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Game Theory. Topic 2 Simultaneous Games. “ Loretta ’ s driving because I ’ m drinking and I ’ m drinking because she ’ s driving. ”. - The Lockhorns. Review. Understanding the game Noting if the rules are flexible Anticipating our opponents ’ reactions Thinking one step ahead

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game theory

Game Theory

Topic 2

Simultaneous Games

“Loretta’s driving because I’m drinking and I’m drinking because she’s driving.”

- The Lockhorns

  • Understanding the game
  • Noting if the rules are flexible
  • Anticipating our opponents’ reactions
  • Thinking one step ahead
  • Where does this lead us?
    • We’ve defined the “game” but not the outcome
  • The likely outcome of a game when rational, strategic agents interact
    • Each player is playing his or her best strategy given the strategy choices of all other players
    • No player has incentive to change his or her action unilaterally
  • Outline:
    • Model interactions as games
    • Identify the equilibria
    • Decide when they are likely to occur
cigarette advertising on tv



Cigarette Advertising on TV
  • All US tobacco companies advertised heavily on TV
  • Surgeon General issues official warning
      • Cigarette smoking may be hazardous
  • Cigarette companies fear lawsuits
      • Government may recover healthcare costs
  • Companies strike agreement
      • Carry the warning label and cease TV advertising in exchange for immunity from federal lawsuits.
strategic interaction
Strategic Interaction
  • Players: Reynolds and Philip Morris
  • Strategies: Advertise or Not Advertise
  • Payoffs: Companies’ Profits
  • Environment:
    • Each firm earns $50 million from its customers
    • Advertising costs a firm $20 million
    • Advertising captures $30 million from competitor
  • How to represent this game?
what to do
What to Do?

If you are advising Reynolds, what strategy do you recommend?

best replies
Best Replies
  • A strategy is a best reply to some opponents’ strategy if it does at least as well as any other strategy
  • si is a best replyto s-i if

for every si’

solving the game
Solving the Game
  • Best reply for Reynolds:
      • If Philip Morris advertises:
      • If Philip Morris does not advertise:
  • A strategy is dominantif it outperforms all other strategies no matter what opposing players do
  • Games with dominant strategies are easy to solve
    • No need for “what if …” thinking
  • sistrictly dominates si’ if

for every s-i

(the payoff is strictly higher for any strategies of the other players)

  • siweakly dominates si’ if

for every s-i, and

for some s-i

  • A strategy si is strictly dominatedif some strategy si’ strictly dominates it
  • A strategy si is the dominant strategy if it strictly dominates all other strategies
  • A strategy si is weakly dominated if some strategy si’ weakly dominates it
  • A strategy si is the weakly dominant strategy if it weakly dominates all others

If you have a dominant strategy, and no ability to agree on an alternate course of action, use it

If your opponent has a dominant strategy, and no ability to agree on an alternate course of action, then expect her to play it

prisoner s dilemma
Prisoner’s Dilemma


  • Both players have a dominant strategy
  • The equilibrium results in lower payoffs for each player


prisoner s dilemma1
Prisoner’s Dilemma
  • Both players have a dominant strategy (s1,s1)

u11 > u21u12 > u22

  • The equilibrium results in lower payoffs for each player

u22 > u11

  • The above two statements imply:

u12 > u22 >u11 > u21

cigarette advertising
Cigarette Advertising
  • After the 1970 agreement:
    • Cigarette advertising decreased by $63 million
    • Industry Profits rose by $91 million
prisoner s dilemma2
Prisoner’s Dilemma
  • The dominant strategy will be played
how to win a bidding war by bidding less
How to Win a Bidding War by Bidding Less?
  • The battle for Federated (1988)
      • Parent of Bloomingdales
  • Current share price ≈ $60
  • Expected post-takeover share price ≈ $60
  • Macy’s offers $70/share
    • contingent on receiving 50% of the shares
  • Do you tender your shares to Macy’s?
how to win a bidding war continued
How to Win a Bidding War (continued)
  • Robert Campeau bids $74 per share not contingent on amount acquired
  • Campeau’s Mixed Scheme:
    • If less than 50% tender their shares, each receives:

$74 per share

    • If more than 50% tender their shares, (if X% tender), each receives:
the federated game
The Federated Game
  • To whom do you tender your shares?
how to win a bidding war
How to Win a Bidding War
  • Each player has a dominant strategy: Tender shares to Campeau
  • Resulting Price:

(½ x 74) + (½ x 60) = $67

  • BUT: Macy’s offered $70 !
dominant strategies
Dominant Strategies

“The biggest, looniest deal ever. ”

– Fortune Magazine, July 1988

on Campeau’s acquisition of Federated Stores

prisoner s dilemma examples
Prisoner’s Dilemma Examples
  • Pricing by Firms
    • High or low prices?
    • Value menus and loyalty programs
  • Divorce
    • Hire attorneys or proceed amicably?
  • Nuclear Weapons
    • Build or don’t build weapons?
  • State governments
    • Inducements to attract business to a state
dominated strategies
Dominated Strategies
  • Two restaurants compete
    • Can charge price of $30, $50, or $60
  • Customer base consists of tourists and natives
    • 600 tourists pick randomly
    • 400 natives select the lowest price
  • Marginal costs are $10
tourists natives
Tourists & Natives
  • Example scenario:
    • Restaurant 1: $50, Restaurant 2: $60
    • Restaurant 1 gets:

300 tourists + 400 natives

= 700 customers x ($50-$10) = $28K

    • Restaurant 2 gets:

300 tourists + 0 natives

= 300 customers x ($60-$10) = $15K

tourists natives1
Tourists & Natives

in thousands of dollars

R. 2

iterated deletion of dominated strategies
Iterated Deletion of Dominated Strategies
  • Does any player have a dominated strategy?
    • Eliminate the strictly dominated strategies
    • Reduce the size of the game
    • Repeat: Iterate the above procedure
no dominated strategies
No Dominated Strategies
  • Often there are no dominated strategies
  • Some games may have multiple equilibria
  • Equilibrium selection becomes an issue
  • Method:

For each player, find the best response to every strategy of the other player

  • An outcome in which every player is playing a best response to the strategies of all other players.
  • An equilibriumis a strategy profile s such that si is a best reply to s-i for all i.
games of coordination
Games of Coordination
  • Complements & technology adoption
    • Two complementing firms
    • Must use same technology, but each firm has a preferred technology
    • Equilibrium does not offer a unique prediction
    • Commit (or go first) to win!

Firm 2

games of assurance
Games of Assurance
  • Joint research ventures
    • Each firm may invest $50,000 into an R&D project
    • Project succeeds only if both invest
    • If successful, each nets $75,000

Firm 2

games of chicken
Games of Chicken
  • Entry into small markets

Firm 2

the right game to play
The Right Game to Play
  • Why do we “solve” games?
  • To know which one to play!
    • How do internal corporate changes impact the outcome of strategic interaction?
  • Some games are better than others
capacity constraints
Capacity Constraints
  • Can decreasing others’ added value increase our profits?
  • Can decreasing total industry value increase our profits?
multiple equilibria
Multiple Equilibria
  • What is the predictive power of game theory when there are multiple equilibria?
    • Sometimes nothing ?
    • Refinements
      • Focal points
      • Efficiency
      • Evolutionary stability
      • Fairness
      • Risk dominance
  • Games have predictable outcomes
    • Notice dominant & dominated strategies
  • Select the right game to play
  • Looking ahead:
    • Sequential Games:

How do games unfold over time?