The concepts that we have used so far make sense, but they are a bit too restrictive and limit the set of games for which we may make reasonable predictions. A better approach, in a sense that we should clarify, is Nash equilibrium. Before formalizing the concept, imagine a game in which there is one sensible prediction of the outcome. If the prediction makes sense, all players should find it acceptable, in the sense that they would not be willing to deviate from the prescribed strategy, if no other player deviates as well: The equilibrium should be stable.
DEFINITION 14.3 (Nash equilibrium) An outcome 
is a Nash equilibrium if no player would gain anything by choosing another strategy, provided that the other players do not change their strategy; in other words, no player i has an incentive to deviate from strategy 
. Let us denote
Table 14.5 Finding Nash equilibrium for the game in Table 14.3.
the set of strategies played by all players but i as
This can be used to streamline notation as follows:
Then, for any player i, is the best response to :
The best way to grasp Nash equilibria is by a simple example, in which we also illustrate a possible way to find them for two-players games with discrete sets of actions.
Example 14.8 To find a Nash equilibrium, we can:
- Consider each player in turn, and find her best strategy for each possible strategy of the opponent; for the row player, this means considering each column (strategy of the column player) and find the best row (best response of row player) by marking the corresponding payoff; rows and columns are swapped for the column player.
- Check if there is any cell in which both payoffs are marked; all such cells are Nash equilibria.
Let us apply the idea to the game in Table 14.3; the results are shown in Table 14.5. If we start with row player 1, we see that her best response to left is top, as 1 > 0; then we underline the payoff 1 in the top-left cell; by the same token, player 1 should respond with top to center, and with bottom to right Then we proceed with column player 2; we notice that her best response to top is center, since 2 = max{0, 2,1}; if player 1 plays bottom, player 2 should choose left. In this game, there is no need to break ties, as for each row there is exactly one payoff preferred by the column player, and for each column there is exactly one payoff preferred by the row player. The outcome (top, center) is the only cell with both payoffs marked, and indeed it is a Nash equilibrium. To better grasp what Nash equilibrium is about, note that no player has an incentive to deviate from this outcome, if the other player does not.
We see that the Nash equilibrium in the example is the same outcome that we predicted by iterated elimination of (strongly) dominated strategies. Indeed, it can be proved that:
- If iterated elimination of (strictly) dominated strategies results in a unique outcome, then this is a Nash equilibrium.
- If a Nash equilibrium exists, it is not eliminated by the iterated elimination of (strictly) dominated strategies.
We should note that Nash equilibria may not exist,15 and they need not be unique (see problem 14.1). However, they are a more powerful tool than elimination of dominated strategies, as they may provide us with an answer when the use of dominated strategies fail, without contradicting the prediction of iterated elimination, when this works. In other words, since Nash equilibria are based on less restrictive assumptions than iterated elimination, it fails less often.16 Last but not least, the prediction suggested by Nash equilibria makes sense and sounds plausible enough.
Now it is natural to wonder about more complicated settings, like the case in which each player has a continuum of strategies at her disposal. To this aim, it is useful to interpret Nash equilibria in terms of best response functions.
DEFINITION 14.4 (Best response function) The best response function for each player i is a function
mapping the strategies of the other players into the strategy maximizing the payoff of i in response to s−i.
We immediately see that a Nash equilibrium is a solution of a system of equations defined by the best response functions of all players. For a two-player game, we should essentially solve the following system of two equations
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