Chapter 12

Choices Involving Strategy

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 (or, each “player”) makes at least one decision, and

• Each player’s welfare depends on others’ choices as well as his own choice

• A game• Includes any situation in which strategy plays a role• Military planning, dating, auctions, negotiation, oligopoly

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One-stage and multiple-stage games

• Two types of games:• One-stage game: each player 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

Figure 12.1: How to Describe a one-stage Game

• Essential features of a one-stage game:• Players• Actions or strategies• Payoffs

• Represented in a simple table

• The game is called:• Battle of Wits

• From The Princess Bride by S. Morgenstern

• Matching Pennies

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This could be a metaphor for a battle in a war, or for a tennis rally.

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 play specified strategies

• A strategy is a player’s dominant strategy if it is the player’s best response, no matter what strategies are chosen by other players

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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

Roger

Deny Squeal

-2

-2

-1

-6

-6

-1

-5

-5

Osk

ar

De

ny

Sq

uea

l

(a) Oskar’s Best Response

Roger

Deny Squeal

-2

-2

-1

-6

-6

-1

-5

-5

Osk

ar

De

ny

Sq

uea

l

(b) Roger’s Best Response

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Figure 12.4: Best Responses to the Provost’s Nephew

Osk

ar

De

ny

Sq

uea

l

(a) Oskar’s Best Response

Roger

Deny Squeal

-2

0

-1

-6

-6

-1

-5

-5

Osk

ar

De

ny

Sq

uea

l

(b) Roger’s Best Response

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Roger

Deny Squeal

-2

0

-1

-6

-6

-1

-5

-5

Thinking Strategically: Iterative Deletion of Dominated Strategies

• Even if the strategy to choose is not obvious, one 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 this allows us to solve games even when no player has a dominant strategy

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Who will do what?

It may be possible to predict the outcome of a game by means of the iterative deletion of dominated strategies

Betty

Left Center Right

Al

Top 3, 1 2, 3 10, 2

High 4, 5 3, 0 6, 4

Low 2, 2 5, 4 12, 3

Bottom 5, 6 4, 5 9, 7

High is dominated, for Al, by BottomLe

ft is

dom

inat

ed,

for

Bet

ty,

by R

ight

Top is dominated, for Al, by Low

Bottom is dominated, for Al, by Low

Guessing Half the Median

• This is another example of predicting the outcome of a game by means of the iterative deletion of dominated strategies

• Five players (actually, any odd number will do)• Each picks a number up to a given maximum• Each player’s penalty is the difference

between his chosen number and half the median of the players’ chosen numbers

• Prove that each player’s choice is one (1)

Second-Price Sealed-Bid Auctions: truth is the weakly dominant strategy!

• The highest bidder wins the auction

• But pays the second-highest bid

• Why is this auction special?

• Every bidder has a weakly dominant strategy: bid what the object’s value is to you

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

-5 12-15

Tony

Action Romantic

Maria

Action

5

2

-1

-1

Romantic

1

1

2

5

Figure 12.9: The Battle of the Sexes

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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

• To find the Nash Equilibrium:• 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

• Can you find the Nash Equilibrium?

• There is none• if only pure (or, non-

random) strategies are allowed

• But if each player tosses a coin to pick his strategy, these randomized strategies are the Nash Equilibrium of this game.

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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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“Battle of Wits” has a Nash Equilibrium in Mixed Strategies

VizziniLeft Right Mixed (q, 1-q)

Wesley

Left 1, -1 -1, 1 q 1 + (1 - q) (-1),

q (-1) + (1 - q) 1

Right -1, 1 1, -1 q (-1) + (1 - q) 1,

q 1 + (1 - q) (-1)

Mixed (p, 1-p) p 1 + (1 - p) (-1),p (-1) + (1 - p) 1

p (-1) + (1 - p) 1,p 1 + (1 - p) (-1)Given that Vizzini is playing a mixed

strategy, Wesley will also play a mixed strategy only if his two pure strategies, Left and Right, have equal payoffs. That is, q 1 + (1 - q) (-1) = q (-1) + (1 - q) 1. This yields q = 0.5.

Similarly, given that Wesley is playing a mixed strategy, Vizzini will also play a mixed strategy only if his two pure strategies, Left and Right, have equal payoffs. That is, p (-1) + (1 - p) 1 = p 1 + (1 - p) (-1). This yields p = 0.5.

In the Nash equilibrium of this game, both players will play each strategy with a 50% probability.

Weakly-dominated strategies

BettyL R

Al T 6, 10 8, 10B 5, 1 11, 5

There are two Nash equilibria: (T, L) and (B, R). But only the latter does not have a weakly dominated strategy.Therefore, (B, R) is a better prediction.

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

• One can always find a Nash equilibrium in a multi-stage game of perfect information by using backward induction

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l r

L R

Al’s move

Betty’s move

(8, 10)

(5, 1) (11, 5)

Nash Equilibrium, but not Subgame Perfect

Subgame Perfect Nash Equilibrium

A Two-Stage Game• This game has two Nash

equilibria:• Al chooses r and Betty

chooses R, and • Al chooses l and Betty

chooses L.• But only the former is

subgame perfect• In other words, only the

former satisfies backward induction

• The (l, L) equilibrium is based on a non-credible threat by Betty to play L if the opportunity arose.

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

• A credible threat of permanent punishment for non-cooperative behavior can be strong enough incentive to foster cooperation

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The grim strategy may enforce cooperation

Round 1 Round 2 3 4 5 6 …

Marge 2 2 2 2 2 2 …

Homer 2 2 2 2 2 2 …

Round 1 Round 2 3 4 5 6 …

Marge 2 2 0 1 1 1 …

Homer 2 2 3 1 1 1 …

If Marge and Homer both play the grim strategy, their payoffs are:

If Marge plays the grim strategy but Homer decides to loaf in Round 3, their payoffs are:

Assuming Homer cares sufficiently about the future losses that would occur if he decides to loaf, it is a Nash equilibrium of the repeated Spouses’ Dilemma when both Marge and Homer play the grim strategy. Therefore, cooperation is possible in indefinitely repeated games.

Asymmetric Information

• Now the true payoffs are not necessarily known to all players

• Each player knows his own payoffs but not necessarily the payoffs of the other players

• In these games, each player’s decision can reveal some of his information to the other players

• And each player can try to mislead the other players

Winner’s Curse

• A Ford Mustang is offered for sale by second-price auction

• There are three bidders: Melissa, Olivia, and Elvis• There is a 50% chance that the car has a serious

mechanical problem, in which case it is worth $2,000 to all bidders

• And there is a 50% chance that the car is problem-free, in which case it is worth $10,000 to all bidders

• Melissa knows whether or not there is a problem• Elvis and Olivia have no idea; they are willing to pay

$6,000, the expected value of the car• What will happen at the auction?

Winner’s Curse

• If Elvis and Olivia do not know that Melissa knows the true condition of the car, the Nash equilibrium outcome is that Melissa bids the true value of the car and the others each bid $6,000• Winner’s Curse: If either Elvis or Olivia win, they

have overpaid!

• If Elvis and Olivia know that Melissa knows the true condition of the car, the Nash equilibrium outcome is that Melissa bids the true value of the car and the others each bid $2,000• No Winner’s Curse

Reputation