Dirk BergemannDepartment of EconomicsYale UniversityEconomics 121b: Intermediate MicroeconomicsProblem Set 10: Mixed Strategy and Repeated Game4/7/10This problem set is due 4/14.1. Mixed Strategy Nash Equilibrium. Find the unique, mixed strategyequilibrium, of the matching pennies game:BobHead TailAnn Head 1; 1 1; 1Tail 1; 1 1; 1(a) First draw the best response function of Ann and Bob in a two-dimensional graph.(b) Then solve for the mixed strategy equilibrium algebraically.2. Find all, pure and mixed strategy equilibria of the “Hawk-Dove”game:Defend AttackDefend 3; 3 1; 4Attack 4; 1 0; 0(a) First draw the best response function of the row and the columnplayer in a two-dimensional graph.(b) Then identify the pure and mixed strategy equilibria algebraically(guided by the geometric representation).3. Two owners i = 1; 2 of a stand on the New Haven farmers’ market sellapples. The e¤ort that the y put into selling the apples is ei. They canchoose any e¤ort between 0 and 1. The revenue that they make is anincreasing function of both owners’e¤ort: R(e1; e2) = 2pe1+ e2. Eachowner receives one half of this revenue. For each owner i the cost of e¤orteiare Ci(ei) = 0:5 (ei)2. Thus, owner i’s net utility is:ui(e1; e2) =pe1+ e2 0:5 (ei)2:(a) For e ach owner i write down the …rst order condition for the optimalchoice of eigiven the other owner’s choice ej. Show that the secondderivative of utility with respect to eiis negative.(b) Divide the two …rst order conditions by each other and conclude thatin a Nash equilibrium the two e¤ort levels have to be identical.1(c) Denote the common equilibrium e¤ort level by e . Substitute e1=e2= e into the …rst order condition and solve for e . (Hint: beginby squaring both sides of the …rst order condition.)(d) If both owners chose the e¤ort level 1, i.e. e1= e2= 1, would theowners be better o¤ or worse o¤ than in the Nash equilibrium?4. The game shown below is played repeatedly in an in…nite numbe r of pe-riods t = 1; 2; :::. Both players maximize the present discounted value oftheir payo¤s. Both players have the same discount factor :L RT 4,4 2,12B 12,2 3,3(a) For the above game identify the Nash equilibrium of the static game.Does each player have a dominant strategy?(b) Now consider the following strategy in a repeated game. The twoplayers start by playing (T; L) in period 1. In periods t > 1 playersplay (T; L) if both players played (T; L) in all previous periods. Oth-erwise, players play (B; R). What is the lowest value of  such thatthese strategies form a subgame perfect Nash equilibrium?5. The game shown below is played repeatedly in an in…nite numbe r of pe-riods t = 1; 2; :::. Both players maximize the present discounted valueof their payo¤s. Both players have the same discount factor . Playersstart by playing (T; L) in period 1. In periods t > 1 players play (T; L)if both players played (T; L) in all previous periods. Otherwise, playersplay (B; R). What is the lowest values of  for which it is true that the sestrategies form a subgame perfect Nash equilibrium?L RT 2,2 0,0B 0,0 1,16. Reading Assignment: NS Chapter