Principles of Game Theory Lecture 3: Simultaneous Move Games.

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Transcript of Principles of Game Theory Lecture 3: Simultaneous Move Games.

Principles of Game Theory

Lecture 3: Simultaneous Move Games

Administrative

• Problem sets due by 5pm• Piazza or ~gasper/GT?

• Quiz 1 is Sunday • Beginning or end of class?

• Questions from last time?

Review

• Simultaneous move situations• Backward induction (rollback)• Strategies vs Actions

Normal form games

• Simultaneous move games• Many situations mimic situations of 2+ people

acting at the same time• Even if not exactly, then close enough – any situation

where the player cannot condition on the history of play.

• Referred to as Strategic or Normal form games

• Two components to the game1. The strategies available to each player

2. The payoffs to the players• “Simple” games often represented as a matrix of

payoffs.

Cigarette Advertising example

• 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.

1964

1970

Strategic Interaction:Cigarette Advertising

• Players?• Reynolds and Philips Morris

• Strategies:• Advertise or Not

• Payoffs• Companies’ Profits

• Strategic Landscape• Firm i can earn $50M from customers• Advertising campaign costs i $20M• Advertising takes $30M away from competitor j

Strategic Form Representation

Philip Morris

No Ad Ad

Reynolds No Ad 50 , 50

Ad

PLAYERS

STRATEGIES

PAYOFFS

Strategic Form Representation

Philip Morris

No Ad Ad

Reynolds No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

PLAYERS

STRATEGIESPAYOFFSPAYOFFS

What would you suggest?

• If you were consulting for Reynolds, what would you suggest?

• Think about best responses to PM• If PM advertises?• If PM doesn’t?

Philip Morris

No Ad Ad

Reynolds

No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

Nash Equilibrium

• Equilibrium • Likely outcome of a game when rational

strategic agents interact• Each player is playing his/her best strategy

given the strategy choices of all other players• No player has an incentive to change his or

her strategy unilaterally

Mutual best response.Not necessarily the best outcome for both

players.

Dominance

• A strategy is (strictly/weakly) dominant if it (strictly/weakly) outperforms all other choices no matter what opposing players do.• Strict >• Weak ≥

• Games with dominant strategies are easy to analyze

If you have a dominant strategy, use it.

If your opponent has one, expect her to use it.

Solving using dominance

• Both players have a dominant strategy

• Equilibrium outcome results in lower payoffs for each player• Game of the above form is often called the

“Prisoners’ Dilemma”

Philip Morris

No Ad Ad

Reynolds

No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

Equilibrium

Optimal

Pricing without Dominant Strategies

• Games with dominant strategies are easy to analyze but rarely are we so lucky.

Example:

• Two cafés (café 1 and café 2) compete over the price of coffee: $2, $4, or $5

• Customer base consists of two groups• 6000 Tourists: don’t know anything about the city but

want coffee• 4000 Locals: caffeine addicted but select the cheapest

café

• Cafés offer the same coffee and compete over price • Tourists don’t know the price and ½ go to each café

Café price competition

• Example scenario: • Café 1 charges $4 and café 2 charges $5:• Recall: tourists are dumb and don’t know where to go• Café 1 gets:

• 3000 tourists + 4000 locals = 7K customers * $4 = 28K

• Café 2 gets• 3000 tourists + 0 locals = 3K customers * $5 = 15K

• Draw out the 3x3 payoff matrix given • $2, $4, or $5 price selection (simultaneous selection)• 6K tourists and 4K locals.

Café price competition

Café 2

$2 $4 $5

Café 1

$2 10 , 10 14 , 12 14 , 15

$4 12 , 14 20 , 20 28 , 15

$5 15 , 14 15 , 28 25 , 25

• No dominant strategy

Dominated Strategies

• A player might not have a dominant strategy but may have a dominated strategy• A strategy, s, is dominated if there is some

other strategy that always does better than s.Café 2

$2 $4 $5

Café 1

$2 10 , 10 14 , 12 14 , 15

$4 12 , 14 20 , 20 28 , 15

$5 15 , 14 15 , 28 25 , 25

Dominance solvable

• If the iterative process of removing dominated strategies results in a unique outcome, then we say that the game is dominance solvable.

• We can also use weak dominance to “solve” the game, but be careful

Player 2

Left Right

Player 1 Up 0,0 1,1

Down 1,1 1,1

Weakly Dominated Strategies

Player 2

Left Right

Player 1 Up 0,0 1,1

Down 1,1 1,1

• (Down, Right) is an equilibrium profile

• But so is (Down, Left) and (Up, Right).• Why?

• Recall our notion of equilibrium: No player has an incentive to change his or her strategy unilaterally

Fictitious Play

• Often there are not dominant or dominated strategies.

• In such cases, another method for finding an equilibrium involves iterated “what-if..” fictitious play:

Best Response Analysis

• Similarly you can iterate through each strategy and list the best response for the opponent.

• Then repeat for the other player.

• Mutual best responses are eq

Multiple Equilibria

• We’ve said nothing about there always being a unique equilibrium. Often there isn’t just one:

Equilibrium Selection

• With multiple equilibria we face a very difficult problem of selection:

Equilibrium Selection

• With multiple equilibria we face a very difficult problem of selection:• Imagine Harry had different preferences:

Equilibrium Selection

• With multiple equilibria we face a very difficult problem of selection:• Classic issues of coordination:

No equilibrium in pure strategies

• Nor must there exist an equilibrium in pure strategies• Pure strategies means no randomization

(penalty kicks)• We’ll talk about general existence later

Player 2

Rock Paper Scissors

Player 1

Rock 0,0 -1,1 1,-1

Paper 1,-1 0,0 -1,1

Scissors

-1,1 1,-1 0,0

Multiple players

• While a X b matrixes work fine for two players (with relatively few strategies – a strategies for player 1 and b strategies for player 2), we can have more than two players: a X b X … X z

Homework

• Study for the quiz

• Next time: more mathematical introduction to simultaneous move games • Focus on section 1.2 of Gibbons

Equilibrium Illustration

The Lockhorns: