Question

Problem 1: Consider the following simultaneous move game with two players, denoted by 1 and 2: 1 2 T B L 1,0 0,2 M R 0,1 5,0 2,1 1,0 1. Is there a strategy for any of the players which a player would never choose? 2. If there is a strategy which a player never chooses (it is called, a dominated strategy), and this fact is known among the players, find the equilibria of the game. Hint: In a mixed strategy equilibrium, think about how probabilities should be assigned if there exists a strategy which a player never chooses.

Answer 1

1) When player 1 chooses T, it is best for player 2 to choose M. When player 1 chooses B, it is best for player 2 to choose L . Hence player 2 never chooses R. R is a dominated strategy.

2) Since R is never chosen, it will not be a part of equilibrium strategy, we can assign probability =0 to R .

Now , we need to find out the Best response of the two players to find out the MSNE.

Assume player 1 randomizes by assuming that player 2 chooses L with probability ' q' and M with probability ' 1-q '.

For player 1 to be indifferent between between T and B, the expected payoff from two strategies is equal.

Payoff from T = Payoff from B ===> q(1) + (1-q)(0) = (0)(q) + 2(1-q)

Hence q = 2/3

Similarly , assuming that the player 2 randomizes , he assumes that player 1 plays T with startegy ' p ' and strategy B with a probability ' 1-p '. For player 2 to be indifferent between choosing L and M , the expected pay from T = expected payoff from B.

2(1-p) + (0)(p) = (1)(p) + (1)(1-p) ===> p=1/2.

Any points beyond the point of indifference of the two players will make the player prefer one strategy over other.

The Best response Function of the two players hence become :

q> 2/3 BR, EST BOUT q=23 qc 43 BR₂ as L LORM p <Y2 p=Y2 m p>/

We can plot the above best response function to find the intersection of the two. To plot we have q on vertical axis and p on horizontal axis. When q >2/3 , plyer 1 chooses T . To plot this we take choosing T as choosing p=1 and plot it on the graph. And when q< 2/3 , he chooses B which means p= 0. For q = 2/3, p can take any value as the player is indifferent between choosing T or B.

T.

From the graph, we see that the only point of intersection is at E. Which means both the players randomize and player 1 plays with probability 1/2 and 1/2 and player 2 plays with probability 2/3 and 1/3 for L and M respectively. There is no pure strategy Nash Equilibrium.