Game Theory Mike ShorLecture 3 “Loretta’s driving because I’m drinking and I’m drinking because she’s driving.” - The Lockhorns
Review • Understanding the game • Noting if the rules are flexible • Anticipating our opponents’ reactions • Thinking one step ahead • Where does this lead us? • We’ve defined the “game” but not the outcome
Equilibrium • The likely outcome of a game when rational, strategic agents interact • Each player is playing his or her best strategy given the strategy choices of all other players • No player has incentive to change his or her action unilaterally • Outline: • Model interactions as games • Identify the equilibria • Decide if they are likely to occur
Equilibrium Illustration The Lockhorns
1964 1970 Cigarette Advertising on TV • 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 • Players: Reynolds and Philip Morris • Strategies: Advertise or Not Advertise • Payoffs: Companies’ Profits • Strategic Landscape: • 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?
PAYOFFS Representing a Game PLAYERS STRATEGIES
What to Do? If you are advising Reynolds, what strategy do you recommend?
Solving the Game • Best reply for Reynolds: • If Philip Morris advertises: • If Philip Morris does not advertise:
Dominance • A strategy is dominantif it outperforms all other choices no matter what opposing players do • Games with dominant strategies are easy to play • No need for “what if …” thinking
DominanceA Technical Point • Strict Dominance: Advertise is strictly dominant forReynoldsif: • Profit (Ad , Ad) > Profit (No , Ad) • Profit (Ad , No) > Profit (No , No) • Weak Dominance: Advertise is weakly dominant if: • Some inequalities are weak (), • At least one is strong(>) • By “dominant” we will mean “strictly dominant”
Dominance COMMANDMENT If you have a dominant strategy, use it. Expect your opponent to use her dominant strategy if she has one.
Prisoner’s Dilemma • Both players have a dominant strategy • The equilibrium results in lower payoffs for each player Optimal Equilibrium
Cigarette Advertising • After the 1970 agreement: • Cigarette advertising decreased by $63 million • Industry Profits rose by $91 million • Prisoner’s Dilemma • An equilibrium is NOT necessarily efficient • Players can be forced to accept mutually bad outcomes • Bad to be playing a prisoner’s dilemma, but good to make others play
How to Win a Bidding War by Bidding Less? • The battle for Federated (1988) • Parent of Bloomingdales • Current share price ≈ $60 • Expected post-takeover share price ≈ $60 • Macy’s offers $70/share • contingent on receiving 50% of the shares • Do you tender your shares to Macy’s?
How to Win a Bidding War (continued) • Robert Campeau bids $74 per share not contingent on amount acquired • Campeau’s Mixed Scheme: • If less than 50% tender their shares, each receives: $74 per share • If more than 50% tender their shares, (if X% tender), each receives:
The Federated Game • To whom do you tender your shares?
How to Win a Bidding War • Each player has a dominant strategy: Tender shares to Campeau • Resulting Price: (½ x 74) + (½ x 60) = $67 • BUT: Macy’s offered $70 !
Dominant Strategies “The biggest, looniest deal ever. ” – Fortune Magazine, July 1988 on Campeau’s acquisition of Federated Stores • What if players do not have dominant strategies?
Pricing without Dominant Strategies • Two bars (bar 1, bar 2) compete • Can charge price of $2, $4, or $5 • Customer base consists of tourists and natives • 6,000 tourists pick a bar randomly • 4,000 natives select the lowest price bar • Marginal costs are close to zero
Tourists & Natives • Example scenario: • Bar 1 charges $4, Bar 2 charges $5 • Bar 1 gets: 3,000 tourists + 4,000 natives = 7,000 customers x $4 = $28K • Bar 2 gets: 3,000 tourists + 0 natives = 3,000 customers x $5 = $15K
Tourists & Natives in thousands of dollars Bar 2
Successive Elimination of Dominated Strategies • Does any player have a dominant strategy? • Does any player have a dominated strategy? • A strategy is dominated if there is some other strategy which always does better • Eliminate the dominated strategies • Reduce the size of the game • Iterate the above procedure • What is the equilibrium?
Dominance CAVEAT Expect your opponent to use her dominant strategy if she has one. BUT Be sure you understand your opponents’ true payoffs. (Do you know what really motivates them?)
No Dominated Strategies • Often there are no dominated strategies • Some games may have multiple equilibria • Equilibrium selection becomes an issue • Method: For each player, find the best response to every strategy of the other player • Games of Coordination • Games of Assurance • Games of Chicken
Games of Coordination • Joint ventures and supplier choice • Two firms engaged in joint venture • Must use the same supplier, but each firm has a preferred supplier Firm 2
Games of Coordination • Solving: Firm 2
Games of Assurance • Joint research ventures • Each firm may invest $50,000 into an R&D project • Project succeeds only if both invest • If successful, each nets $75,000 Firm 2
Games of Chicken • Entry into small markets Firm 2
The Right Game to Play • Why do we “solve” games? • To know which one to play! • How do internal corporate changes impact the outcome of strategic interaction? • Some games are better than others
Your Value to a Game • Your added value = the size of the pie when you’re in the game minus the size of the pie when you are not • Added value limits how much you can get • You cannot receive much more than your added value • Added value provides benchmark • You should receive close to your added value • Change the Game! • You can increase your payoffs by increasing your added value OR decreasing the added value of other players.
Capacity Constraints • Can decreasing others’ added value increase our profits? • Can decreasing total industry value increase our profits?
Summary • Games have predictable outcomes • Notice dominant & dominated strategies • Select the right game to play • Seemingly internal corporate changes can impact the outcome of strategic interaction • Looking ahead: • Sequential Games: How do games unfold over time?