**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

- Martin Osborne, ``What is game theory?''
- Perloff, Chapter 13 ``Oligopoly and Monopolistic Competition''
- Quizzes
- Applications

- Stiglitz and Walsh, Chapter 19 ``Strategic Behavior''
- Narrated lecture with graphs
- Quizzes
- FAQs and pitfalls

- Krugman and Wells, Chapter 15 “Oligopoly”
- Chapter text
^{3} - Animated graphs

- Chapter text

# More information

If you want to learn more before taking Econ 402…

- Ben Polak's course is free online, complete with lecture videos
- Mike Shor's MBA course has free, entertaining MBA-level lecture notes and readings. He also has a thorough game theory website called gametheory.net.