November 15, 2015

29. Game Theory - Summary - Intermediate Microeconomics - Varian

29. Game Theory

29.1 The Payoff Matrix of a Game
29.2 Nash Equilibrium
29.3 Mixed Strategies
29.4 The Prisoner’s Dilemma
29.5 Repeated Games
29.6 Enforcing a Cartel
29.7 Sequential Games
29.8 A Game of Entry Deterrence

Game theory is concerned with the general analysis of strategic interaction.

In this chapter we discuss the basics of the subject and explore how it works and how it can be used to study economic behavior in oligopolistic markets.

Game Theory
Game theory studies strategic interaction, developed by von Neu-
mann and Morgenstern around 1950

How to depict payoffs of game from different strategies
1. two players
2. two strategies
3. example

Dominant strategy
Each person has a strategy that is best no matter what the
other person does
Nice when it happens, but doesn’t happen that often

Nash equilibrium
1. what if there is no dominant strategy?
2. in this case, look for strategy that is best if the other player
plays his best strategy
3. note the “circularity” of definition
4. appropriate when you are playing against a “rational” oppo-
5. each person is playing the best given his expectations about
the other person’s play and expectations are actually con-
6. example

7. Nash equilibrium in pure strategies may not exist.

8. but if allow mixed strategies (and people only care about
expected payoff), then Nash equilibrium will always exist

Prisoner’s dilemma
1. 2 prisoners, each may confess (and implicate other) or deny
2. gives payoff matrix

3. note that (confess, confess) is unique dominant strategy
equilibrium, but (deny, deny) is Pareto efficient
4. example: cheating in a cartel
5. example: agreeing to get rid of spies
6. problem — no way to communicate and make binding agree-

Repeated games
1. if game is repeated with same players, then there may be
ways to enforce a better solution to prisoner’s dilemma
2. suppose PD is repeated 10 times and people know it
a) then backward induction says it is a dominant strategy to
cheat every round
3. suppose that PD is repeated an indefinite number of times
a) then may pay to cooperate
4. Axelrod’s experiment: tit-for-tat

Example – enforcing cartel and price wars

Sequential game — time of choices matters

I. Example: entry deterrence
1. stay out and fight
2. excess capacity to prevent entry — change payoffs
3. see Figure 29.7.
4. strategic inefficiency

29.1 The Payoff Matrix of a Game

1. A game can be described by indicating the payoffs to each of the players
for each configuration of strategic choices they make.

2. A dominant strategy equilibrium is a set of choices for which each
player’s choices are optimal regardless of what the other players choose.

29.2 Nash Equilibrium

3. A Nash equilibrium is a set of choices for which each player’s choice is
optimal, given the choices of the other players.

29.4 The Prisoner’s Dilemma

4. The prisoner’s dilemma is a particular game in which the Pareto efficient
outcome is strategically dominated by an inefficient outcome.

5. If a prisoner’s dilemma is repeated an indefinite number of times, then
it is possible that the Pareto efficient outcome may result from rational

6. In a sequential game, the time pattern of choices is important. In these
games, it can often be advantageous to find a way to precommit to a
particular line of play.

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