Nash Equilibrium: Finite Action Set N players go hunting; each can choose either action S = "hunt stag\" or H = \"hunt hare\". Ifall JV players choose S, then each gets payoff 2. If any player chooses H, then each harehunter gets payoff 1, and each staghunter gets payoff zero. (Interpretation, the stag is only caught if all N players work together). {3] Suppose rst that N = 2 (is, that there are two players}. W'rite out the payo matrix, and nd all NE. (b) Now let N = 3. Show that there are exactly two NE: [1) everyone hunts stag, and (ii) everyone hunts hare. Nash Equilibrium: Continuous Action Set. Two roommates each need to choose to clean their apartment, and each can choose any amount oftiIne t,- 2 0 to clean. cheir choices are t,- and ti, then player i's payoff is given by {l tti t3} {This payoff function implies that the more one roommate cleans, the less valuable is cleaning for the other roommate}. (a) Find a NE pair of cleaning times, (t1, 3;]. (b) Now suppose that P1 gets to choose her cleaning time t1 rst, then P2 sees and responds to this with his choice t2. \"That are the [SPNE} cleaning times? Consider a twoplayer secondpr'tce auction each submits (simultaneously)I a sealed bid, and whoever bids the most wins, but pays the laser's bid. Losers pay nothing. The object is worth $10 to P1 and $5 to P2, and both know this. (a) Show that the best possible NE payo' to P1 is 10, and nd an equilibrium where this is his payoff. (Note, this means he pays zero for the object}. (b) Show that the worst possible NE payoff to P1 is D, and nd a NE where this is his payo". (Note, this means that he pays 10 for the object). Mixed Strategies. Consider the classic childhood game, rockpaperscissors, with the fol lowing payoffs (assuming paper beats rock, rock beats scissors, scissors beat paper): a P 3 s 0,0 ~1,1 1,1 P 1,-1 0,0 1,1 3 -1,1 1,-1 0,0 (a) Show that there is no pure strategy NE. (b) Show that there is a mixed strategy NE in which each player randomizes over all three actions, choosing each with probability . his in from a textbook, and confuses some students: it means that if student 1 chooses t; and student 2 chooses in student 1's peyo' is {10 up; ti, and student 2': peyoffia {10 t1}t2 3%. (He-telling Model). Consider the following game of political competition, with two candidates, P1 and P2. A continuum of voters are uniformly distibuted on the interval from t] to 1, where CI and 1 represent (respectively) the left-wing and right-wing ideology extremes. Each candidate 3' = 1, 2 chooses his policy position :81- 6 [0,1]. Assume that each voter votes for the candidate ciosest to him, ipping a coin if indifferent. Candidates get payo 1 from winning, 0 from losing, and 11- from a tie (whoever gets the most votes, wins}. 2 Show: the only NE is x; :2\"; = 11