Finding Nash Equilibria
The Best Respone Method
When a game does not have any dominant or dominated strategies, or when the iterated deletion of dominated strategies does not yield a unique outcome, we find equilibria using the best response (also called best reply) method. Note that this method will always find all of the Nash equilibria (in pure strategies—we'll learn about mixed strategies later) even if the game does have dominant or dominated strategies.
Consider a two-player game with five strategies for each player:
| Player 2 | ||||||
|---|---|---|---|---|---|---|
| V | W | X | Y | Z | ||
| Player 1 | A | 9 , 9 | 7 , 1 | 5 , 6 | 3 , 4 | 1 , 1 |
| B | 7 , 8 | 5 , 2 | 3 , 6 | 1 , 4 | 3 , 3 | |
| C | 5 , 6 | 3 , 3 | 1 , 8 | 9 , 7 | 1 , 5 | |
| D | 3 , 9 | 1 , 9 | 9 , 4 | 7 , 9 | 5 , 9 | |
| E | 1 , 2 | 9 , 8 | 7 , 7 | 5 , 6 | 3 , 7 | |
We will start by finding the best responses for Player 1. That is, for every possible strategy of Player 2, we will identify Player 1's best response.
For example, if Player 2 plays V, Player 1 can earn 9 (from strategy A), 7 (from strategy B), 5 (from strategy C), 3 (from strategy D), or 1 (from strategy E). As 9 is his highest possible payoff, we conclude that Player 1's best response to V is to play A.
Underline this highest payoff in the strategic form (the game matrix):
| Player 2 | |||
|---|---|---|---|
| V | W | ||
| Player 1 | A | 9 , 9 | |
| B | |||
Now that we have determined Player 1's best response to Player 2 playing V, we do the same for Player 2's other strategies, underlining the payoff that results from Player 1's best response to each.
| Player 2 | ||||||
|---|---|---|---|---|---|---|
| V | W | X | Y | Z | ||
| Player 1 | A | 9 , 9 | 7 , 1 | 5 , 6 | 3 , 4 | 1 , 1 |
| B | 7 , 8 | 5 , 2 | 3 , 6 | 1 , 4 | 3 , 3 | |
| C | 5 , 6 | 3 , 3 | 1 , 8 | 9 , 7 | 1 , 5 | |
| D | 3 , 9 | 1 , 9 | 9 , 4 | 7 , 9 | 5 , 9 | |
| E | 1 , 2 | 9 , 8 | 7 , 7 | 5 , 6 | 3 , 7 | |
In the above, we have found that:
- If Player 2 plays V, Player 1's best response is A, earning 9
- If Player 2 plays W, Player 1's best response is E, earning 9
- If Player 2 plays X, Player 1's best response is D, earning 9
- If Player 2 plays Y, Player 1's best response is C, earning 9
- If Player 2 plays Z, Player 1's best response is D, earning 5
Next, we do the same thing to identify Player 2's best responses to every strategy of Player 1. For example, if Player 1 plays E, Player 2's, best response is W with a payoff of 8. Other strategies would earn 2 (V), 7 (X), 6 (Y), and 7 (Z).
Sometimes, there is more than one best response, if more than one strategy earns the same highest payoff. For example, if Player 1 plays D, Player 2 can earn 9 (from strategy V), 9 (from strategy W), 4 (from strategy X), 9 (from strategy Y), or 9 (from strategy Z). Thus, the maximum Player 2 can earn is 9, but several strategies can achieve this. In this case, each of those strategies is a best response, and we underline the payoff associated with each of them. Thus, we have:
| Player 2 | ||||||
|---|---|---|---|---|---|---|
| V | W | X | Y | Z | ||
| Player 1 | A | 9 , 9 | 7 , 1 | 5 , 6 | 3 , 4 | 1 , 1 |
| B | 7 , 8 | 5 , 2 | 3 , 6 | 1 , 4 | 3 , 3 | |
| C | 5 , 6 | 3 , 3 | 1 , 8 | 9 , 7 | 1 , 5 | |
| D | 3 , 9 | 1 , 9 | 9 , 4 | 7 , 9 | 5 , 9 | |
| E | 1 , 2 | 9 , 8 | 7 , 7 | 5 , 6 | 3 , 7 | |
Recall that an equilibrium is defined as a strategy for each player so that all players are simultaneously playing their best responses to other players' strategies. This is equivalent to saying that a pair of strategies in the above game is in equilibrium if both payoffs are underlined. In the above, we find three equilibria: (A,V), (E,W), and (D,Z). You should convince yourself that in all three cases, neither player has an incentive to deviate, or change her strategy unilaterally.
A common error
One of the most common errors in the analysis of equilibria in games is confusing the equilibrium strategies with equilibrium payoffs. Consider the following game:
| Player 2 | |||
|---|---|---|---|
| X | Y | ||
| Player 1 | A | 3 , 3 | 1 , 1 |
| B | 2 , 4 | 3 , 3 | |
The above game has a unique equilibrium, which is (A,X). That is, in equilibrium, Player 1 plays A and Player 2 plays X. The equilibrium is not (3,3), which are the payoffs the players earn in equilibrium. To see why this distinction is important, note that (B,Y) also yields a payoff of 3 for each player, but is not an equilibrium. In fact, strategy Y for player 2 is dominated.