The Hangman's Paradox and Newcomb's Paradox as Psychological Games
We present a (hopefully) fresh interpretation of the Hangman's Paradox and Newcomb's Paradox by casting the puzzles in the language of modern game theory, instead of in the realm of epistemology. Game theory moves the analysis away from the formal logic of the puzzles toward more practical problems, such as: On what day would the executioner hang the prisoner if he wanted to surprise him as much as possible? How should a surprise test be administered? We argue that both the Hangman's Paradox and Newcomb's Paradox are analogous to a well-known phenomenon in game theory, that giving a player an additional attractive (even dominant) strategy may make him worse off. In the Hangman's Paradox, the executioner is determined to surprise the prisoner as much as possible, yet he cannot surprise him at all because he cannot commit in advance to a random schedule. The possibility of changing his mind (i.e., the presence of alternative strategies) superficially would seem to help the executioner, but because it changes the expectations of the prisoner, in the end it works dramatically to his disadvantage. In Newcomb's Paradox, a man given an extra dominant choice is worse off because it changes God's expectations about what he will do. Our analysis cannot be couched in terms of the standard Nash framework of games, but must instead be put in a recent extension called psychological games, where payoffs may depend on beliefs as well as on actions.
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