Game Theory - Pure/Mixed Strategies - Nash Equilibrium Practice question-- can y

ID: 1189062 • Letter: G

Question

Game Theory - Pure/Mixed Strategies - Nash Equilibrium

Practice question-- can you explain?

Each of three players is deciding between the pure strategiesLeft and Right in a simultaneous move game. If all players choose Left each one receives a payoff of 1. If all players choose Right, each receives a payoff of 4. In all other possibilities all players receive a payoff of 0. Find all pure Nash equilibria of the game. Can you find a mixed strategy Nash equilibrium such that exactly one player play mixed strategy? If yes, what is it? Can you find a mixed strategy Nash equilibrium such that exactly two players play mixed strategy? If yes, what is it? Can you find a mixed strategy Nash equilibrium such that all three players play mixed strategy? If yes, what is it?

Explanation / Answer

A) There are two pure Nash equilibria of this game. The first is where all are choosing left, and the second where all are choosing right.

B) Let p1 be the probability that player 1 chooses left, p2 be the probability that player 2 chooses left, and p3 be the probability that player 3 chooses left.

Strategy profile
(player 1, player 2, player 3)

probability

A’s payoff

B’s payoff

C’s payoff

LLL

p1p2p3

LLR

p1p2 (1p3)

LRL

p1(1p2)p3

LRR

p1(1p2) (1p3)

RLL

(1p1)p2p3

RLR

(1p1)p2(1p3)

RRL

(1p2) (1p2)p3

RRR

(1p1) (1p2) (1p2)

Expected payoff of A = 1p1p2p3 + 0 + 0 + 0 + 0 + 0 + 0 + 4 (1p1) (1p2) (1p2)

= p1p2p3 + 4 (1p1) (1p2) (1p2)

Similarly,

Expected payoff of B = p1p2p3 + 4 (1p1) (1p2) (1p2)

Expected payoff of C = p1p2p3 + 4 (1p1) (1p2) (1p2)

Suppose only A is playing the mixed strategy game, that is, each of p2 and p3 is either 0 or 1.

From this we conclude the following:

When player 2 and player 3 choose left, then p2 = p3 = 1. Therefore, the expected payoff of A for choosing left is 1 and for choosing right is 0.

When player 2 and player 3 choose right, then p2 = p3 = 0. Therefore, the expected payoff of A for choosing left is 0 and for choosing right is 4.

When player 2 chooses right and player 3 choose left, then p2 = 0, p3 = 1. Therefore, the expected payoff of A for choosing left is 0 and for choosing right is 0.

When player 2 chooses left and player 3 choose right, then p2 = 1, p3 = 0. Therefore, the expected payoff of A for choosing left is 0 and for choosing right is 0.

The expected payoff matches at the last two cases but it does not take any positive value there. This means that there does not exist any mixed strategy Nash equilibrium such that exactly one player play mixed strategy.