Game Theory

1. Game Theory “Loretta’s Driving Because I’m Drinking and I’m Drinking Because She’s Driving” - The Lockhorns Cartoon Mike Shor Lecture 3

2. Review • Understand the game you are in • Note if the rules are flexible • Anticipate your opponents’ reactions • Understand the assumptions • Recognize that not everyone else understands them Game Theory - Mike Shor

3. Game Theory - Mike Shor

4. Equilibrium • Nash Equilibrium: • A set of strategies, one for each player, such that each player’s strategy is best for her given that all other players are playing their equilibrium strategies • Best Response: • The best strategy I can play given the strategy choices of all other players • Everybody is playing a best response • No incentive to unilaterally change my strategy Game Theory - Mike Shor

5. Identifying the Equilibrium • Pure strategy equilibrium • Consider mixed later • Dominance • Dominance solvable • Only one dominant strategy • Successive elimination of dominated strategies • Cell-by-cell inspection Game Theory - Mike Shor

6. Cigarette Advertising on TV • All US tobacco companies advertised heavily on TV • Surgeon General issues official warning • Cigarette smoking may be hazardous • Cigarette companies’ reaction • Fear of potential liability lawsuits • Companies strike agreement • Carry the warning label and cease TV advertising in exchange for immunity from federal lawsuits. 1964 1970 Game Theory - Mike Shor

7. Strategic Interactions • Players: Reynolds and Philip Morris • Strategies: { Advertise , Do Not Advertise } • Payoffs: Companies’ Profits • 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? Game Theory - Mike Shor

8. Normal (Strategic) Form PLAYERS STRATEGIES PAYOFFS Game Theory - Mike Shor

9. Normal Form • Best reply for Reynolds: • If Philip Morris advertises: advertise • If Philip Morris does not advertise: advertise • Regardless of what you think Philip Morris will do Advertise! Game Theory - Mike Shor

10. Dominant Strategy A strategy that outperforms all other choices no matter what opposing players do • Firm 1’s strategies: { A, B, C } • Firm 2’s strategies: { X, Y, Z } • C is strictly dominant for Firm 1 if: • P(C,X)>P(A,X) P(C,X)>P(B,X) • P(C,Y)>P(A,Y) P(C,Y)>P(B,Y) • P(C,Z)>P(A,Z) P(C,Z)>P(B,Z) • C is weakly dominant for Firm 1 if: • Some inequalities are weak (), at least one is strong(>) Game Theory - Mike Shor

11. Dominance Solvable • If each player has a dominant strategy, the game is dominance solvable • What is the equilibrium of the cigarette advertising game? COMMANDMENT If you have a dominant strategy, use it. Expect your opponent to use her dominant strategy if she has one. Game Theory - Mike Shor

12. Cigarette Advertising • After the 1970 agreement, cigarette advertising decreased by $63 million • Profits rose by $91 million • Prisoner’s Dilemma • An equilibrium is NOT necessarily efficient • What if the game is not dominance solvable? Game Theory - Mike Shor

13. A Strategic Situation Two firms competing over sales • Time and The Economist must decide upon the cover story to run some week. • The big stories of the week are: • A presidential scandal (labeled S), and • A proposal to deploy US forces to Grenada (G) • Neither knows which story the other magazine will choose to run Game Theory - Mike Shor

14. One Dominant Strategy • Who has a dominant strategy? • Assume it will be played! • Other player can plan accordingly. Game Theory - Mike Shor

15. Dominated Strategies • For The Economist: G dominant = S dominated • Dominated Strategy: • There exists another strategy which always does better regardless of opponents’ actions Game Theory - Mike Shor

16. Successive Elimination of Dominated Strategies • If a strategy is dominated, eliminate it • The size and complexity of the game is reduced • Eliminate any dominant strategies from the reduced game • Continue doing so successively Game Theory - Mike Shor

17. Example: Tourists & Natives • Two bars (bar 1, bar 2) compete • Can charge price of $2, $4, or $5 • 6000 tourists pick a bar randomly • 4000 natives select the lowest price bar Bar 2 Game Theory - Mike Shor

18. Successive Elimination of Dominated Strategies • Does any player have a dominant strategy? • Does any player have a dominated strategy? • Eliminate the dominated strategies • Reduce the normal-form game • Iterate the above procedure • What is the equilibrium? Game Theory - Mike Shor

19. Bar 2 Successive Elimination of Dominated Strategies Bar 2 $2 $4 $5 $2 10 , , 10 14 , , 12 14 , , 15 Bar 1 Bar 1 $4 20 , , 20 28 , , 15 12 , , 14 $5 15 , , 28 25 , , 25 15 , , 14 Game Theory - Mike Shor

20. No Dominated Strategies • Often there are no dominated strategies • Or: reducing the game is not sufficient • There may be multiple equilibria • Method: Cell-by-cell inspection • Ask: Is each player playing the best response to the other player? Game Theory - Mike Shor

21. Types of Games • Games of Assurance • Games of Coordination • Games of Chicken Game Theory - Mike Shor

22. Games of Assurance • Two firms each earning $45,000 • Both can invest the $45,000 into R&D • R&D successful only if both invest • If R&D successful, each earns $95,000 Firm 2 Game Theory - Mike Shor

23. Cell-by-cell Inspection • Consider { Invest , Don’t } • Both players have an incentive to change their strategy: NOT an equilibrium Firm 2 Game Theory - Mike Shor

24. Assurance Outcomes • Two equilibria exist • Both firms prefer (I ,I) to (D,D) • Payoffs of 50 to each firm instead of 45 • However, investing is risky • Must have assurances • How to achieve assurance? • Strategic moves: commit to choosing I • Sequential moves: leader chooses the equilibrium Game Theory - Mike Shor

25. Games of Coordination • Joint ventures and the choice of supplier • Two firms engaged in joint venture • Must use the same supplier, but each firm has a preferred supplier Firm 2 Game Theory - Mike Shor

26. Coordination Outcomes • Two equilibria exist • Firms prefer different equilibria • How to achieve the most desirable outcome for you? • Strategic moves: commit to choosing A • Sequential moves: leader chooses the equilibrium Game Theory - Mike Shor

27. Summary • You must put yourself in your rival’s shoes • Recognize dominant and dominated strategies • Anticipate that your opponent will recognize them as well Game Theory - Mike Shor