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2. Consider the following game involving two players and a chance move by nature'. China and Russia have the nuclear technology to destroy each other. 'Nture' tosses a fair coin so that with probability China moves first and Russia moves second, and with probabiltiy Russia moves first and America moves second. For now, assume that both countries observe nature's choice so they know whether they are first or second. The country who moves first decides whether to fire its missiles or to wait. If it fires, the game ends: the country who fired gets a payoff of -1, and the other country gets -4. If the first country waits, then the second country chooses whether to fire or to wait. If it fires, then the game ends, it gets -1 and the other country gets - 4. If it waits, then both countries get 0. Assume that each country maximizes its expected payoff. Treat this as one game, rather than two games. Figure 1 is the game tree representing this game. The first payoff refers to China and the second payoff refers to Russia. There's no payoff for 'nature'. (a) What makes this a game of perfect information? Write down all possible strategies for China and for Russia. (b) Find and explain any pure-strategy subgame perfect equilibria (SPE), making clear what constitutes a subgame. Are there any Nash equilibria which are not SPE? If no, explain why not. If yes, give an example. (c) Now suppose that neither Russia or China observes the move by nature, or each other's move. So if a country is called upon to move, it does not know whether it is the first mover or whether it is the second mover and the other country has chosen to wait. Again, treat this as one game. Draw a game tree similiar to figure 1 but for this new game. Indicate clearly which nodes are in the same information sets. (d) Write down the set of all strategies for China and for Russia in the game from part (2c). Find and explain carefully two symmetric pure strategy SPEs in this game that have very different outcomes. China fire wait fire Russia wait Nature (0,0) 1/2 (-4,-1) fire Russia Wait China fine wait (0,0) Figure 1: Game Tree of the Nuclear Threat Game: Perfect Information 2. Consider the following game involving two players and a chance move by nature'. China and Russia have the nuclear technology to destroy each other. 'Nture' tosses a fair coin so that with probability China moves first and Russia moves second, and with probabiltiy Russia moves first and America moves second. For now, assume that both countries observe nature's choice so they know whether they are first or second. The country who moves first decides whether to fire its missiles or to wait. If it fires, the game ends: the country who fired gets a payoff of -1, and the other country gets -4. If the first country waits, then the second country chooses whether to fire or to wait. If it fires, then the game ends, it gets -1 and the other country gets - 4. If it waits, then both countries get 0. Assume that each country maximizes its expected payoff. Treat this as one game, rather than two games. Figure 1 is the game tree representing this game. The first payoff refers to China and the second payoff refers to Russia. There's no payoff for 'nature'. (a) What makes this a game of perfect information? Write down all possible strategies for China and for Russia. (b) Find and explain any pure-strategy subgame perfect equilibria (SPE), making clear what constitutes a subgame. Are there any Nash equilibria which are not SPE? If no, explain why not. If yes, give an example. (c) Now suppose that neither Russia or China observes the move by nature, or each other's move. So if a country is called upon to move, it does not know whether it is the first mover or whether it is the second mover and the other country has chosen to wait. Again, treat this as one game. Draw a game tree similiar to figure 1 but for this new game. Indicate clearly which nodes are in the same information sets. (d) Write down the set of all strategies for China and for Russia in the game from part (2c). Find and explain carefully two symmetric pure strategy SPEs in this game that have very different outcomes. China fire wait fire Russia wait Nature (0,0) 1/2 (-4,-1) fire Russia Wait China fine wait (0,0) Figure 1: Game Tree of the Nuclear Threat Game: Perfect Information