Question 1: Mixed strategies
Consider the following two normal-form games, Game A and Game B:
Game A
Player 2 Left [q] Right [1-q] 666,777 555,222 222, 555 777,666
Top [r] Player 1 Bottom [1 - r]
Game B
Player 2 Left [q] Right [1-q] 1,1 0,0 0,0 0,0
Top [r] Player 1 Bottom [1 - r]
(a) For each one of the two games, state, using math notation, each player's (mixed-strategy) best-response correspondence. You should use the r and q notation indicated in the game matrices.
(b) For each one of the two games, draw the graphs of the best-response correspondences that you derived under (a) in a diagram with q on the horizontal axis and r on the vertical axis. With the help of this diagram, identify the full set of Nash equilibria. What are the Nash equilibria?
Hint. If you failed to derive the best-response correspondences under (a), you are allowed to derive the Nash equilibria using other techniques (doing this, in your private notes, may be a good idea regardless, as a way of checking the result you obtained with the help of the best-response correspondences).
A Nash equilibrium is said to be strict if each player has a unique best-response to the other players' equilibrium strategies. That is, in a two-player normal-form game (using the notation defined in the set of slides L4), a Nash equilibrium (p1**,p2**) is strict if both of the following two conditions hold:
v1(p1**,p2**)>v1(p1,p2**) for all feasible p1!=p1**,
v2(p1**,p2**)>v2(p1**,p2) for all feasible p2!=p2**.
(c) Can a non-degenerate mixed-strategy Nash equilibrium be strict? ("Non-degenerate" means that the player does not play one pure strategy with probability one.) Explain why or why not.