Suppose that two players are playing the following game. Player A can choose either Top or Bottom, and Player B can choose either Left or Right. The payoffs are given in the following table:
Where the number on the left is the payoff to Player A, and the number on the right is the payoff to Player B.
A) (1 point) Does player A have a dominant strategy, and if so what is it?
B) (1 point) Does player B have a dominant strategy and if so what is it?
C) (4 points) For each of the following say true if the strategy combination is a Nash equilibrium, and false if it is not a Nash equilibrium:
i) Player A plays Top and Player B plays Left
ii) Player A plays Bottom and Player B plays Left
iii) Player A plays Top and Player B plays Right
iv) Player A plays Bottom and Player B plays Right
D) (2 points) If each player plays her maximin strategy what will be the outcome of the game? (Give your answer in terms of the strategies each player chooses)
1. For each game, draw a graph with player 1ââ¬â¢s best response function (choice of p as a function of q), and player 2ââ¬â¢s best response function (choice of q as a function of p), with p on the horizontal axis and q on the vertical axis.
2. Using this graphs, find all the Nash equilibriums for the game, both pure and mixed strategy Nash equilibriums (if any). Label these equilibriums on the corresponding graph.
3. In those games that have multiple pure strategy Nash equilibriums, how do the expected payoffs from playing the mixed strategy Nash equilibrium compare with the payoffs from playing the pure strategy Nash equilibriums? Which type of strategy (mixed or pure) would players prefer to play in these games?
Problem 2
Two people are involved in a dispute. Player 1 does not know whether player 2 is strong or weak; she assigns probability ñ to player 2 being strong. Player 2 is fully informed. Each player can either fight or yield. Each player obtains a payoff of 0 is she yields (regardless of the other personââ¬â¢s action) and a payoff of 1 if she fights and her opponent yields. If both players fight, then their payoffs are (-1; 1) if player 2 is strong and (1;-1) if player 2 is weak. The Bayesian game is the following, depending on the type of player 2:
Y
F
Y
F
Y
0, 0
0, 1
Y
0, 0
0, 1
F
1, 0
-1, 1
F
1, 0
1, -1
Player 2 is strong (ñ)
Player 2 is weak (1-ñ)
Player 2 is strong (ñ)
After writing all the strategies and payoffs in the same matrix, find the Bayesian Nash equilibriums, depending on the value of ñ (ñ ââ°Â¤ 1/2 or ñ ââ°Â¥1/2).
