1. SEEM 35301 Game Theory Games of strategy Sequential games Simultaneous decisions Dominated strategies Nash equilibrium Prisoners’ dilemma

2. SEEM 35302 Sequential decisions Previously …. Sequential decisions with uncertainty Decision trees … with “chance” nodes but … “God does not play dice” – Albert Einstein “Subtle is the Lord, but malicious He is not.” What about your competitors?

3. SEEM 35303 A sequential “game” Decisions made in sequence. Your decision depends on decision made previously by others, and others’ decisions follow and depend on yours, etc. all all. Outcome/payoff depends on all decisions made by all.

4. SEEM 35304 Lucy Van Pelt vs. Charlie Brown Lucy Van Pelt holds a football on the ground and invites Charlie Brown to run up and kick it. At the last moment, Lucy pulls the ball away. Charles Brown, kicking air, lands on his back, and this gives Lucy great perverse pleasure.

5. SEEM 35305 This time …

6. SEEM 35306 Representing Decisions in a Game Tree Charlie Reject AcceptLucy Pull Ball Away Let Charlie kick ,,  ,,

7. SEEM 35307 And then …

8. SEEM 35308 Games of Strategy Vijay Krishna (Harvard Business School): Any situation where the choices of two or more rational decision makers together leads to gains and losses for them is called a game. A game may simultaneously involve elements of both conflict and co-operation among the decision makers.

9. SEEM 35309 Market Competition - HDTV both Vizio considers entering a market now monopolised by Samsung. Samsung can decide to respond by being accommodating or aggressively fight a price war. Profit outcomes for both firms depends on the strategies of both firms. As Vizio, you can analyse this problem using Decision Analysis by estimating probabilities of Samsung’s responses.

10. SEEM 353010 Market Entry – Decision Tree for Vizio Vizio Keep out Enter Samsung Accommodate Fight price war $0 to Vizio $100,000 to Vizio -$200,000 to Vizio p 1-p How to estimate the probabilities? What does p depend on? If no information, p=0.5? Then Vizio will not enter market.

11. SEEM 353011 Game Tree Representation Probability of Samsung’s response will depend on Samsung’s payoff in the different scenarios Vizio Samsung 0, 10 -7, 2 5, 8 Do not enter Enter Market Aggressive Accommodate

12. SEEM 353012 Market Entry – Game Tree Model Vizio Keep out Enter Samsung Accommodate Fight price war $0 to Vizio $300,000 to Samsung $100,000 to Vizio $100,000 to Samsung -$200,000 to Vizio -$100,000 to Samsung

13. SEEM 353013 Analysing Game Trees Rule 1: Look Ahead and Reason Back! For this market alone, Vizio should choose enter because Samsung (rationally) will accommodate. If Samsung worries that Vizio may enter other markets in the region after this, Samsung may take a tough stand. Vizio should not enter. The “payoff” should include all “benefits”.

14. SEEM 353014 Look Ahead & Reason Back 1. Formulate the game tree of the situation. Identify your own and opponent’s strategy at each stage. This assumes: Your opponent’s strategy can be observable. Strategy must be irreversible. 2. Evaluate payoffs at the “leaves” of the tree. Think about what will happen at the end. 3. Reason backward through the tree. Identify the best strategy for each player at each stage, starting at the end. Note the essence of a game of strategy is interdependence. Your decision affects your opponent’s decision and your opponent’s decision affects yours.

15. SEEM 353015 More complex games White-1 P-K4 P-Q4 Black-1 P-QB4 White-2 Theoretically, can map out all possible chess moves and then select the best sequence of moves to win the game!

16. SEEM 353016 Chess - Human vs. Computers Good chess players can “see” 14 moves ahead! (1968) David Levy: “No computer can beat him in 10 years” Deep Blue Chess playing machine built by IBM in the 1990’s 2 to 2.5 million moves per second. (1996) Deep Blue 1 lost to world chess champion Gary Kasparov. (1997) Deep Blue 2 defeated Kasparov.

17. SEEM 353017 Deep Blue vs. Kasparov 1996, game 1. The final position. abcdefgh 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 abcdefgh

18. SEEM 353018 Homework? Draw the game tree for TIC-TAC-TOE. Sure-win strategies?

19. SEEM 353019 The Election of the Chief Executive for Hong Kong The next Chief Executive of Hong Kong SAR Government will be “elected”. Mr. B is Beijing’s favourite candidate. Ms. C (the potential challenger) considers entering the race. Mr. B must determine whether to launch a preemptive advertising campaign against Ms. C (expensive) or not (cost-saving). Ms. C must determine whether to enter the race.

20. SEEM 353020 Election Game Tree B No Ad Advertise Out C In 1, 1 Out C In 3, 2 2, 4 4, 2 B’s, C’s payoff The larger the better

21. SEEM 353021 Game Tree B No Ads Ads Out C In 1, 1 Out C In 3, 2 2, 4 4, 2 B’s, C’s payoff

22. SEEM 353022 Game Tree B No Ads Ads Out C In 1, 1 Out C In 3, 2 2, 4 4, 2 B’s, C’s payoff

23. SEEM 353023 Advantage due to Order of Decisions? First-mover advantage? Mr. B (first) sets the stage for Ms. C (second). Mr. B can look ahead to Ms. C’s optimal response and make the move to his advantage. Can Ms. C improve her situation by acting first?

24. SEEM 353024 Game Tree C Out In No Adv B Adv 1, 1 No Adv B Adv 4, 2 2, 3 2, 4 C’s, B’s payoff

25. SEEM 353025 Better off being first? Is there a first-mover advantage? What about adoption of new technology? Better off as a technology leader? Better off as a technology follower?

26. SEEM 353026 Simultaneous Decisions In the chess example, the sequence of decisions alternate between the players. In other situations, the decision may not be sequential but simultaneous. Tic-tac-toe (sequential) Stone-paper-scissors (simultaneous) In simultaneous games, the payoffs to the players are still interdependent on chosen strategies of ALL players.

27. SEEM 353027 Time vs. Newsweek Each week, these magazines decide on what story to put on the cover. They do not know the other’s decision until publication. Suppose there are two “hot” stories: (A): Anna Chapman, the Russian Spy, (B): British Petroleum Oil Spill damage Newsstand buyers only purchase if story is on cover. 70% interested in (A) and 30% in (B). Purchases evenly split the if both magazines have the same story.

28. SEEM 353028 Matrix Representation of Game Payoff for Time Newsweek AB Time A35 = 70/270 B3015 = 30/2 What should Time do?

29. SEEM 353029 Matrix Representation of Game for Newsweek Payoff to Newsweek AB Time A3530 B7015 No matter what Time does, Newsweek is better off putting (A) as cover story.

30. SEEM 353030 Dominant Strategies Payoffs Newsweek Time, Newsweek AB Time A35, 3570, 30 B30, 7015, 15 Choosing (A) is a dominant strategy for both Time and Newsweek!

31. SEEM 353031 Dominant Strategy A dominant strategy is one that makes a player better off than he would be if he used any other strategy, no matter what strategy his opponent uses. A strategy is dominated if there is another strategy that under no circumstances leads to a lower payoff, and sometimes yields a better payoff. Note: For some games, there may be no dominant strategy for some players.

32. SEEM 353032 Properties of a dominant strategy 1: A dominant strategy dominates your other strategies, NOT your opponent! Even with your dominant strategy, your payoff could be smaller than your opponents. 2: A dominant strategy does not requires that the worst possible outcome of the dominant strategy is better than the best outcome of an alternative strategy.

33. SEEM 353033 Pricing example Time’s Newsweek’s Price Sales $2$3 Time’s Price $24 million8 million $30 million5 million Suppose there are just two possible pricing choices: $3 (a profit margin of $2 per copy) and $2 ($1 per copy). Customers will always buy the lower-priced magazine. Profits are split equally between the two. The total readership is 5 million if the price is $3, and rises to 8 million if the price is only $2.

34. SEEM 353034 Analysing Games Rule 2: If you have a dominant strategy, use it ! Rule 3: Eliminate any dominated strategies from consideration, and do so successively!

35. SEEM 353035 Eliminating Dominated Strategies - Example American ship at A, Iraqi ship at I. Iraqi plans to fire a missile at American ship; American ship plans to fire a defense missile to neutralize the attack (simultaneously). Missiles programmed to (possibly) turn every 20 seconds. If missile not neutralised in 60 seconds, American ship sinks! E I A F C B DH G

36. SEEM 353036 Possible strategies (paths) For American, A2, A3 dominated by A4, A6, A7 dominated by A8, A1 is dominated by A8, A5 is dominated by A4, Only A4 and A8 not dominated. Similarly for Iraqi. Only I1 and I5 are not dominated.

37. SEEM 353037 Simplified Game American Iraqi vs. Iraqi I1- IFCB I5- IHGD American A4- ABED OH A8- ADEB HO E I A F C B D H G No dominant strategy for either player!

38. SEEM 353038 Nash Equilibrium A set of strategies constitute a Nash Equilibrium if: no player can benefit by changing her strategy while the other players keep their strategies unchanged. Each player’s strategy is the “best-response” to the other players’ set of strategies.

39. SEEM 353039 Dominant Strategy Equilibrium Higher viewership means more advertising revenues for both TV stations. Each TV station has a dominant strategy. In this case, the equilibrium for this game is obvious. TVB payoff ATV ATV payoff Soap- Opera News & Analysis TVB Soap- Opera 55%, 45% 52%, 48% News & Analysis 50%, 50% 45%, 55%

40. SEEM 353040 Dominant Strategy Equilibrium If, in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of (dominant) strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game.

41. SEEM 353041 Nash Equilibrium Payoffs Newsweek Time, Newsweek AB Time A42, 2870, 30 B30, 7018, 12 No dominant strategy for Newsweek. Unique Nash equilibrium.

42. SEEM 353042 Example - “Chicken” H > C > D No dominant strategy Two Nash equilibria James Dean SwerveDon’t swerve MadSwerveC, CC, H MaxDon’t swerve H, CD, D

43. SEEM 353043 Choosing among Multiple Equilibria Some games have multiple equilibria. “Rule of the road” Hong Kong, Britain, Australia, Japan (left) China, Europe, Mexico, USA (right) The social convention of the locale determines which equilibrium to choose. Brit Drive on left Drive on right Yank Drive on left, D, D Drive on right D, D, Sweden switch from left to right in 1967.

44. SEEM 353044 In-class exercise (Texas A&M) 12 Each of you owns a production plant and can choose to produce 1 or 2 units of a product. More total production will lower price and hence profit. What would you do? # of “1” Payoff to “1” firms Payoffs to “2” firms 0$0.50 1$0.04$0.54 2$0.08$0.58 ::: 29$1.16$1.66 30$1.20$1.70 ::: 59$2.36$2.86 60$2.40$2.90 :::

45. SEEM 353045 Is a Nash equilibrium “good” for the players? Just because a game has an equilibrium does not mean that those strategies are “best” for the players. Prisoners’ dilemma: Two burglars, Bob and Al, are captured at the scene of a burglary and interrogated separately by the police. Each has to choose whether or not to confess. Outcomes: If neither man confesses, then both will serve only one year. If both confesses, both will go to prison for 10 years. However, if one burglar confesses and implicates the other, and the other burglar does not confess, the one who has collaborated with the police will go free, while the other burglar will go to prison for 20 years.

46. SEEM 353046 Prisoners’ dilemma Punishment Al confessdeny Bob confess10,100,20 deny20,01,1 For each player, the dominant strategy is to confess! Unique Nash equilibrium! Both play the dominant strategy but create mutually disastrous outcome! Both would be better off by denying! 1950 – Dresher & Flood (Rand) A. W. Tucker

47. SEEM 353047 Cartels Companies or countries form an alliance to jointly make price and production decisions. World Trade Organisation (WTO) / General Agreement on Tariffs and Trade (GATT) The Organization of Petroleum Exporting Countries (OPEC) is a cartel. the mission of OPEC is to coordinate and unify the policies of its Member Countries and ensure the stabilization of oil markets in order to secure a regular supply of petroleum to consumers, a steady income to producers …

48. SEEM 353048 OPEC – Maintaining a Cartel Total output:4mb6mb8mb Price per barrel: $25$15$10 Extraction costs: Iran:$2/barrel; Iraq:$4/barrel Dominant strategy: produce at higher level !

49. SEEM 353049 Ensuring Co-operation The dominant strategy equilibrium results in each producing 4 million barrels and achieving 56 million in total joint profit. Suppose OPEC countries have agreed to maintain production at 2 mb per day. If members produces 2 million barrels each (as agreed), they will make 88 million in total joint profit. Is it possible to achieve cooperation, when the dominant strategy is to cheat?

50. SEEM 353050 Detection of Cheating Co-operation is difficult when the reward for cheating is high. How to tell if some member cheated and produced more? The price is US$25 per barrel only if members maintained low production. If price drops below $25, then someone has cheated! What if demand actually decreased?

51. SEEM 353051 Identifying cheaters In a two-player game, an honest party knows who cheated. Still, the cheating party may deny they cheated. When there are many players, even when cheating has been detected, it may be difficult to identify who cheated ! If voluntary cooperation is not possible, how about making use of punishment? In a one-period game, there is no solution to achieve reciprocal co-operation.

52. SEEM 353052 Punishment – Credible Threat? If the game repeats, cooperation may be enforced. Suppose Iran begins to cheat and produces 4 million barrel per day secretly. Iran’s profit goes up from 46 to 52 million per day. When Iraq finds out, Iraq also produces 4 million barrels. Iran’s profit goes down to 46 to 32 million per day. Assume it takes a month for Iraq to know. Iran’s total profit through cheating: 6x30= 180 million Iraq retaliates by increasing production. Iran’s cheating gain will be wiped out in 13 days (i.e., 180 million / 14 million)

53. SEEM 353053 Competition or Collusion? DVD player vendors: Fortress Broadway wholesale: $1500, retail: $3000 Broadway: lowest price guarantee: “refund double the price difference” Should Fortress cut its price to $2750? What will the consumer do? How will Broadway respond?

54. SEEM 353054 “Implicit” Cartel If Fortress tries to increase its market share by lowering its price to $2750. Customers will buy from Broadway and claim from a $500 rebate. The “selling price” for Broadway is effectively $2500; lower than Fortress’ price of $2750. In response, Broadway will not give away rebates but lower its price to $2750. Fortress becomes worse off … so why bother? Collusion is enforced by “announcing” the punishment!

55. SEEM 353055 Sustaining Co-operation Mechanism must Detect cheating and Deter cheating. Which Punishment? Simplicity and clarity Easy for potential cheaters to evaluate consequences. Certainty Players have confidence that defection will be punished and co-operation rewarded. Severity not to “fit the crime” but for deterrence! Risk of mistakes?

56. SEEM 353056 Tit-for-tat Exodus 21:22-25 If men who are fighting hit a pregnant woman and she gives birth prematurely but there is no serious injury, the offender must be fined whatever the woman’s husband demands. But if there is a serious injury, you are to take life for a life, eye for eye, tooth for tooth, hand for hand, burn for burn, wound for wound, bruise for bruise.

57. SEEM 353057 Tit-for-tat Strategy Co-operates in the first period, thereafter mimics the rival’s action from previous rounds Clarity ( simple to implement ) Niceness ( does not initiates cheating ) Provocability ( it never lets cheating go unpunished ) Forgiveness ( does not hold a grudge, willing to restore cooperation ) “Chain reaction” of mistakes?

58. SEEM 353058 Hatfields and McCoys This is one of best-documented stories on inter-family feud (1878 – 1891).18781891 Early settlers in the Tug Valley on the Kentucky and West Virginia border. Feud started over the disputed ownership of a pig! Kentucky West Virginia

59. SEEM 353059 Tit-for-tat with misperception Mis-perception leads to perpetual retaliation! Nuclear conflict? Cuban missile crisis (1962).

60. SEEM 353060 Tit-for-tat Strategy When misperceptions are possible, in the long run tit-for-tat will spend half the time cooperating and half of it defecting. When the probability of a misperception is small, it will take a lot longer for this phenomenon to occur. When the probability is 50%, whatever you do will not have any affect on your opponent.  Opponent will perceive aggression with 0.5 probability. When the probability is 50%, there is no hope of achieving co-operation. One should always attack! Feud never ends … Should one be more forgiving?

61. SEEM 353061 中庸之道 (The Moderate Chinese Way) Tit-for-tat is quick to punish opponent who has a long history of cooperation. Other responses: (Matthew 5:38): “But I tell you, do not resist an evil person. If someone strikes you on the right cheek, turn to him the other also.” A more forgiving tit-for-tat: Begin cooperating Continue cooperating, but keep count of how many times the other side appears to be have defected while you have cooperated. If the this percentage becomes unacceptable, revert to tit- for-tat.

62. SEEM 353062 Summary Games of strategy Sequential games Simultaneous decisions Dominated strategies Nash equilibrium Prisoners’ dilemma

63. SEEM 353063 Multiple equilibria Intranet Users Example AdvancedProven Supplier Advanced 20,200,0 Proven 0,05,5 Nash equilibria