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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
- Where does this lead us?
- We’ve defined the “game” but not the outcome

Equilibrium

- 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

1970

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

- 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?

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

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

- Best reply for Reynolds:
- If Philip Morris advertises:
- If Philip Morris does not advertise:

Dominance

- 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

Dominance

- 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

Dominance

- 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

Dominance

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

Optimal

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

Equilibrium

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

- After the 1970 agreement:
- Cigarette advertising decreased by $63 million
- Industry Profits rose by $91 million

Prisoner’s Dilemma

- The dominant strategy will be played

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)

- 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

- To whom do you tender your shares?

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

“The biggest, looniest deal ever. ”

– Fortune Magazine, July 1988

on Campeau’s acquisition of Federated Stores

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

- 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

- 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

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

- 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

Equilibrium

- 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.

Equilibrium Illustration

The Lockhorns

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

- 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

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

- Can decreasing others’ added value increase our profits?
- Can decreasing total industry value increase our profits?

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

Summary

- Games have predictable outcomes
- Notice dominant & dominated strategies
- Select the right game to play
- Looking ahead:
- Sequential Games:

How do games unfold over time?

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