1\2 X Y Z A 8,6 12,5 5,2 B 4,8 8,10 10,12 Suppose Player 2 believes Player 1 will pay a mixed strategy θ1 (A,B) = (θ, 1- θ). Find a range of the frequency of play of A (i.e., values of θ) that makes the pure strategy Y a best response. The minimum value of θ = _____ The maximum value of θ = _____

1\2 X Y Z A 8,6 12,5 5,2 B 4,8 8,10 10,12 Suppose Player 2 believes Player 1 will pay a mixed strategy θ1 (A,B) = (θ, 1- θ). Find a range of the frequency of play of A (i.e., values of θ) that makes the pure strategy Y a best response. The minimum value of θ = _ The maximum value of θ = _

Managerial Economics: A Problem Solving Approach

5th Edition

ISBN:9781337106665

Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor

Publisher:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor

Chapter18: Auctions

Section: Chapter Questions

Problem 4MC

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Suppose there are two players playing a game with east or west and south and nerth ways. Find the expected Nash equilibrium by using the concept of probabilities. Player X Left[L) Right|R) Player Y Up(U) (5,6) (0,8) (4,6) Down[D) (0,9)
(ii) A mixed strategy profile (p, q) is one in which p = (p,P2.... P) is the mixed strategy of player 1, and q- (g1, q2,..q4) is the mixed strategy of player 2. Show that if p, >0 in a Nash equilibrium profile (p*, q*), the player 2 must also play i with strictly positive probability q'; > 0. (State clearly any theorem you use to show this. You are not required to justify the theorem.) %3D
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.)
1\2 Y Z A 7,4 3,5 B 8,10 1,9 C 3,12 5,4 Suppose Player 1 believes Player 2 will pay a mixed strategy θ2 (Y,Z) = (θ, 1- θ). Find a range of the frequency of play of Y (i.e., values of θ) that makes the pure strategy A a best response. The minimum value of θ = ________ The maximum value of θ = ________
Two players play the following game for infinite times. For the player to continue to cooperate what would be the ranges of their discount factor, δ_1 and δ_2, respectively? cooperate betray cooperate (10,20) (-25,30) betray (15, -22) (-12, -18)
If a strategy profile is a Nash equilibrium, there is at least one player that could achieve higher payoffs by deviating.(a) True. (b) False.
Player 2 X Y 2, -1 1, 1 A 0,0 0, 7 -1, -3 Player 1 B C -1, -2 (e) Let BV (V) be the best response vector to strategies V. Here V = (Strategy of Player 1) For instance, Player 1's best response to Y is A, and Player 2's best response to A is X, so BV (;) = (4). Find BV^), BV(§), BV(*).
Find the mixed strategy Nash equilibrium by using the concept of probabilities. Player X West{W} East{E} Player Y (0,10) South{S} (7,8) (4,8) (0,12) North{N}
(a) Calculate the safety levels of both players.(b) Find the set of all Nash equilibria (pure and mixed).
For the friend-foe game, recall that there were 3 Nash equilibria possible, but the equilibria set didn't include the cooperative outcome, for which both players would win. Friend Foe Friend 500,500 0,1000 Foe 1000,0 0.0 a) If the game is played répeatedly. propose a play strategy that will enforce cooperation. For what valucs of o (discount factor) the equilibrium will be (Friend. Friend)?
With what probability does player 1 play Down in the mixed strategy Nash equilibrium? (Input your answer as a decimal to the nearest hundredth, for example: 0.14, 0.56, or 0.87). PLAYER 1 Up Down PLAYER 2 Left 97,95 47, 33 Right 8,43 68,91