Matching Pennies is a finite, zero-sum, two-player game, and game theory should be able to solve this game and provide the proper strategy for two rational players. The solution turns out to be more complex than the cake division problem: players do not choose a single, fixed strategy, but select a mixed strategy. In a mixed strategy, players choose one of their options accord-have enough evidence to convict the pair on the principal charge. They plan to sentence both to a year in prison on a lesser charge. Simultaneously, the police offer each prisoner a Faustian bargain. If he testifies against his partner, he will go free while the partner will get three years in prison on the main charge. Oh yes, there is a catch. ...If both prisoners testify against each other, both will be sentenced to two years in jail. The prisoners are given a little time to think this over, but in no case may either learn what the other has decided until he has irrevocably made his decision. Each is informed that the other prisoner is being offered the very same deal. Each prisoner is only concerned with his own welfare—with minimizing his own prison sentence. [9]
The payoff grid shows the utilities for both prisoners, listing Prisoner A first and Prisoner B second. For ease of use, the utilities are displayed as years of jail time rather than as positive and negative utility values. Look closely at this situation. First of all, is it a game in the game theory sense? Yes, it is: there are two rational players who are only interested in their own welfare (and not in abstract concepts such as cooperation or loyalty), both players have to choose a strategy simultaneously, and each possible outcome of their decision is measured in discrete numbers—in this case, in terms of years of jail time. Because the two prisoners want to minimize their sentence, they desire the lowest possible number. Next question: Is Prisoner's Dilemma a zero-sum game? The answer is no. Look at the upper left and lower right cells. With these outcomes, both "players" in our game do not have inverse outcomes. The gains of one player are not equal to the losses of another player in every case, so the game is not a zero-sum problem. Now think about what each prisoner might decide to do in this situation. First of all, it seems like it is better to turn state's evidence, to "defect" rather than "cooperate" with the other prisoner. If the other player cooperates, then the defector receives the best possible outcome, which is to receive no jail time at all. But both players are rational and will be thinking the same thing, which means both prisoners will "defect" and turn state's evidence. This means that both of them will receive two years. But if they both cooperated, they could have received only one year each! Game theorists do not agree on the proper solution to the Prisoner's Dilemma. There are two ways of thinking about this problem. Using a minimax approach, it is clear that it is always better to defect, no matter what the other prisoner does. If you defect and the other prisoner does not, you get the best possible outcome. But if the other prisoner decides to defect, then it is a good thing you did too, because you saved yourself from the worst possible outcome. According to this logic, both players will defect and the rational outcome is the lower right cell of the payoff grid. The other approach is to say that because both players are rational and because the payoff grid is symmetrical, both players will make the same choice. This means that the two players are choosing between the upper left and the lower right cells. Given this choice, two rational players will end up choosing the better of their two options, the upper left, where they receive only one year of jail time. The Prisoner's Dilemma remains an unsolved game theory problem. It clearly demonstrates that even very simple sets of rules can provide incredibly complex decision-making contexts, which raise questions not just about mathematics and game design, but about society and ethics as well. [9]Ibid. p. 118.
