max II₂ (2i, j): 72120 Cj (ci + c₂)² X2i X21 + x2j where T2i (T25) denotes contest investment by i (j) in stage 2. Following Nti (1999), first-order optimality conditions determine the unique pure strategy Nash equilibrium. Equilibrium investments satisfy Ri -R-C2is = C₁ R. (ci + c₂)²¹ (1)

max II₂ (2i, j): 72120 Cj (ci + c₂)² X2i X21 + x2j where T2i (T25) denotes contest investment by i (j) in stage 2. Following Nti (1999), first-order optimality conditions determine the unique pure strategy Nash equilibrium. Equilibrium investments satisfy Ri -R-C2is = C₁ R. (ci + c₂)²¹ (1)

Exploring Economics

8th Edition

ISBN:9781544336329

Author:Robert L. Sexton

Publisher:Robert L. Sexton

Chapter15: Oligopoly And Strategic Behavior

Section: Chapter Questions

Problem 15P

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max II2i(2, ₁) = #2120 X2i X2i -R-C2i + x2j where x2; (2) denotes contest investment by i (j) in stage 2. Following Nti (1999), first-order optimality conditions determine the unique pure strategy Nash equilibrium. Equilibrium investments satisfy C₂ x2 = (a + c)²R; x2 = (a + c)² R. X2i Ꭱ c₂)²¹
(a) Consider a 2-player zero-sum game with the following payoff matrix, in which a E R is some fixed value. The matrix is given, as usual, from the perspective of the row player. C1 C2 α C3 5 T1 1₂-1 2 -4 T3 2 -2 1 (i) For what range of values for a will (r₁, C₁) be a pure Nash equilibrium of this game? Justify your answer. (ii) Treating a as a constant, give a linear program to find the optimal mixed strategy for the row player. (iii) Suppose that you were given some optimal mixed strategy x for the row player. Explain how you could then determine an optimal mixed strategy y for the column player without needing to solve another linear program.
Cournot duopoly game. There are two firms each with cost zero. Each firm chooses its output. The demand function is p(q) = 1- q1 - q2. Because the price is negative for q; > 1, we may take the strategy set of each player to be Ai -[0, 1]. The Nash equilibrium strategy profile is (а) (3, 3) (b) No Nash equilibrium (c) (1/2, 1/2) (d) (1/3, 1/3)
Consider two players in the game-theoretic setting below: Player 2 Choice A Choice B Choice 20 |-30 A -20 30 Player 1 Choice |-30 40 B 30 -40 What is the (pure-strategy) Nash equilibrium? There are more than one (pure-strategy) Nash equilibria. O (P1, P2) = (A, B) is the only (pure-strategy) Nash equilibrium. O (P1, P2) = (A, A) is the only (pure-strategy) Nash equilibrium. O There is no (pure-strategy) Nash equilibrium. O (P1, P2) = (B, A) is the only (pure-strategy) Nash equilibrium. (P1, P2) = (B, B) is the only (pure-strategy) Nash equilibrium.
In the following symmetric general sum game (2, 2) (0,0) (0,0) (0,0) (0,0) (2, 2) (0,0) (2,2) (0,0) (i) Find all pure Nash equilibria. (ii) Find all mixed Nash equilibria in which all probabilities are positive. (vi) Which of these are evolutionary stable strategies?
1 \ 2 a b c A 2,2 2,1 0,0 B 3,1 3,0 1,3 C 1,3 3,2 3,1 a) Eliminate iteratively all strictly dominated strategies. b) Find all Nash equilibria. c) What is the player 1’s best response to the strategy σ2 = (z , ½ , ½ - z) of player 2, where z is a parameter that lies in the closed interval [0, ½]. d) Compute the payoffs to player 1 and to player 2 for their respective strategy profiles σ1 = (0,½,½) and σ2 = (½,0,½).
Consider two players, P1 and P2, each can choose A or B. The payoff matrix as a result of the players' choices are listed below: P1 A B 25 35 A 25 O There is no pure-strategy Nash equilibrium. O (P1, P2) (B. A) O (P1, P2) - (A. A) (P1, P2) - (B. B) O (P1, P2) - (A, B) O There are multiple pure-strategy Nash equilibria. 35 Identify the pure-strategy Nash equilibrium (P1, P2). P2 55 45 B 35 45
Consider two players, P1 and P2, each can choose A or B. The payoff matrix as a result of the players' choices are listed below: P1 O (P1, P2) (A, A) O (P1, P2) = (A,A) O (P1, P2) = (A, A) A Identify the pure-strategy Nash equilibrium (P1, P2). (P1, P2) = (A, A) B There are multiple pure-strategy Nash equilibria. O There is no pure-strategy Nash equilibrium. -25 35 A 25 -35 P2 35 -45 B -35 45
Consider the game in strategic form defined by the following payoff functions: U1(t1,t2) = [10 + w]tįt2 - tỉ, U2(tı,t2) = [10 + w]t,t2 - t. %3D Set the parameter w equal to 1 (a) This game has a unique (pure strategy) Nash equilibrium (t¡, t;) in which t¡ = t;. Find the value of t;. (If your answer is a fraction, report it in lowest terms.) (b) To which strategy tą is the strategy ti = 4 a best response? (If your answer is a fraction, report it in lowest terms.)
Players 1, 2, and 3 are playing a game in which the strategy of player i isdenoted yi and can be any nonnegative real number. The payoff function for player 1 is V1(y1,y2,y3) = y1 + y1y2 - (y1)2,for player 2 is V2(y1,y2,y3) = y2 + y1y2 - (y2)2,and for player 3 is V3(y1,y2,y3) = (10 - y1 - y2 - y3)y3.These payoff functions are hill shaped. Find a Nash equilibrium. (Hint: Thepayoff functions are symmetric for players 1 and 2.)
Suppose B = [bj] is the m x n payoff matrix of a 2-person zero-sum simultaneous game a. Let a' be the 2-person zero-sum simultaneous game with payoff matrix B' = [b,g) = [b.j + a] for some constant a e R. Show that if (, T) is a Nash equilibrium of a and the value of a is v*, then ( , 7') is also a Nash equilibrium of a' and the value of a' is v* + a.
a) Find the Nash equilibria in the game (in pure and mixed strategies) and the associated payoffs for the players. b) Now assume that the game is extended in the following way: in the beginning Player 1 can decide whether to opt out (this choice is denoted by O) or whether to play the simultaneous-move game in a) (this choice is denoted by G). If Player 1 opts out (plays O) then both Player 1 and Player 2 get a payoff of 4 each and the game ends. If Player 1 decides to play G, then the simultaneous-move game is played. Find the pure-strategy Nash equilibria in this extended version of the game. (Hint: note that Player 1 now has 4 strategies and write the game up in a 4x2 matrix.) c) Write the game in (b) up in extensive form (a game tree). Identify the subgames of this game.