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Game Theory

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Game Theory

Mike ShorLecture 3

“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 if they are likely to occur

The Lockhorns

1964

1970

- All US tobacco companies advertised heavily on TV
- Surgeon General issues official warning
- Cigarette smoking may be hazardous

- Government may recover healthcare costs

- Carry the warning label and cease TV advertising in exchange for immunity from federal lawsuits.

- Players:Reynolds and Philip Morris
- Strategies:Advertise or Not Advertise
- Payoffs:Companies’ Profits
- Strategic Landscape:
- 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?

PAYOFFS

PLAYERS

STRATEGIES

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

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

- A strategy is dominantif it outperforms all other choices no matter what opposing players do
- Games with dominant strategies are easy to play
- No need for “what if …” thinking

- Strict Dominance:
Advertise is strictly dominant forReynoldsif:

- Profit (Ad , Ad) > Profit (No , Ad)
- Profit (Ad , No) > Profit (No , No)

- Weak Dominance:
Advertise is weakly dominant if:

- Some inequalities are weak (),
- At least one is strong(>)

- By “dominant” we will mean “strictly dominant”

COMMANDMENT

If you have a dominant strategy, use it.

Expect your opponent to use her dominant strategy if she has one.

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

Optimal

Equilibrium

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

- Prisoner’s Dilemma
- An equilibrium is NOT necessarily efficient
- Players can be forced to accept mutually bad outcomes
- Bad to be playing a prisoner’s dilemma, but good to make others play

- The battle for Federated (1988)
- Parent of Bloomingdales

- contingent on receiving 50% of the shares

- 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:

- If less than 50% tender their shares, each receives:

- To whom do you tender your shares?

- Each player has a dominant strategy: Tender shares to Campeau
- Resulting Price:
(½ x 74) + (½ x 60) = $67

- BUT: Macy’s offered $70 !

“The biggest, looniest deal ever. ”

– Fortune Magazine, July 1988

on Campeau’s acquisition of Federated Stores

- What if players do not have dominant strategies?

- Two bars (bar 1, bar 2) compete
- Can charge price of $2, $4, or $5

- Customer base consists of tourists and natives
- 6,000 tourists pick a bar randomly
- 4,000 natives select the lowest price bar

- Marginal costs are close to zero

- Example scenario:
- Bar 1 charges $4, Bar 2 charges $5
- Bar 1 gets:
3,000 tourists + 4,000 natives

= 7,000 customers x $4 = $28K

- Bar 2 gets:
3,000 tourists + 0 natives

= 3,000 customers x $5= $15K

in thousands of dollars

Bar 2

- Does any player have a dominant strategy?
- Does any player have a dominated strategy?
- A strategy is dominated if there is some other strategy which always does better
- Eliminate the dominated strategies
- Reduce the size of the game
- Iterate the above procedure

- A strategy is dominated if there is some other strategy which always does better
- What is the equilibrium?

Bar 2

CAVEAT

Expect your opponent to use her dominant strategy if she has one.

BUT

Be sure you understand your opponents’ true payoffs.

(Do you know what really motivates them?)

- 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

- Games of Coordination
- Games of Assurance
- Games of Chicken

- Joint ventures and supplier choice
- Two firms engaged in joint venture
- Must use the same supplier, but each firm has a preferred supplier

Firm 2

- Solving:

Firm 2

- 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

- Entry into small markets

Firm 2

- 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

- Your added value =
the size of the pie when you’re in the game

minus

the size of the pie when you are not

- Added value limits how much you can get
- You cannot receive much more than your added value

- You should receive close to your added value

- You can increase your payoffs by increasing your added value OR decreasing the added value of other players.

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

- Games have predictable outcomes
- Notice dominant & dominated strategies

- Select the right game to play
- Seemingly internal corporate changes can impact the outcome of strategic interaction

- Looking ahead:
- Sequential Games:
How do games unfold over time?

- Sequential Games: