Game Theory

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

# 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).