4 Normal form games Simultaneous move games Two components to the game Many situations mimic situations of 2+ people acting at the same timeEven 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 gamesTwo components to the gameThe strategies available to each playerThe payoffs to the players“Simple” games often represented as a matrix of payoffs.
5 Cigarette Advertising example All US tobacco companies advertised heavily on TVSurgeon General issues official warningCigarette smoking may be hazardousCigarette companies fear lawsuitsGovernment may recover healthcare costsCompanies strike agreementCarry the warning label and cease TV advertising in exchange for immunity from federal lawsuits.19641970
6 Strategic Interaction: Cigarette Advertising Players?Reynolds and Philips MorrisStrategies:Advertise or NotPayoffsCompanies’ ProfitsStrategic LandscapeFirm i can earn $50M from customersAdvertising campaign costs i $20MAdvertising takes $30M away from competitor j
7 Strategic Form Representation PAYOFFSPLAYERSPhilip MorrisNo AdAdReynolds50 , 50STRATEGIES
9 What would you suggest?If you were consulting for Reynolds, what would you suggest?Think about best responses to PMIf PM advertises?If PM doesn’t?Philip MorrisNo AdAdReynolds50 , 5020 , 6060 , 2030 , 30
10 Nash Equilibrium Equilibrium Likely outcome of a game when rational strategic agents interactEach player is playing his/her best strategy given the strategy choices of all other playersNo player has an incentive to change his or her strategy unilaterallyMutual best response.Not necessarily the best outcome for both players.
11 DominanceA 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 analyzeIf you have a dominant strategy, use it.If your opponent has one, expect her to use it.
12 Solving using dominance Philip MorrisNo AdAdReynolds50 , 5020 , 6060 , 2030 , 30Both players have a dominant strategyEquilibrium outcome results in lower payoffs for each playerGame of the above form is often called the “Prisoners’ Dilemma”OptimalEquilibrium
13 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 $5Customer base consists of two groups6000 Tourists: don’t know anything about the city but want coffee4000 Locals: caffeine addicted but select the cheapest caféCafés offer the same coffee and compete over priceTourists don’t know the price and ½ go to each café
14 Café price competition Example scenario:Café 1 charges $4 and café 2 charges $5:Recall: tourists are dumb and don’t know where to goCafé 1 gets:3000 tourists locals = 7K customers * $4 = 28KCafé 2 gets3000 tourists + 0 locals = 3K customers * $5 = 15KDraw out the 3x3 payoff matrix given$2, $4, or $5 price selection (simultaneous selection)6K tourists and 4K locals.
16 Dominated Strategies Café 2 $2 $4 $5 Café 1 10 , 10 14 , 12 14 , 15 A player might not have a dominant strategy but may have a dominated strategyA strategy, s, is dominated if there is some other strategy that always does better than s.Café 2$2$4$5Café 110 , 1014 , 1214 , 1512 , 1420 , 2028 , 1515 , 1415 , 2825 , 25
17 Dominance solvableIf 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 carefulPlayer 2LeftRightPlayer 1Up0,01,1Down
18 Weakly Dominated Strategies Player 2LeftRightPlayer 1Up0,01,1Down(Down, Right) is an equilibrium profileBut 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
19 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:
20 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
21 Multiple EquilibriaWe’ve said nothing about there always being a unique equilibrium. Often there isn’t just one:
22 Equilibrium Selection With multiple equilibria we face a very difficult problem of selection:
23 Equilibrium Selection With multiple equilibria we face a very difficult problem of selection:Imagine Harry had different preferences:
24 Equilibrium Selection With multiple equilibria we face a very difficult problem of selection:Classic issues of coordination:
25 No equilibrium in pure strategies Nor must there exist an equilibrium in pure strategiesPure strategies means no randomization (penalty kicks)We’ll talk about general existence laterPlayer 2RockPaperScissorsPlayer 10,0-1,11,-1
26 Multiple playersWhile 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
27 Homework Study for the quiz Next time: more mathematical introduction to simultaneous move gamesFocus on section 1.2 of Gibbons