This game is based on the Hangman's Paradox. Suppose that a judge, Player 1, sen- tences...

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This game is based on the Hangman's Paradox. Suppose that a judge, Player 1, sen- tences a prisoner, Player 2, to be hanged in one the next 7 days (T is an integer). Moreover, the judge tells the prisoner that the day of hanging will be a surprise. Formally, there are T possible periods in the game. In any of the first T - 1 periods of the game, Player 1 chooses whether to hang (H) or not (N), and Player 2 simulta- neously chooses (i.e. guesses) whether he will be hanged (h) or not (n). If Player 1 chooses H in any of the first T - 1 periods, the game ends at the end of that period. Otherwise, the game proceeds to the next period. [Note: at the end of each period, players' actions in that period become publicly observable.] In the 7th period, Player 1 has only one action available: H (player 2 still has his two actions). Note that there is a terminal node only after a period in which Player 1 chooses H. Payoffs at any terminal node are (-1, 1) if player 2 chose h in that period (hence, he was not surprised) and (1, -1) if player 2 chose n in that period (hence, he was surprised). (a) Argue that there are proper subgames in this game (a subgame is a proper sub- game if it is not the game itself). (10 pts) (b) Suppose T = 2. Is there a Nash equilibrium that is not subgame perfect? (10 pts) (c) Suppose T = 3. Is there a Nash equilibrium that is not subgame perfect? (10 pts) Notice that in the game as described so far, neither player's terminal utility is affected by 2's guess in any period that 1 chooses N. We incorporate this factor by now modifying payoffs so that at each terminal node, we augment a disutility to player 2 and an extra utility to player 1 of > 0 for every period in which 2 played h but 1 played N during the path taken to that terminal node. This game is based on the Hangman's Paradox. Suppose that a judge, Player 1, sen- tences a prisoner, Player 2, to be hanged in one the next 7 days (T is an integer). Moreover, the judge tells the prisoner that the day of hanging will be a surprise. Formally, there are T possible periods in the game. In any of the first T - 1 periods of the game, Player 1 chooses whether to hang (H) or not (N), and Player 2 simulta- neously chooses (i.e. guesses) whether he will be hanged (h) or not (n). If Player 1 chooses H in any of the first T - 1 periods, the game ends at the end of that period. Otherwise, the game proceeds to the next period. [Note: at the end of each period, players' actions in that period become publicly observable.] In the 7th period, Player 1 has only one action available: H (player 2 still has his two actions). Note that there is a terminal node only after a period in which Player 1 chooses H. Payoffs at any terminal node are (-1, 1) if player 2 chose h in that period (hence, he was not surprised) and (1, -1) if player 2 chose n in that period (hence, he was surprised). (a) Argue that there are proper subgames in this game (a subgame is a proper sub- game if it is not the game itself). (10 pts) (b) Suppose T = 2. Is there a Nash equilibrium that is not subgame perfect? (10 pts) (c) Suppose T = 3. Is there a Nash equilibrium that is not subgame perfect? (10 pts) Notice that in the game as described so far, neither player's terminal utility is affected by 2's guess in any period that 1 chooses N. We incorporate this factor by now modifying payoffs so that at each terminal node, we augment a disutility to player 2 and an extra utility to player 1 of > 0 for every period in which 2 played h but 1 played N during the path taken to that terminal node.

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Answers a There are proper subgames in this game because at each period before the final period T Pl... View the full answer