Question

Exercise 3: In class we discussed the Nash Equilibrium solution concept and how it is useful to eliminate non-rational behavior, but with the disadvantage of not necessarily leading to a unique solution. One possible refinement that can be considered is the following: "If more than one Nash Equilibrium exist, but one of them makes every player better off than under the other Nash Equilibria, then surely that is the Nash Equilibrium that will be played." Consider playing the following simultaneous move game, where numbers are in monetary amounts: A 1 (9001,9001) 9000, 0) B (0 ,9000) (9000,9000) a. Obtain the Nash Equilibria and apply the above refinement to determine which of them will be played. b. Would you play according to (a), or does this game point towards a potential weakness in the refinement? Explain.

Answer 1

Solution:

a)

Dominant strategy is a strategy for a player i.e. best response to all strategy profile of another player. If all players in a game have a dominant strategy then we say that Game has dominant strategy equilibrium.

Nash equilibrium is a strategy profile such that for each player given strategy, it is the best response. Nash equilibrium is a set of strategies such that each player is doing their best given the strategy of another player.

If Player 1 (Rows) chooses A then best response of Player 2 is to choose I because the payoff of 9001 is greater than 9000.

If Player 2 (Columns) chooses I then the best response of Player 1 is to choose A because the payoff of 9001 is better than 9000.

Therefore, (A; I) is the Nash Equilibrium

If Player 1 (Rows) chooses B then best response of Player 2 is to choose II because the payoff of 9000 is greater than 0.

If Player 2 (Columns) chooses II then the best response of Player 1 is to choose B because the payoff of 9000 is better than 0.

Therefore, (B; II) is the Nash Equilibrium

After refinement, players have to play with A and I because payoff under this case is greater than the B; II.

b)

No, players will not play according to part (a) i.e. A; I because player is not able to observe the action of another player so both will play by choosing their dominant strategy which states that both will choose B and II.