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(*).
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- 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.)Several firms collude in an oligopoly where the Industry Supply is given by QS = 6 - 5P and Industry Demand is given by, QD = 5P – 5. If the probability that this collusion will continue is given by (1/P*), what is the minimum number of firms required to break the tacit collusion?Consider the game of Chicken in which each player has the option to “get out of the way” and “hang tough” with payoffs: Get out of the way Hang tough Get out of the way 2,2 1,3 Hang tough 3,1 00 a. Find all pure strategy Nash equilibria, if they exist b. Let k be the probability that player 1 chooses “hang tough” and u be the probability that player two chooses “hang tough.” Find the mixed stragety Nash equilibria, if they exist
- Assume the following game situation: If Player A plays UP and Player B plays LEFT then Player A gets $2 and Player B gets $4. If Player A plays UP and Player B plays RIGHT then Player A gets $3 and Player B gets $6. If Player A plays DOWN and Player B plays LEFT then Player A gets $5 and Player B gets $2. If Player A plays DOWN and Player B plays RIGHT then Player A gets $1 and Player B gets $1. What is the mixed strategy expected payout for Player B? 1 40/15 39/15 11/2Consider the following game 1\2 Y Z A 10,3 3,9 B 8,5 6,1 Suppose Player 2 holds the following belief about Player 1: θ1 (A,B) = (9/10,1/10) What is the expected payoff from playing ‘Y’ ? What is the expected payoff from playing ‘Z’ ? Based on these beliefs, player 2 should respond by playing _____Q56 A Nash equilibrium is an outcome... a. Achieved by cooperation between players in the game. b. That is achieved by collusion where no party has an incentive to change their behaviour. c. Where each player's strategy depends on the behaviour of its opponents. d. That is achieved when players in the game have jointly maximized profits and divided those profits according to market share of each player. e. Where each player's best strategy is to maintain its present behaviour given the present behaviour of the other players.
- Using the payoff matrix, suppose this game is infinitely repeated and that the interest rate is sufficiently “low.” Identify trigger strategies that permit players 1 and 2 to earn equilibrium payoffs of 140 and 180, respectively, in each period1. Find best responses of one of the firms to the strategies of its counterpart. Which of the strategies is not a best response to any pure strategy? 2. Let firm 2 play strategies (0, 1, 2) with probabilities (q1, q2, q3), q1 +q2 +q3 = 1. Calculate the expected payoff of firm 1 from playing each of the three strategies. Show that the strategy which is not the best response to any pure strategies is strictly dominated by the other two strategies if firm 2 plays mixed strategies.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)
- 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).Which of the following is FALSE for the grim trigger strategy and the infinite horizon repeated Prisoner's Dilemma game illustrated above? A. In the grim trigger strategy profile, if a player chooses D in a period, then both players chooses D forever after that period B. The threshold discount factor for sustaining cooperation under grim trigger strategy depends on the utility numbers in the stage game C. If all utility numbers remain the same but 3 is replaced by 5 in the stage game, then cooperation CANNOT be sustained in this game for all possible values of the discount factor.two players, a and b are playing an asymmetrical game. there are n points on the game board. each turn player a targets a pair of points and player b says whether those two points are connected or unconnected. a can target each pair only once and the game ends when all pairs have been targeted. player b wins if a point is connected with all other points on the very last turn, while player a wins if any point is connected with all other points on any turn but the very last one or if no point is connected to all other points after the last turn. for what values of n does either player have a winning strategy?
