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HW 2 (+28) Game theory 1. (+6) For the following game, answer the following. Left Center Right Up (0,0) (5,3) (3,6) Down (4,6) (1,2) (4,3) a. (+4) Identify all pure-strategy Nash equilibria, or state that no such equilibrium exists. b. (+2) Is this game zero-sum? Why or why not? 2. (+12) A dictator possesses nuclear weapons, and has threatened to use them unless the UN satisfies its demands. In that case, other countries may retaliate or not. If the UN satisfies the dictator’s demands, the payoffs (?????? ,????) are (−20,20). If the UN does not satisfy the dictator’s demands, the dictator could decide not to use its nuclear weapons, in which case payoffs are (0,0). If the dictator uses nuclear weapons and the UN does not retaliate, payoffs are (−100,−1). If they do retaliate, payoffs are (−120,−500). a. (+6) Identify all pure-strategy Nash equilibria of this game, or state that no such equilibrium exists. (Remember, an equilibrium is a list of contingent strategies for each player. That is, the dictator must decide whether to nuke or not in the event that the UN does not satisfy. Also, if the UN does not satisfy then it must decide whether to retaliate or not in the event that the dictator nukes.) b. (+4) Tell which of the above equilibria (if any) are sub-game perfect. c. (+2) On the basis of this analysis, what should the UN do? Why? 3. (+8) Shepsle and Bonchek (1997, p. 148) write, Some political observers have taken the infrequency with which presidents have used the veto as an indicator of cooperation between the legislative and executive branches. In this view, infrequent use of the veto corresponds to a high degree of harmony between the branches. In his 1994 State of the Union speech, President Bill Clinton proudly announced the end of gridlock and the beginning of a new period of interbranch harmony by pointing to the fact that not once had he vetoed a piece of congressional legislation during 1993. Those authors then argue in favor of an alternative view, which is that Congress amends its laws to the point that they know it will not be vetoed. Write down a game such that, in equilibrium, Congress amends a bill and vetoes do not occur. To be fully specified, such a game should include strategies for both Congress and the President and outcome utilities for every combination of strategies. (If the game is sequential, a strategy must specify a player’s behavior plan for the entire game, possibly including multiple decision nodes). Identify equilibrium strategies and demonstrate that neither Congress nor the President has an incentive to deviate to a different strategy. 4. (+2) In a particular industry, each worker can work 40 hours or 80 hours. If others work 40, an individual can profit by working 80 hours. If others work 80, an individual loses substantially by only working 40 hours. In equilibrium, all workers therefore work 80 hours. If all worked 40 instead, however, then all would be happier. What game is this situation most similar to: matching pennies, prisoner’s dilemma, traveler’s paradox, battle of sexes, stag hunt, or chicken? HW 2 (+28) Game theory