Game Theory

Perloff 13.2, 13.5; WB 17, 18.
We restrict attention to games where each player chooses between two actions.

The essays by Osborne, linked below, are recommended reading.

Normal-form games

There are two players, 1 and 2, who can each choose from two actions. Their payoffs depend on which actions are chosen.

A pure strategy is a decision by the player to take one of the actions. A mixed strategy is a decision by the player to take each action with some probability.1

A pair of strategies, one for each player, is a Nash equilibrium (NE) if neither player is better off switching to another strategy. Nash showed that there is always an NE, though it may be in mixed strategies (an MSNE).

Finding the NEs

For any strategy of the other player, find the pure strategies that are best responses to that strategy — give the highest payoff. Any mixed strategies built from these best responses are also best responses. Wherever two strategies are best responses to each other, there is a NE. It is convenient to graph best response functions and see where they intersect.

Extensive-form games

The situation is as above, but player 1 moves first. A node is a point where an action must be chosen. A path is a sequence of actions that could be played.

A (behavioral) strategy is a decision by the player to take each action with some probability at each of the player's nodes.

NE is defined in the same way. Sometimes NE strategies in extensive-form games involve “unreasonable play off the equilibrium path.” Instead, we use subgame-perfect NE (SPNE) which requires that actions at every node be a best response to strategies played for “later” nodes.

Finding the SPNEs

Backward induction. Find what the second player wants to do at each node; collapse the payoffs; and then solve the first player's problem.

Application to oligopoly

The high-low output games discussed in class are two-action simplifications of the Cournot and Stackelberg models of oligopoly. Our oligopoly models do not fit the description above — there are an infinite number of actions (output choices) for each firm — but the methods of best responses and backward induction still work.2

For entry games, we revert to simple two-action games that are analyzed above (though it is implied that payoffs come from Cournot play).

More resources

More information

If you want to learn more before taking Econ 402


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