Dirk BergemannDepartment of EconomicsYale UniversitySolutions by Olga TimoshenkoEconomics 121b: Intermediate MicroeconomicsProblem Set 10: Mixed Strategy and Repeated Game4/7/10This problem set is due 4/14.General RemarksOverall the students did well on this Problem Set. Some of the typicalmistakes include failure to label the graph with the best responses in question1 (either the axes or whose BR was depicted in the graph) as well as to provideclear algebraic solution for the best response. It is not enough to calculateexpected utility of a player from both strategies and set them equal to eachother to determine a mixed strategy of the opponent and fully characterize thebest resp onse. Please, see solutions below for a complete argument.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.SolutionLet α be the probability with which Ann pays Head (H) in a mixedstrategy. Similarly, let β be the probability with which Bob paysHead (H) in a mixed strategy.First, we will find Ann’s best response to Bob’s mixed strategy (β, 1−β). Ann’s utility from playing Head (H) isuA(H) = 1β + (−1)(1 − β) = 2β − 1Ann’s utility from playing Tail (T) isuA(T ) = (−1)β + 1(1 − β) = 1 − 2βAnne will play H with probability 1, i.e. α = 1, if uA(H) ≥ uA(T ),or β ≥ 1/2. Similarly, Ann will play T with probability 1, i.e. α = 0,if uA(H) ≤ uA(T ), or β ≤ 1/2. When β = 1 uA(H) = uA(T ), thus1any α between 0 and 1 will be Ann’s b est response to Bob’s strategy(1/2, 1/2). Thus, Ann’s best response isα = 1 if β > 1/2α = 0 if β < 1/2α ∈ [0, 1] if β = 1/2Ann’s best response is depicted in the figure below.Similar calculations lead to Bob’s best response beingβ = 1 if α < 1/2β = 0 if α > 1/2β ∈ [0, 1] if α = 1/2Bob’s best response is depicted in the figure below.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91BETAALPHA BEST RESPONSE OF ANNBEST RESPONSE OF BOB(b) Then solve for the mixed strategy equilibrium algebraically.Solution Ann’s problem is to choose a mixed strategy (α, 1−α) giventhe mixed strategy of Bob to maximize the expected utility. Annsolvesmaxα∈[0,1]αuA(H) + (1 − α)uA(T ) = α(2β − 1) + (1 − α)(1 − 2β)2maxα∈[0,1]2β + α(4β − 2)The objective function is linear in α. Thus, when 4β − 2 > 0, α = 1;when 4β − 2 < 0, α = 0; and when 4β − 2 = 0, α ∈ [0, 1]. Thus, weobtain best response as the one found in part (a). The calculationsfor Bob’s best response are symmetric.A mixed strategy profile (α, 1 − α); (β, 1 − β) constitute a mixedstrategy Nash Equilibrium if given opponent’s strategy, the player isplaying its best response. From the best responses found in part (a),the only mixed strategy Nash Equilibrium is (1/2, 1 /2); (1/2, 1/2).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 column playerin a two-dimensional graph.SolutionLet α b e the probability with which Row Player (RP) pays Defend(D) in a mixed strategy. Similarly, let β b e the probability withwhich Column Player (CP) pays Defend (D) in a mixed strategy.First, we will find RP’s best resp onse to CP’s mixed strategy (β, 1 −β). RP’s utility from playing D isuRP(D) = 3β + (1)(1 − β) = 1 + 2βRP’s utility from playing Attack (A) isuRP(A) = 4β + 0(1 − β) = 4βRP will play D with probability 1, i.e. α = 1, if uP R(D) ≥ uP R(A),or β ≤ 1/2. Similarly, RP will play A with probability 1, i.e. α = 0,if uRP(D) ≤ uRP(A), or β ≥ 1/2. When β = 1 uRP(D) = uRP(A),thus any α between 0 and 1 will be RP’s best response to CP’sstrategy (1/2, 1/2). Thus, RP’s best response isα = 1 if β < 1/2α = 0 if β > 1/2α ∈ [0, 1] if β = 1/23Similar calculations lead to CP’s best response beingβ = 1 if α < 1/2β = 0 if α > 1/2β ∈ [0, 1] if α = 1/2The best responses of b oth player are depicted in the figure below.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91BETAALPHA BEST RESPONSE OF COLUMN PLAYERBEST RESPONSE OF ROW PLAYER(b) Then identify the pure and mixed strategy equilibria algebraically (guidedby the geometric representation).SolutionA strategy profile (α, 1−α); (β, 1−β) constitutes a Nash Equilibriumif given opponent’s strategy, the player is playing its best response.From the figure above, the best responses intersect three times yield-ing two pure strategy Nash Equilbria [ (Attack, Defend) and (Defend,Attack)] and one mixed strategy equilibrium [(1/2,1/2);(1/2,1/2)].3. Two owners i = 1, 2 of a stand on the New Haven farmers’ market sellapples. The effort that they put into selling the apples is ei. They canchoose any effort between 0 and 1. The revenue that they make is anincreasing function of both owners’ effort: R(e1, e2) = 2√e1+ e2. Eachowner receives one half of this revenue. For each owner i the cost of efforteiare Ci(ei) = 0.5 (ei)2. Thus, owner i’s net utility is:ui(e1, e2) =√e1+ e2− 0.5 (ei)2.4(a) For each owner i write down the first order condition for the optimalchoice of eigiven the other owner’s choice ej. Show that the secondderivative of utility with respect to eiis negative.SolutionMaximization problem of owner i ismaxei∈[0,1]√ei+ ej− 0.5(ei)2FOC12√ei+ ej− ei= 0Further,u′i(ei, ej) =12√ei+ ej− eiu′′i(ei, ej) = −(14(ei+ ej)3/2+ 1)Since eiand ejare non-negative, u′′< 0. Thus, the solution to theFOC is a maximizer.(b) Divide the two first order conditions by each other and conclude thatin a Nash equilibrium the two effort levels have to be identical.SolutionDenote E = ei+ ejand rewrite the FOCs for owner 1 and 2 in thefollowing way12√E= e112√E= e2The two equations determine the Nash equilibrium effort levels ofboth owners. Notice that by dividing the equations we obtaine1= e2Thus, in the NE the effort levels of both owners are the same e1=e2= e∗.(c) Denote the common equilibrium effort level by e∗. Substitute e1=e2= e∗into the first order condition and solve for e∗. (Hint: beginby squaring both sides of the first order condition.)5SolutionPlug e∗into the FOC to obtain12√2e= ee3=18e =12Compute the utility of an owner in