principles of game theory lecture 3: simultaneous move games
TRANSCRIPT
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: