Game Theory

1. Game Theory Topic 2 Simultaneous Games “Loretta’s driving because I’m drinking and I’m drinking because she’s driving.” - The Lockhorns

2. 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

3. 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 when they are likely to occur

4. 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.

5. Strategic Interaction • Players: Reynolds and Philip Morris • Strategies: Advertise or Not Advertise • Payoffs: Companies’ Profits • Environment: • 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?

6. PAYOFFS Strategic Form of a Game PLAYERS STRATEGIES

7. What to Do? If you are advising Reynolds, what strategy do you recommend?

8. Best Replies • A strategy is a best reply to some opponents’ strategy if it does at least as well as any other strategy • si is a best replyto s-i if for every si’

9. Solving the Game • Best reply for Reynolds: • If Philip Morris advertises: • If Philip Morris does not advertise:

10. Dominance • A strategy is dominantif it outperforms all other strategies no matter what opposing players do • Games with dominant strategies are easy to solve • No need for “what if …” thinking

11. Dominance • sistrictly dominates si’ if for every s-i (the payoff is strictly higher for any strategies of the other players) • siweakly dominates si’ if for every s-i, and for some s-i

12. Dominance • A strategy si is strictly dominatedif some strategy si’ strictly dominates it • A strategy si is the dominant strategy if it strictly dominates all other strategies • A strategy si is weakly dominated if some strategy si’ weakly dominates it • A strategy si is the weakly dominant strategy if it weakly dominates all others

13. Dominance If you have a dominant strategy, and no ability to agree on an alternate course of action, use it If your opponent has a dominant strategy, and no ability to agree on an alternate course of action, then expect her to play it

14. Prisoner’s Dilemma Optimal • Both players have a dominant strategy • The equilibrium results in lower payoffs for each player Equilibrium

15. Prisoner’s Dilemma • Both players have a dominant strategy (s1,s1) u11 > u21u12 > u22 • The equilibrium results in lower payoffs for each player u22 > u11 • The above two statements imply: u12 > u22 >u11 > u21

16. Cigarette Advertising • After the 1970 agreement: • Cigarette advertising decreased by $63 million • Industry Profits rose by $91 million

17. Prisoner’s Dilemma • The dominant strategy will be played

18. 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?

19. 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:

20. The Federated Game • To whom do you tender your shares?

21. 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 !

22. Dominant Strategies “The biggest, looniest deal ever. ” – Fortune Magazine, July 1988 on Campeau’s acquisition of Federated Stores

23. Prisoner’s Dilemma Examples • Pricing by Firms • High or low prices? • Value menus and loyalty programs • Divorce • Hire attorneys or proceed amicably? • Nuclear Weapons • Build or don’t build weapons? • State governments • Inducements to attract business to a state

24. Dominated Strategies • Two restaurants compete • Can charge price of $30, $50, or $60 • Customer base consists of tourists and natives • 600 tourists pick randomly • 400 natives select the lowest price • Marginal costs are $10

25. Tourists & Natives • Example scenario: • Restaurant 1: $50, Restaurant 2: $60 • Restaurant 1 gets: 300 tourists + 400 natives = 700 customers x ($50-$10) = $28K • Restaurant 2 gets: 300 tourists + 0 natives = 300 customers x ($60-$10) = $15K

26. Tourists & Natives in thousands of dollars R. 2

27. Iterated Deletion of Dominated Strategies • Does any player have a dominated strategy? • Eliminate the strictly dominated strategies • Reduce the size of the game • Repeat: Iterate the above procedure

28. Iterated Deletion of Dominated Strategies R. 2

29. 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

30. Equilibrium • An outcome in which every player is playing a best response to the strategies of all other players. • An equilibriumis a strategy profile s such that si is a best reply to s-i for all i.

31. Equilibrium Illustration The Lockhorns

32. Games of Coordination • Complements & technology adoption • Two complementing firms • Must use same technology, but each firm has a preferred technology • Equilibrium does not offer a unique prediction • Commit (or go first) to win! Firm 2

33. 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

34. Games of Chicken • Entry into small markets Firm 2

35. 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

36. Capacity Constraints • Can decreasing others’ added value increase our profits? • Can decreasing total industry value increase our profits?

37. Multiple Equilibria • What is the predictive power of game theory when there are multiple equilibria? • Sometimes nothing ? • Refinements • Focal points • Efficiency • Evolutionary stability • Fairness • Risk dominance

38. Summary • Games have predictable outcomes • Notice dominant & dominated strategies • Select the right game to play • Looking ahead: • Sequential Games: How do games unfold over time?