Chapter 12
Choices Involving Strategy
McGraw-Hill/Irwin
Copyright © 2008 by The McGraw-Hill Companies, Inc. All Rights Reserved.
Main Topics
What is a game?
Thinking strategically in one-stage
games
Nash equilibrium in one-stage games
Games with multiple stages
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What is a Game?
 A game is a situation in which each member of a group
makes at least one decision, and cares both about his
own choice and about others’ choices
 Includes any situation in which strategy plays a role
 Military planning, dating, auctions, negotiation, oligopoly
 Two types of games:
 One-stage game: each participant makes all choices before
observing any choice by any other player
 Rock-Paper-Scissors, open-outcry auction
 Multiple-stage game: at least one participant observes a
choice by another participant before making some decision of
her own
 Poker, Tic-Tac-Toe, sealed-bid auction
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Figure 12.1: How to Describe a
Game
Essential features of
a one-stage game:
Players
Actions or strategies
Payoffs
Represented in a
simple table
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Thinking Strategically:
Dominant Strategies
Each player in the game knows that her payoff
depends in part on what the other players do
Needs to make a strategic decision, think about her
own choice taking other players’ view into account
A players’ best response is a strategy that
yields her the highest payoff, assuming other
players behave in a specified way
A strategy is dominant if it is a player’s only
best response, regardless of other players’
choices
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The Prisoners’ Dilemma: Scenario
Players: Oskar and Roger, both students
The situation: they have been accused of
cheating on an exam and are being questioned
separately by a disciplinary committee
Available strategies: Squeal, Deny
Payoffs:
If both deny, both suspended for 2 quarters
If both squeal, both suspended for 5 quarters
If one squeals while the other denies, the one who
squeals is suspended for 1 quarter and the one who
denies is suspended for 6 quarters
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Figure 12.3: Best Responses to
the Prisoners’ Dilemma
(a) Oskar’s Best Response
(b) Roger’s Best Response
Roger
Roger
Squeal
Deny
Squeal
-2
-1
-2
-1
-6
-1
-5
-5
-2
-6
-6
Squeal
-6
Oskar
-2
Squeal
Oskar
Deny
Deny
Deny
-1
-5
-5
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Thinking Strategically: Iterative
Deletion of Dominated Strategies
 Even if the strategy to choose is not obvious, can
sometimes identify strategies a player will not choose
 A strategy is dominated if there is some other strategy
that yields a strictly higher payoff regardless of others’
choices
 No sane player will select a dominated strategy
 Dominated strategies are irrelevant and can be
removed from the game to form a simpler game
 Look again for dominated strategies, repeat until there
are no dominated strategies left to remove
 Sometimes allows us to solve games even when no
player has a dominant strategy
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Nash Equilibrium in
One-Stage Games
 Concept created by mathematician John Nash,
published in 1950, awarded Nobel Prize
 Has become one of the most central and important
concepts in microeconomics
 In a Nash equilibrium, the strategy played by each
individual is a best response to the strategies played
by everyone else
 Everyone correctly anticipates what everyone else will do and
then chooses the best available alternative
 Combination of strategies in a Nash equilibrium is stable
 A Nash equilibrium is a self-enforcing agreement:
every party to it has an incentive to abide by it,
assuming that others do the same
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Figure 12.8: Nash Equilibrium in
the Prisoners’ Dilemma
Roger
Deny
Squeal
-2
-1
Deny
-2
Oskar
-6
-6
-5
Squeal
-1
-5
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Nash Equilibria in Games with
Finely Divisible Choices
Concept of Nash equilibrium also applies to
strategic decisions that involve finely divisible
quantities
Determine each player’s best response
function
A best response function shows the
relationship between one player’s choice and
the other’s best response
A pair of choices is a Nash equilibrium if it
satisfies both response functions
simultaneously
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Figure 12.10: Free Riding in
Groups
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Mixed Strategies
 When a player chooses a strategy without randomizing
he is playing a pure strategy
 Some games have no Nash equilibrium in pure
strategies, in these cases look for equilibria in which
players introduce randomness
 A player employs a mixed strategy when he uses a
rule to randomize over the choice of a strategy
 Virtually all games have mixed strategy equilibria
 In a mixed strategy equilibrium, players choose
mixed strategies and the strategy each chooses is a
best response to the others players’ chosen strategies
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Games with Multiple Stages
In most strategic settings events unfold over
time
Actions can provoke responses
These are games with multiple stages
In a game with perfect information, players
make their choices one at a time and nothing is
hidden from any player
Multi-stage games of perfect information are
described using tree diagrams
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Figure 12.13: Lopsided Battle of
the Sexes
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Thinking Strategically:
Backward Induction
 To solve a game with perfect information
 Player should reason in reverse, start at the end of the tree
diagram and work back to the beginning
 An early mover can figure out how a late mover will react, then
identify his own best choice
 Backward induction is the process of solving a
strategic problem by reasoning in reverse
 A strategy is one player’s plan for playing a game, for
every situation that might come up during the course of
play
 Can always find a Nash equilibrium in a multi-stage
game of perfect information by using backward
induction
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Cooperation in Repeated Games
Cooperation can be sustained by the threat of
punishment for bad behavior or the promise of
reward for good behavior
Threats and promises have to be credible
A repeated game is formed by playing a
simpler game many times in succession
May be repeated a fixed number of times or
indefinitely
Repeated games allow players to reward or
punish each other for past choices
Repeated games can foster cooperation
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Figure 12.16: The Spouses’
Dilemma
 Marge and Homer
simultaneously choose
whether to clean the
house or loaf
 Both prefer loafing to
cleaning, regardless of
what the other chooses
 They are better off if
both clean than if both
loaf
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Repeated Games: Equilibrium
Without Cooperation
When a one-stage game is repeated, the
equilibrium of the one-stage game is one Nash
equilibrium of the repeated game
Examples: both players loafing in the Spouses’
dilemma, both players squealing in the Prisoners’
dilemma
If either game is finitely repeated, the only
Nash equilibrium is the same as the one-stage
Nash equilibrium
Any definite stopping point causes cooperation
to unravel
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Repeated Games: Equilibria With
Cooperation
If the repeated game has no fixed stopping
point, cooperation is possible
One way to achieve this is through both
players using grim strategies
With grim strategies, the punishment for
selfish behavior is permanent
Credible threat of permanent punishment for
non-cooperative behavior can be strong
enough incentive to foster cooperation
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