Question

Find a value of x such that the following game has a unique Nash equilibrium:

Left Right

Up 2,4 1,6

Down x,8 3,7

Answer 1

Answer

A game will have unique Nash equilibrium if at least one of the player has a unique dominant strategy i.e. a player has a strategy which is independent of what other player is choosing.

We can see from above table that If Player 1 chooses Up, then the player 2 will choose Right because player 2 will get higher payoff if he chooses right. We can also see from above table that If Player 1 chooses Down, then the player 2 will choose Left because player 2 will get higher payoff if he chooses Left.

Hence If player 1 chooses Up, then player 2 is choosing right and if player 1 is choosing down then player 2 is choosing left and hence, player 2's strategy is depending on Player 1 strategy and hence, Player 2 is not having any dominant strategy.

Thus, In order to have unique Nash equilibrium Player 1 must have a dominant strategy.

We can see from above table that If Player 2 chooses Right, then the player 1 will choose Down because player 1 will get higher payoff if he chooses right. Hence In order to have dominant strategy for Player 1. If If Player 2 chooses Left, then the player 1 should again choose Down. We can also see from above table that If Player 2 chooses Left, then the player 1 will choose Down if x > 2 or x = 2.

Hence, In order to have a unique Nash equilibrium x > 2 (or x can be equal to 2 as well).

Thus Player 1 will always choose Down and f Player 1 chooses Down, then the player 2 will choose Left because player 2 will get higher payoff if he chooses Left.

Hence unique Nash equilibrium will be (Down , Left) and payoff will be (x,8).