14.122 Problem Set #11. Find a game which is not solvable by pure strategy iterated strict dominance butwhich does have an unique pure strategy Nash equilibrium.2. Consider the model of Cournot competition discussed in class where the inversedemand function is P (q)=1− q and the firms have zero marginal costs. Show that itis strictly dominated for the firms to produce any quantity greater than12Write the setof strategies which are not strictly dominated for the firms as an interval [S1, S1]. Findthe interval of strategies [S2, S2] which are not strictly dominated when a firm’s opponentchooses a quantity in [S1, S1]. Prove by induction that when the iterated strict dominanceprocess is continued the set of strategies remaining at stage 2k is the interval [S2k, S2k]whereS2k=kj=114j.S2k=12−12k−1j=114j.Conclude that the game is solvable by iterated strict dominance.3. Consider the following game-theoretic model of the equilibrium determination ofthe cleanliness (and effort distribution) of an apartment shared by two roommates. In thegame, the two roommates simultaneously choose the effort, e1e2, to spend on apartmentcleaning. They each get utility from the cleanliness of the apartment (which is a functionof the sum of the efforts) and disutility from the effort they personally expend. Player 1places a higher valuation on cleanliness. Specifically, assume that e1and e2are chosen fromthe set of nonnegative real numbers and thatu1(e1,e2)=k log(e1+ e2) − e1u2(e1,e2) = log(e1+ e2) − e2,where k>1(a) Find the two players best response functions.(b) Find the pure strategy Nash equilibria of the game. How does the equilibriumdistribution of effort reflect the differences in the players’ tastes.(c) Try to write down a modification of the model above in which the outcome seemsmore fair.4. Write out a formal specification (strategy sets, payoff functions, etc.) of the twoplayer version of the game described in problem 1.8 of Gibbons. What is the pure strategyNash equilibrium of the two player game?15. Find all of the Nash equilibria of the following games.1.8.Consider a population of voters uniformly distributed alongthe ideological spectrum from left = 0) to right = 1). Each ofthe candidates for a single office simultaneously chooses a cam-paign platform (i.e., a point on the line between x = 0 and x = 1).The voters observe the candidates’ choices, and then each votervotes for the candidate whose platform is closest to the voter’sposition on the spectrum.If there are two candidates and theychoose platforms = and = for example, then allvoters to the left of x = vote for candidate 1, all those tothe right vote for candidate 2, and candidate 2 wins the elec-tion with 55 percent of the vote. Suppose that the candidatescare only about being elected-they do not really care about theirplatforms at all. If there are two candidates, what is the pure-strategy Nash equilibrium. If there are three candidates, exhibita pure-strategy Nash equilibrium. (Assume that any candidateswho choose the same platform equally split the votes cast for thatplatform, and that ties among the leading vote-getters are resolvedby coin flips.) See Hotelling (1929) for an early model along