# Best Responses and Safety in Numbers

We know that a Nash equilibrium is a set of strategies, one for each player, such that no player has incentive to change his or her strategy given what the other players are doing. How can we find Nash equilibria in complicated games with many strategies? This lesson introduces a simple algorithm to find all of a game’s pure strategy Nash equilibria. All it requires is some time to go through and mark all of a player’s best responses.

Takeaway Points

1. Review: A player’s best response is the strategy (or strategies) that generate the greatest payoff for him or her given what the other players are doing.
2. In larger games, it may prove helpful to mark best responses with asterisks (*) in the payoff matrix.
3. Best responses allow for indifference. For example, if the best payoff a player can earn in response to a particular opposing strategy is 0, then all instances of 0 receive the asterisk.
4. After doing this for all strategies, if all the payoffs in a particular cell have an asterisk next to them, then that strategy profile is a pure strategy Nash equilibrium.
5. Any outcome that does not have asterisks for all of its payoffs are not equilibria.

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