Econ 407.1 Fall 2016 Homework 3 Due at the beginning of class, Tuesday, November 1 1. (3 points) Find a Nash equilibrium of the following game:UD L1, ?2?2, 3 R2, 43, ?1 Show your derivation clearly. 2. (5 points) Find all Nash equilibrium of the following game:UD L3, 12, ?1 R4, ?25, 2 3. Let A = {a, b} and I = {Curly, Larry, Moe}, with preferences 1st2nd Curlyab Larryab Moeba The choice will be made by majority vote, with ties resolved by coin-flip. Each individuali has a utility function,ui (first choice) = 8, ui (second choice) = 0. Each individual experiences a cost c ? 0 if they vote, and no cost if they do not vote. Eachindividual chooses whether or not to vote, and which alternative to vote for if they vote.Each individual seeks to maximize expected utility minus voting cost.a) (1 point) For what values of c is there a pure-strategy Nash equilibrium in whicheveryone votes?b) (1 point) For what values of c is there a pure-strategy Nash equilibrium in which noone votes?c) (4 points) Suppose c = 3. Find a Nash equilibrium. 1 4. Suppose a public good level must be chosen in a society with three individuals, I ={1, 2, 3}. The public good level x costs x units of the private good to produce, so the setof allocations is the set A = {(x; t1 ,?t2 , t3 ) : x > 0 and t1 + t2 + t3 = x}. Each individualhas a utility function ui (x, yi ) = ?i x + yi , with ?i = i for each i. Each individual i hasan endowment of the private good? wi , so given an allocation (x; t1 , t2 , t3 ), each individualiâs utility is ui (x, wi ? ti ) = ?i x + wi ? ti .a) (3 points) Find the set of Pareto efficient allocations.b) (3 points) For the voluntary contribution mechanism, find the Nash equilibrium levelof the public good and each individualâs contribution. 2