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 • 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

4. Grey’s Anatomy vs. The Donald NBC delays 'Apprentice' premiere By Nellie Andreeva 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,…

5. Grey’s Anatomy vs. The Donald 'Grey' move has NBC red Peacock shifts 'Apprentice' back By Nellie Andreeva 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.

6. Game Theory and Movie Release Dates The Imperfect Science of Release Dates New York Times 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.

7. 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?

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 • Parenting

9. Game Theory Secrets for ParentsWall Street Journal July, 2014 The party is over, and you're down to the last bit of cake. All three of your children want it. If you're familiar with game theory, you might think of the classic strategy in which one person cuts the cake and the other chooses the slice. But how do you divide it three ways without anyone throwing a fit? Game theory is, in essence, the science of strategic thinking—a way of making the best decision possible based on the way you expect other people to act. It was once the domain of Nobel Prize-winning economists and big thinkers on geopolitics, but now parents are getting in on the act. Though game theory assumes, as a technical matter, that its players are rational, it applies just as well to not-always-rational children. A key lesson in game theory, says Barry Nalebuff, a professor at the Yale School of Management, is to understand the perspective of the other players. It isn't about what you would do in another person's shoes, he says; it's about what they would do in their shoes. "Good game theory," he says, "appreciates the quirks and features that make us unique and takes us as we are." The same could be said of good parenting. So how to deal with the problem of dividing a piece of cake into three equal shares? Try this: After the first child cuts and the second one chooses, each child further cuts his or her own slice into thirds. The third child then chooses a third of a slice from each plate. It might get messy, but all three should feel fairly treated.

10. Game Theory Secrets for ParentsWall Street Journal July, 2014 Credible Punishments: In game theory as in parenting, you have to deliver on your threats, like actually turning off the TV if you said you were going to, even if it punishes you too. Joshua Gans, an economist at the University of Toronto and the author of "Parentonomics," offers advice for gaining a credible reputation at home. When his children were young and would disobey, he would say, "I'm thinking of a punishment." It's much easier to pretend to think of a punishment than to come up with a new one every time, he notes—or, worse, to issue a noncredible threat in the heat of the moment. ("That's it, I'm canceling Christmas!") Once he earned his credibility, he found that he had only to close his eyes and count to 10, and his children would spring into action. The Bedtime Ultimatum: For shortening the bedtime routine with several children to tuck in, one parent advises using an ultimatum game of take-it-or-leave-it. Before bed, just have the children play rocks, paper, scissors and allow the winning child to choose the book. If the others don't agree with the choice, no one gets a story. Sleep Training 101: Game theory can work from the earliest days of parenthood. Prof. Nalebuff applies the concept of backward induction to help new mothers get some sleep. If a mother repeatedly gets up in the middle of the night with the child, he explains, eventually the child will only respond to the mother comforting him. Instead, mothers should look forward and reason backward: If you ever want your husband to get up in the middle of the night, then you have to get him involved at the very start. Everyone, it seems, needs to be sleep trained.

11. 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).

12. 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.

13. 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?

14. 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?

15. 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.

16. 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

17. 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.)

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

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

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

21. 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?

22. 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?

23. 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.

24. 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.

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

26. 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.

27. 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.

28. 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.

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

30. 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.

31. 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.

32. 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.

33. 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”.

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

35. 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)

36. 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.)

37. 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

38. Example 7: With “Downstream” Actions Potential Entrant Don’tEnter Enter Incumbent Firm Potential Entrant PE1 PIM1 Incumbent Firm Incumbent Firm PID1 PIM2 Potential Entrant and so on…. The present discounted value of profits for the incumbent and potential entrant depends on their strategies. PE2 Suppose each period the incumbent sets the optimal price as a monopolist and maximizes the present discounted value of profits which is +10. and so on….

39. 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?

40. 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.

41. 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)

42. 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)

43. 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

44. 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.

45. 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.)

46. Example 9: Limit Pricing When a firm sets it price and output so that there is not enough demand left for another firm to enter the market profitably.

47. Incumbent (suppose monopolist) Example 9: Lower Price, PL Monopoly Price, PM Potential Potential Entrant Entrant Don’t Don’t Enter Enter Enter Enter Incumbent Incumbent Hard Soft PL PM Hard Soft PL PM Ball Ball Ball Ball PE: -1 +5 0 0 -1 +5 0 0 Inc: 8+1 8+5 8+8 8+10 10+1 10+5 10+8 10+10 Note: Incumbent’s profits are $10 per period if set monopoly price and $8 per period if set lower price. What price the incumbent sets initially does not influence second period profits for incumbent or potential entrant. For simplicity, second period payoffs are not discounted.

48. Incumbent (suppose monopolist) Example 9: Lower Price, PL Monopoly Price, PM Potential Potential Entrant Entrant Don’t Don’t Enter Enter Enter Enter Incumbent Incumbent Hard Soft PL PM Hard Soft PL PM Ball Ball Ball Ball PE: -1 +5 0 0 -1 +5 0 0 Inc: 8+1 8+5 8+8 8+10 10+1 10+5 10+8 10+10 Note: Incumbent’s profits are $10 per period if set monopoly price and $8 per period if set lower price. What price the incumbent sets initially does not influence second period profits for incumbent or potential entrant. For simplicity, second period payoffs are not discounted.

49. Incumbent (suppose monopolist) Example 9a: Lower Price, PL Monopoly Price, PM Potential Potential Entrant Entrant Don’t Don’t Enter Enter Enter Enter Incumbent Incumbent Hard Soft PL PM Hard Soft PL PM Ball Ball Ball Ball PE: -1 -.5 0 0 -1 -.5 0 0 Inc: 8+1 8+5 8+8 8+10 10+1 10+5 10+8 10+10 Note: Incumbent’s profits are $10 per period if set monopoly price and $8 per period if set lower price. What price the incumbent sets initially does not influence second period profits for incumbent or potential entrant. For simplicity, second period payoffs are not discounted.

50. Incumbent (suppose monopolist) Example 9a: Lower Price, PL Monopoly Price, PM Potential Potential Entrant Entrant Don’t Don’t Enter Enter Enter Enter Incumbent Incumbent Hard Soft PL PM Hard Soft PL PM Ball Ball Ball Ball PE: -1 -.5 0 0 -1 -.5 0 0 Inc: 8+1 8+5 8+8 8+10 10+1 10+5 10+8 10+10 Note: Incumbent’s profits are $10 per period if set monopoly price and $8 per period if set lower price. What price the incumbent sets initially does not influence second period profits for incumbent or potential entrant. For simplicity, second period payoffs are not discounted.