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##### Game Theory

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**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?