Game Theory

1. Game Theory

2. Game Theory Definition Thestudy of strategic decision making. More formally, it is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. Game Theory is used to analyze how firms interact but has many other applications.

3. Other Applications of Game Theory • National Defense – Terrorism and Cold War • Movie Release Dates and Program Scheduling • Auctionshttp://en.wikipedia.org/wiki/Spectrum_auctionhttp://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction • Sports – Cards, Cycling, and race car driving • Politics – positions taken and $$/time spent on campaigning • Nanny Monitoring • Group of Birds Feeding • Mating Habits

4. Game Theory and Terrorism Game theory helps insurers to judge the risks of terror Financial Times Jenny Wiggins September 8, 2004 Shortly after September 11 2001, a small group of companies that specialise in assessing risk for the insurance industry launched US terrorism risk models. These combine technology and data to predict likely terrorist targets and methods of attack, and possible losses to life and property. They are aimed at the insurance and reinsurance industry, which already uses similar models to assess potential losses from natural catastrophes such as hurricanes and earthquakes. "Most major commercial insurers and reinsurers are using terrorism modelling today," says Robert Hartwig, chief economist at the Insurance Information Institute.

5. Game Theory and Terrorism (cont.) Andrew Coburn, director of terrorism research at RMS, says the company can pinpoint possible targets because it believes terrorists make rational decisions. "Their methods and targeting are very systematic," he says. RMS uses game theory - analytical tools designed to observe interactions among people - in its models. It argues that, as security increases around prime targets, rational terrorists will seek out softer targets. Industry participants, however, say the predictive abilities of the models are limited, given the difficulty of foreshadowing human behaviour. The development of the models has attracted the interest of the US government…

6. Game Theory and Randomization Random Checks Newsweek October 22, 2007 Security officials at Los Angeles International Airport now have a new weapon in their fight against terrorism: randomness. Anxious to thwart future terror attacks in the early stages while plotters are casing the airport, security patrols have begun using a computer program called ARMOR (Assistant for Randomized Monitoring of Routes) to make the placement of security checkpoints completely unpredictable.

7. Game Theory and Randomization (cont.) Randomness isn't easy. Even when they want to be unpredictable, people follow patterns. That's why the folks at LAX turned to the computer scientists at USC. The idea began as an academic question in game theory: how do you find a way for one "agent" (or robot or company) to react to an adversary who has perfect information about the agent's decisions? Using artificial intelligence and game theory, researchers wrote a set of algorithms to randomize the actions of the first agent. Academic colleagues couldn't appreciate how the technology could be useful. "It was very disappointing," says Milind Tambe, the USC engineering professor who led the ARMOR team.

8. Other Applications of Game Theory • National Defense – Terrorism and Cold War • Movie Release Dates and Program Scheduling • Auctions http://en.wikipedia.org/wiki/Spectrum_auction http://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction • Sports – Cards, Cycling, and race car driving • Politics – positions taken and $$/time spent on campaigning • Nanny Monitoring • Group of Birds Feeding • Mating Habits

9. Grey’s Anatomy vs. The Donald NBC delays 'Apprentice' premiere By Nellie Andreeva Dec 20, 2007 NBC is taking the premiere of "Celebrity Apprentice" out of the cross-hairs of the last original episode of ABC's "Grey's Anatomy"... or so it seems.NBC on Wednesday said that it will push the launch of "Apprentice" from Jan. 3 to Jan. 10, expanding "Deal or No Deal" to two hours on Thursday, Jan. 3.The move follows ABC's midseason schedule announcement Friday that included the last original episode of "Grey's" airing Jan. 3,…

10. Grey’s Anatomy vs. The Donald 'Grey' move has NBC red Peacock shifts 'Apprentice' back By Nellie Andreeva Dec 21, 2007 The Thursday night scheduling tango between NBC and ABC continued Thursday morning when ABC officially announced that it will move the last original episode of "Grey's Anatomy" from Jan. 3 to Jan. 10.That led to a reversal in NBC's Wednesday decision to push the premiere of "Celebrity Apprentice" from Jan. 3 toJan. 10 to avoid the first-run "Grey's."NBC said Thursday afternoon that "Apprentice," hosted by Donald Trump, will now launch Jan. 3 as originally planned.

11. Game Theory and Movie Release Dates The Imperfect Science of Release Dates New York Times November 9, 2003 On Dec. 25, which this year happens to be a Thursday, five new movies will be released in theaters -- six, if you count a new Disney IMAX film called ''Young Black Stallion.'' As with the Fourth of July and Thanksgiving, there is a special cachet to opening a film on Christmas Day…. The casual moviegoer rarely ponders why a particular bubbly romantic comedy, serial-killer thriller, literary costume drama or animated talking-farm-animals movie opens on the day it does. Movies come; movies go; movies wind up on video. To those responsible for putting those films on the screen, however, nothing about the timing of their releases is arbitrary.

12. Game Theory and Movie Release Dates (cont.) Last December featured one of the most dramatic games of chicken in recent memory, when two films starring Leonardo DiCaprio were both slated to open on Christmas weekend. Ultimately, Miramax blinked first, moving the release of Martin Scorsese's ''Gangs of New York'' five days earlier and ceding the holiday to the other DiCaprio film, DreamWorks' ''Catch Me if You Can.'' ''We didn't think about moving,'' says Terry Press, the head of marketing for DreamWorks. ''We had been there first, and 'Catch Me if You Can' was perfect for that date.'' This year, DreamWorks chose to schedule a somber psychological drama, ''House of Sand and Fog,'' for the day after Christmas, deferring a bit to Miramax. ''I don't want our reviews to run on the same day as 'Cold Mountain,''' Press says. Ever wonder why a movie theater shows a preview of an upcoming movie that is to be released in 2 years?

13. Other Applications of Game Theory • National Defense – Terrorism and Cold War • Movie Release Dates and Program Scheduling • Auctions http://en.wikipedia.org/wiki/Spectrum_auction http://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction • Sports – Cards, Cycling, and race car driving • Politics – positions taken and $$/time spent on campaigning • Nanny Monitoring • Group of Birds Feeding • Mating Habits

14. Other Applications of Game Theory • National Defense – Terrorism and Cold War • Movie Release Dates and Program Scheduling • Auctionshttp://en.wikipedia.org/wiki/Spectrum_auction http://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction • Sports – Cards, Cycling, and race car driving • Politics – positions taken and $$/time spent on campaigning • Nanny Monitoring • Group of Birds Feeding • Mating Habits

15. Game Theory Terminology • Simultaneous Move Game – Game in which each player makes decisions without knowledge of the other players’ decisions (ex. Cournot or Bertrand Oligopoly). • Sequential Move Game – Game in which one player makes a move after observing the other player’s move (ex. Stackelberg Oligopoly).

16. Game Theory Terminology • Strategy – In game theory, a decision rule that describes the actions a player will take at each decision point. • Normal Form Game – A representation of a game indicating the players, their possible strategies, and the payoffs resulting from alternative strategies.

17. Example 1: Prisoner’s Dilemma(Normal Form of Simultaneous Move Game) Confess (1<2) What is Peter’s best option if Martha doesn’t confess? Confess (6<10) What is Peter’s best option if Martha confess?

18. Example 1: Prisoner’s Dilemma Confess (1<2) What is Martha’s best option if Peter doesn’t confess? Confess (6<10) What is Martha’s best option if Peter Confesses?

19. Example 1: Prisoner’s Dilemma First Payoff in each “Box” is Row Player’s Payoff . Dominant Strategy – A strategy that results in the highest payoff to a player regardless of the opponent’s action.

20. Example 2: Price Setting Game Is there a dominant strategy for Firm B? Low Price Is there a dominant strategy for Firm A? Low Price

21. Nash Equilibrium • A condition describing a set of strategies in which no player can improve her payoff by unilaterally changing her own strategy, given the other player’s strategy. (Every player is doing the best they possibly can given the other player’s strategy.)

22. Example 1: Nash? Nash Equilibrium: (Confess, Confess)

23. Example 2: Nash? Nash Equilibrium: (Low Price, Low Price)

24. Chump, Chump, Chump http://videosift.com/video/Game-Theory-in-British-Game-Show-is-Tense?loadcomm=1

25. EXAMPLE 3: Entry into a fast food market: Is there a Nash Equilibrium(ia)? Yes, there are 2 – (Enter, Don’t Enter) and (Don’tEnter, Enter).Implies, no need for a dominant strategy to have NE. NO Is there a dominant strategy for BK? NO Is there a dominant strategy for McD?

26. EXAMPLE 4: Monitoring Workers Is there a Nash Equilibrium(ia)? Not a pure strategy Nash Equilibrium– player chooses to take one action with probability 1 Randomize the actions yields a Nash = mixed strategy John Nash proved an equilibrium always exists NO Is there a dominant strategy for the worker? NO Is there a dominant strategy for the manager?

27. Mixed (randomized) Strategy • Definition: A strategy whereby a player randomizes over two or more available actions in order to keep rivals from being able to predict his or her actions.

28. Calculating Mixed Strategy EXAMPLE 4: Monitoring Workers • Manager randomizes (i.e. monitors with probability PM) in such a way to make the worker indifferent between working and shirking. • Worker randomizes (i.e. works with probability Pw) in such a way as to make the manager indifferent between monitoring and not monitoring.

29. Example 4: Mixed Strategy 1-PW PW PM 1-PM

30. Manager selects PM to make Worker indifferent between working and shirking (i.e., same expected payoff) • Worker’s expected payoff from working PM*(1)+(1- PM)*(-1) = -1+2*PM • Worker’s expected payoff from shirking PM*(-1)+(1- PM)*(1) = 1-2*PM Worker’s expected payoff the same from working and shirking if PM=.5. This expected payoff is 0 (-1+2*.5=0 and 1-2*.5=0). Therefore, worker’s best response is to either work or shirk or randomize between working and shirking.

31. Worker selects PW to make Manager indifferent between monitoring and not monitoring. • Manager’s expected payoff from monitoring PW*(-1)+(1- PW)*(1) = 1-2*PW • Manager’s expected payoff from not monitoring PW*(1)+(1- PW)*(-1) = -1+2*PW Manager’s expected payoff the same from monitoring and not monitoring if PW=.5. Therefore, the manager’s best response is to either monitor or not monitor or randomize between monitoring or not monitoring.

32. Nash Equilibrium of Example 4 • Worker works with probability .5 and shirks with probability .5 (i.e., PW=.5) • Manager monitors with probability .5 and doesn’t monitor with probability .5 (i.e., PM=.5) Neither the Worker nor the Manager can increase their expected payoff by playing some other strategy (expected payoff for both is zero). They are both playing a best response to the other player’s strategy.

33. Example 4A: What if costs of Monitoring decreases and Changes the Payoffs for Manager 1.5 -.5

34. Nash Equilibrium of Example 4A where cost of monitoring decreased • Worker works with probability .625 and shirks with probability .375 (i.e., PW=.625) • Same as in Ex. 5, Manager monitors with probability .5 and doesn’t monitor with probability .5 (i.e., PM=.5) The decrease in monitoring costs does not change the probability that the manager monitors. However, it increases the probability that the worker works.

35. Example 5: Mixed Strategy and Tennishttp://www.fuzzyyellowballs.com/introducing-the-fyb-strategy-quiz/ Game: Server’s Possible Strategies: (Serve Left , Serve Right) Receiver’s Possible Strategies: (Defend Left , Defend Right) Receiver has a high probability of winning the point if she defends the side the server serves to.

36. Example 5: Mixed Strategy and Tennis(Payoffs are probability of winning point) DL 1-DL SL 1-SL Mixed Strategy Equilibrium (SL =.5 , DL =.67) This results in the probability of the server winning the point to be .65 irrespective of whether he serves to the left or right.

37. Example 5: Mixed Strategy and TennisWhat about the Real World? Minimax Play at Wimbleton Walker and Wooders (AER 2001) http://www.finance.uts.edu.au/staff/johnwooders/WimbledonAER.pdf “We use data from classic professional tennis matches to provide an empirical test of the theory of mixed strategy equilibrium. We find that the serve-and-return play of John McEnroe, Bjorn Borg, Boris Becker, Pete Sampras and others is consistent with equilibrium play.” Results: Probability Server wins is the same whether serve right or left. Which side server serves is not “serially independent”.

38. Example 6 • A Beautiful Mind http://www.youtube.com/watch?v=CemLiSI5ox8

39. Example 6: A Beautiful Mind Nash Equilibria: (Pursue Blond, Pursue Brunnette 1) (Pursue Blond, Pursue Brunnette 2) (Pursue Brunnette 1, Pursue Blond) (Pursue Brunnette 2, Pursue Blond)

40. Sequential/Multi-Stage Games • Extensive form game: A representation of a game that summarizes the players, the information available to them at each stage, the strategies available to them, the sequence of moves, and the payoffs resulting from alternative strategies. (Often used to depict games with sequential play.)

41. Potential Entrant Example 7 Don’tEnter Enter Incumbent Firm Potential Entrant: 0 Incumbent: +10 Price War Share Market (Hard) (Soft) Potential Entrant: -1 +5 Incumbent: +1 +5 What are the Nash Equilibria?

42. Nash Equilibria • (Potential Entrant Enter, Incumbent Firm Shares Market) • (Potential Entrant Don’t Enter, Incumbent Firm Price War) Is one of the Nash Equilibrium more likely to occur? Why? Perhaps (Enter, Share Market) because it doesn’t rely on a non-credible threat.

43. Subgame Perfect Equilibrium • A condition describing a set of strategies that constitutes a Nash Equilibrium and allows no player to improve his own payoff at any stage of the game by changing strategies. (Basically eliminates all Nash Equilibria that rely on a non-credible threat – like Don’t Enter, Price War in Prior Game)

44. Potential Entrant Example 7 Don’tEnter Enter Incumbent Firm Potential Entrant: 0 Incumbent: +10 Price War Share Market (Hard) (Soft) Potential Entrant: -1 +5 Incumbent: +1 +5 What is the Subgame Perfect Equilibrium? (Enter, Share Market)

45. Big Ten Burrito Example 8 Enter Don’t Enter ChipotleChipotle Enter Don’t Enter Don’t Enter Enter BTB: -25 +40 0 0 Chip: -50 0 +70 0

46. Big Ten Burrito Enter Don’t Enter Chipotle Chipotle Enter Don’t Enter Don’t Enter Enter BTB: -25 +40 0 0 Chip: -50 0 +70 0 Use Backward Induction to Determine Subgame Perfect Equilibrium.

47. Subgame Perfect Equilibrium Chipotle should choose Don’t Enterif BTB chooses Enter and Chipotle should choose Enter if BTB chooses Don’t Enter. BTB should choose Enter given Chipotle’s strategy above. Subgame Perfect Equilibrium: (BTB chooses Enter, Chipotle chooses Don’t Enter if BTB chooses Enter and Enter if BTB chooses Don’t Enter.)

48. U.S. Postal Service and Anthrax Is Mail Safer Since Anthrax Attacks? Questions Remain About Post Office Security 5 Years After 5 Died HAMILTON, N.J., Sept. 23, 2006 Five years ago next week, American officials began to suspect that someone was sending anthrax-tainted letters through the mail. Five people eventually died and 17 other became ill as a result. The attacks remain unsolved, but there have been some security upgrades to the nation's postal system. The question remains: are we any safer? The U.S. Postal Service's Tom Day helped design the system that now tests for anthrax at all 280 mail processing centers across the country. He gave CBS News correspondent Bianca Solarzano a tour of the John K. Rafferty Hamilton Post Office Building.

49. U.S. Postal Service and Anthrax (cont.) "This was the first spot where the anthrax was coming out of the envelopes," Day said, pointing to a mail sorting machine. There has been a tunnel-like addition to the machine where letters collected from mail boxes are checked for anthrax. "If anything is escaping from an envelope at this point, we're collecting it and pulling it out through a system right here," Day said. "That, then, goes to this box which is the self contained detection system." The system's cost: $150 million per year. So, after all the improvements, is our mail safe? "I would definitely say the mail in this country is safe," Day said. "Can I give a 100 percent guarantee? The answer is 'no.'"

50. US Postal Service Example 9 Buy Protector Don’t Buy Protector Unstable Person Unstable Person Send Don’t Send Send Don’t Send Anthrax Anth Anth Anthrax USPS: -600 -400 -1000 0 Person: -10 0 +10 0 Subgame Perfect Equilibrium: (US Postal Service Buys Protector; Unstable Person Doesn’t Send Anthrax if USPS Buys Protector and Sends Anthrax if USPS Doesn’t Buy Protector)