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

gamesNash equilibrium in one-stage gamesGames 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 playerRock-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:PlayersActions or strategiesPayoffs

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 doNeeds to make a strategic decision, think about her

own choice taking other players’ view into accountA 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 studentsThe situation: they have been accused of

cheating on an exam and are being questioned separately by a disciplinary committee

Available strategies: Squeal, DenyPayoffs:

If both deny, both suspended for 2 quartersIf both squeal, both suspended for 5 quartersIf 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

Roger

Deny Squeal

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-2

-1

-6

-6

-1

-5

-5

Osk

ar

De

ny

Sq

uea

l

(a) Oskar’s Best Response

Roger

Deny Squeal

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-2

-1

-6

-6

-1

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Osk

ar

De

ny

Sq

uea

l

(b) Roger’s Best Response

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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 inOne-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

Oskar

Deny

-2

-2

-1

-6

Squeal

-6

-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 timeActions can provoke responsesThese 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 behaviorThreats and promises have to be credible

A repeated game is formed by playing a simpler game many times in successionMay be repeated a fixed number of times or

indefinitelyRepeated games allow players to reward or

punish each other for past choicesRepeated 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 gameExamples: 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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