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### Game Theory

“Loretta’s Driving Because I’m Drinking and I’m Drinking Because She’s Driving”

- The Lockhorns Cartoon

Mike Shor

Lecture 3

Review

- Understand the game you are in
- Note if the rules are flexible
- Anticipate your opponents’ reactions
- Understand the assumptions
- Recognize that not everyone else understands them

Game Theory - Mike Shor

Equilibrium Best Response: Everybody is playing a best response

- Nash Equilibrium:
- A set of strategies, one for each player, such that each player’s strategy is best for her given that all other players are playing their equilibrium strategies

- The best strategy I can play given the strategy choices of all other players

- No incentive to unilaterally change my strategy

Game Theory - Mike Shor

Identifying the Equilibrium Dominance Successive elimination of dominated strategies Cell-by-cell inspection

- Pure strategy equilibrium
- Consider mixed later

- Dominance solvable
- Only one dominant strategy

Game Theory - Mike Shor

Cigarette Advertising on TV Cigarette companies’ reaction Companies strike agreement

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

- Fear of potential liability lawsuits

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

1964

1970

Game Theory - Mike Shor

Strategic Interactions

- Players: Reynolds and Philip Morris
- Strategies: { Advertise , Do Not Advertise }
- Payoffs: Companies’ Profits
- 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?

Game Theory - Mike Shor

Normal Form Regardless of what you think Philip Morris will do

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

Advertise!

Game Theory - Mike Shor

Dominant Strategy

A strategy that outperforms all other choices no matter what opposing players do

- Firm 1’s strategies: { A, B, C }
- Firm 2’s strategies: { X, Y, Z }
- C is strictly dominant for Firm 1 if:
- P(C,X)>P(A,X) P(C,X)>P(B,X)
- P(C,Y)>P(A,Y) P(C,Y)>P(B,Y)
- P(C,Z)>P(A,Z) P(C,Z)>P(B,Z)

- C is weakly dominant for Firm 1 if:
- Some inequalities are weak (), at least one is strong(>)

Game Theory - Mike Shor

Dominance Solvable

- If each player has a dominant strategy, the game is dominance solvable
- What is the equilibrium of the cigarette advertising game?

COMMANDMENT

If you have a dominant strategy, use it.

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

Game Theory - Mike Shor

Cigarette Advertising

- After the 1970 agreement, cigarette advertising decreased by $63 million
- Profits rose by $91 million
- Prisoner’s Dilemma
- An equilibrium is NOT necessarily efficient
- What if the game is not dominance solvable?

Game Theory - Mike Shor

A Strategic Situation Neither knows which story the other magazine will choose to run

Two firms competing over sales

- Time and The Economist must decide upon the cover story to run some week.
- The big stories of the week are:
- A presidential scandal (labeled S), and
- A proposal to deploy US forces to Grenada (G)

Game Theory - Mike Shor

One Dominant Strategy

- Who has a dominant strategy?
- Assume it will be played!
- Other player can plan accordingly.

Game Theory - Mike Shor

Dominated Strategies

- For The Economist: G dominant = S dominated
- Dominated Strategy:
- There exists another strategy which always does better regardless of opponents’ actions

Game Theory - Mike Shor

Successive Elimination of Dominated Strategies

- If a strategy is dominated, eliminate it
- The size and complexity of the game is reduced
- Eliminate any dominant strategies from the reduced game
- Continue doing so successively

Game Theory - Mike Shor

Example: Tourists & Natives

- Two bars (bar 1, bar 2) compete
- Can charge price of $2, $4, or $5
- 6000 tourists pick a bar randomly
- 4000 natives select the lowest price bar

Bar 2

Game Theory - Mike Shor

Successive Elimination of Dominated Strategies What is the equilibrium?

- Does any player have a dominant strategy?
- Does any player have a dominated strategy?
- Eliminate the dominated strategies
- Reduce the normal-form game
- Iterate the above procedure

Game Theory - Mike Shor

Successive Elimination of Dominated Strategies

Bar 2

$2

$4

$5

$2

10

,

,

10

14

,

,

12

14

,

,

15

Bar 1

Bar 1

$4

20

,

,

20

28

,

,

15

12

,

,

14

$5

15

,

,

28

25

,

,

25

15

,

,

14

Game Theory - Mike Shor

No Dominated Strategies There may be multiple equilibria Method: Ask:

- Often there are no dominated strategies
- Or: reducing the game is not sufficient

Cell-by-cell inspection

Is each player playing the best response to the other player?

Game Theory - Mike Shor

Games of Assurance

- Two firms each earning $45,000
- Both can invest the $45,000 into R&D
- R&D successful only if both invest
- If R&D successful, each earns $95,000

Firm 2

Game Theory - Mike Shor

Cell-by-cell Inspection

- Consider { Invest , Don’t }
- Both players have an incentive to change their strategy: NOT an equilibrium

Firm 2

Game Theory - Mike Shor

Assurance Outcomes However, investing is risky How to achieve assurance?

- Two equilibria exist
- Both firms prefer (I ,I) to (D,D)
- Payoffs of 50 to each firm instead of 45

- Must have assurances

- Strategic moves: commit to choosing I
- Sequential moves: leader chooses the equilibrium

Game Theory - Mike Shor

Games of Coordination

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

Firm 2

Game Theory - Mike Shor

Coordination Outcomes

- Two equilibria exist
- Firms prefer different equilibria
- How to achieve the most desirable outcome for you?
- Strategic moves: commit to choosing A
- Sequential moves: leader chooses the equilibrium

Game Theory - Mike Shor

Summary

- You must put yourself in your rival’s shoes
- Recognize dominant and dominated strategies
- Anticipate that your opponent will recognize them as well

Game Theory - Mike Shor

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