Game Theory Outline

The following game is for (1) (10 points)

Write down Nash equilibrium outcome(s) (pure strategy) in the above (first-stage) game.

The following game is for (2) – (4) (20 points)

If the game were transformed into a sequential-move game, would you expect that game to exhibit a first-mover advantage, a second-mover advantage, or neither? Explain your reason.
Now consider the possibility of allowing the players with mixed strategies. Suppose that Evert plays DL with probability p and Navratilova plays DL with probability q. Find a Nash equilibrium in mixed strategies for this game.
Write the expected payoffs of the two players in the mixed-strategy equilibrium? Suppose that the order of the payoffs is (Evert, Navratilova)

The following game is for (5) to (7) (20 points)

In the mixed-strategy equilibrium, what is the value of q?
When q = 0.8, what is Evert’s best response?
In the mixed-strategy Nash equilibrium, which pure strategy will not be used by Evert?

The following game is for (8) to (10) (15 points)

This game has an equilibrium with full mixture of all strategies for both players. Suppose that the Kicker plays Left with probability pL and Right with probability pR. The Goalie defends Left with probability qL and Right with probability qR. What is the Kicker’s expected payoffs against the Goalie’s pure strategy of Right?
What is the Goalie’s expected payoffs against the Kicker’s pure strategy of Center? (Note that this is a constant-sum game, in which the sum of the Kicker’s and Goalie’s payoffs equals 100)
Write down equations which can be used to solve the mixed-strategy equilibrium values of pL and pR?

The following game is for (11) to (13) (20 points)

Suppose that Harry chooses Starbucks with probability p, what is Sally’s expected payoff if she chooses Local Latte?
Find Nash equilibrium(s) with pure strategies in the above game?
What are the expected payoffs for these two players in the mixed-strategy Nash equilibrium?

The following game is for (14) to (16) (15 points)

Consider a pricing game between two restaurants, x and y. Each restaurant needs to set prices simultaneously. Their total customers are determined as

Qx = 67 – 2Px + Py

Qy = 67 – 2Py + Px

Suppose that each meal costs the restaurant $4.