Game Theory

1. Game Theory • Game theory models strategic behavior by agents who understand that their actions affect the actions of other agents. • Useful to study • Company behavior in imperfectly competitive markets (such as Coke vs. Pepsi). • Military strategies. • Bargaining/Negotiations. • Biology • A game consists of players, strategies, and payoffs.

2. Battle of Bismarck Sea • We want to model the Battle of the Bismarck Sea. • Two Admirals: Imamura (Japan) and Kenny (US). • Japan is in retreat. • Imamura wants to transport troops in a convoy • from Rabaul to Lae • Kenny wants to bomb Japanese troops. . • North route is two days, Southern route is three days. • It takes one day for Kenny to switch searching routes.

3. Imamura wants to run convoy from Rabaul to Lae

4. Battle of Bismarck Sea Imamura North South -2 -2 North 2 2 Kenny -3 -1 South 1 3 • This representation is called a Normal form Game. • Imamura wants to transport troops. • Kenny wants to bomb Japanese troops. . • North route is two days, Southern route is three days. • It takes one day for Kenny to switch routes.

5. Battle of Bismarck Sea Imamura Players North South -2 -2 North 2 2 Kenny -3 -1 South 1 3 • Imamura wants to transport troops. • Kenny wants to bomb Japanese troops. . • North route is two days, Southern route is three days. • It takes one day for Kenny to switch routes.

6. Battle of Bismarck Sea Imamura Players North South Imamura’s Strategies -2 -2 North 2 2 Kenny -3 -1 South 1 3 • Imamura wants to transport troops. • Kenny wants to bomb Japanese troops. . • North route is two days, Southern route is three days. • It takes one day for Kenny to switch routes.

7. Battle of Bismarck Sea Imamura Players Kenny’s Strategies North South Imamura’s Strategies -2 -2 North 2 2 Kenny -3 -1 South 1 3 • Imamura wants to transport troops. • Kenny wants to bomb Japanese troops. . • North route is two days, Southern route is three days. • It takes one day for Kenny to switch routes.

8. Battle of Bismarck Sea Imamura Imamura’s Payoffs: Each day of Bombing = -1 in payoff North South -2 -2 North 2 2 Kenny -3 -1 South 1 3 • Imamura wants to transport troops. • Kenny wants to bomb Japanese troops. . • North route is two days, Southern route is three days. • It takes one day for Kenny to switch routes.

9. Battle of Bismarck Sea Kenny’s Payoffs: Each day of Bombing = 1 in payoff Imamura Imamura’s Payoffs: Each day of Bombing = -1 in payoff North South -2 -2 North 2 2 Kenny -3 -1 South 1 3 • Imamura wants to transport troops. • Kenny wants to bomb Japanese troops. . • North route is two days, Southern route is three days. • It takes one day for Kenny to switch routes.

10. Battle of Bismarck Sea Imamura North South -2 -2 North 2 2 Kenny -3 -1 South 1 3 Notice that Kenny guessing wrong costs him a day. • Imamura wants to transport troops. • Kenny wants to bomb Japanese troops. . • North route is two days, Southern route is three days. • It takes one day for Kenny to switch routes.

11. The Prisoners’ Dilemma Clyde S C -5 -1 S -5 -15 Bonnie -10 -15 C -1 -10 Bonnie and Clyde are caught. They can confess or be silent.

12. Nash equilibrium • A Nash equilibrium is a set of strategies (s1,s2) • Where each player has no incentive to deviate given the other players’ strategies. • (or) Given other equilibrium strategies a player would choose his equilibrium strategy. • (or) A best response to a best response. • u1(s1,s2)>=u1(z,s2) for all possible z and u2(s1,s2)>=u2(s1,z) for all possible z. • A pure strategy equilibrium is where each player only chooses a particular strategy with certainty. • What are the pure-strategy equilibria in Prisoners’ dilemma?

13. Trench Warfare: a prisoners’ dilemma Mars Not Shoot Shoot -5 Not Shoot -1 -5 -15 Venus -10 -15 Shoot -1 -10 Are we doomed to the bad outcome? Not in trench warfare of WWI. This happens since the game is repeated –that is, played several times.

14. Repeated games: forever punish • Forever punish strategy: if someone cheats, punish forever. • Cheating gives short term gain but long term loss. • Gain of 4 for the time of cheating. • Loss of 5 from the next period on. • Whether this stops cheating depends upon how you value today compared to tomorrow (discount rate). • Forever punish is not so great if there is uncertainty. Why? • Many tribal religions do not have explicit ethical code as part of the religion: no stealing, no murder, etc.

15. Coordination Problem Jim VHS Beta 1 0.5 VHS 1 0 Sean 2 0 Beta 0.5 2 • Jim and Sean want to have the same VCR. • Beta is a better technology than VHS.

16. Information Technology • Phones, Faxes, e-mail, etc. all have the following property: • Network externalities: The more people using it the more benefit it is to each user. • Computers, VCRs, PS2s, also have this property in that both software can be traded among users and the larger the user market, the larger number of software titles are made. • How do markets operate with such externalities?

17. Discussion points • Competitors: VHS vs. Beta, Qwerty vs. Dvorak, Windows vs. Mac, Playstation vs. Xbox. • Does the best always win? • Standardization helps with network externalities. • Drive on left side vs. right side. Out of 206 countries 144 (70%) are rhs. • Left is more nature for an army: swords in right hand, mounting horses. (Napolean liked the other way.) • Sweden switched from left to right in 1967. • Lots of networks: Religions and Languages.

18. Penalty Kick Goalie Dive L Dive R 1 -1 Kick L -1 1 Kicker 1 -1 Kick R 1 -1 • A Kicker can kick a ball left or right. • A Goalie can dive left or right.

19. Mixed Strategy equilibrium • Happens in the Penalty kick game. • Notice that if the Kicker kicks 50-50 (.5L+.5R), the Goalie is indifferent to diving left or right. • If the Goalie dives 50-50 (.5L+.5R), the Kicker is indifferent to kicking left or right. • Thus, (.5L+.5R,.5L+.5R) is a mixed-strategy N.E. • Nash showed that there always exists a Nash equilibrium (which includes mixed strategies).

20. Do you believe it? • Can we empirically test this theory? • Yes! • Study was done with the Italian football league. • Step 1: See if the strategies are really left or right. • Step 2: Calculate payoffs. How? If when the goalie guesses correctly, there is no goal 100% of the time the payoffs are 0 for the kicker and 100 for the goalie. If there no goal 80% of the time, then the payoffs are 20 for the kicker and 80 for the goalie, etc… • Step 3: Calculate the Nash equilibrium. • Step 4: Compare.

21. Do you believe it? • Do they really choose only L or R? Yes. Kickers 93.8% and Goalie 98.9%. • Kickers are either left or right footed. Assume R means kick in “easier” direction. Below is percentage of scoring. • Nash prediction for (Kicker, Goalie)=(41.99L+58.01R, 38.54L+61.46R) • Actual Data =(42.31L+57.69R, 39.98L+60.02R) Dive L Dive R 58.3 94.97 Kick L Kick R 92.91 69.92

22. Parking Enforcement Game Student Driver Park OK Park in Staff -5 -95 Check -5 5 University 5 -5 Don’t 5 0 • Student can decide to park in staff parking. • University can check cars in staff parking lot.

23. What happens? • If the University checks, what do the students do? • If the students park ok, what does the Uni do? • If the uni doesn’t check, what do the students do? • If the students park in the staff parking, what does the uni do? • What is the equilibrium of the game? • What happens if the university makes it less harsh a punishment to only –10. Who benefits from this? Who is hurt by this?

24. Answer • Student parks legally 1/3 of the time and the uni checks 1/10 of the time. • With lower penalty, student parks legally 1/3 of the time and the uni checks 2/3 of the time. • Who’s expected payoff changes? No one.