Principles of Game Theory

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 game • The strategies available to each player • The payoffs to the players • “Simple” games often represented as a matrix of payoffs.

1964 1970 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.

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

PAYOFFS Strategic Form Representation PLAYERS STRATEGIES

PAYOFFS Strategic Form Representation PLAYERS STRATEGIES PAYOFFS

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?

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” Optimal Equilibrium

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 • 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.

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

Weakly Dominated Strategies • (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

Multiple players • While aX bmatrixes work fine for two players (with relatively few strategies – astrategies for player 1 and bstrategies for player 2), we can have more than two players: aX bX … Xz

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:

Principles of Game Theory