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Thinking Strategically
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  • 1. Thinking Strategically Lee Ching Chyi The Chinese University of Hong Kong
  • 2. Chris Childs: “When you get elbows to the head and face, you cannot let it go. I will retaliate every time ” December 11, 1997 Rival guards Michael Jordan of Chicago and Chris Childs of the NY Knicks jostle. Both players received technical fouls but Childs was ejected after he threw the ball into Jordan’s Chest Jordan: “He then threw the ball at me and that really irritated me. Certainly I could have hit him back, it could have been one-, two- or a three-game suspension. But my level head overruled everything ”
  • 3. Charlie Brown & Lucy
  • 4. Charlie Brown & Lucy
  • 5. Bargaining, Game Theory, and Strategic Thinking
    • Bargaining involves interactions between two or more parties.
    • Game Theory
      • A study of interactive behavior among a group of rational players.
      • An important tool for strategic thinking – an essential element of successful bargaining.
  • 6. Types of Interaction
    • Sequential Move
      • Players take turn to make decision (to move).
    • Simultaneous Move
      • All players make decisions at the same time without knowing the decisions made by other players.
  • 7. Sequential Move
    • Game tree
      • A pictorial representation of the game being played.
      • A useful tool to analyze sequential move games.
    • Look Ahead and Reason Backward
      • A powerful way of finding the best move in any stage of a sequential move game.
  • 8. Game Tree: Charlie Brown & Lucy Charlie Lucy Accept Reject Pull Ball Away Let Charlie Kick
  • 9. To Enter or Not to Enter? To Fight or Not to Fight?
    • Suppose the market for vacuum cleaners in an area is dominated by a brand called Fastcleaners, and a new firm, Newcleaners, is deciding whether to enter this market. If Newcleaners enters, Fastcleaners has two choices: accommodate Newcleaners by accepting a lower market share, or fight a price war. If Fastcleaners accommodates the entry, both companies will make a profit of $100,000. However, if Fastcleaners starts a price war, this will cost them $100,000, but can also cause Newcleaners to lose $200,000. If Newcleaners stays out, the profit to them is $0, and to Fastcleaners is $300,000.
    • What will happen ?
  • 10. Game Tree: Entry Deterrence Game Newcleaners Fastcleaners Enter Stay out Accommodate Fight Price War $100,000 to Newscleaners $100,000 to Fastcleaners - $200,000 to Newscleaners - $100,000 to Fastcleaners $0 to Newscleaners $300,000 to Fastcleaners
  • 11. Centipede Game
    • In the Centipede game, there are two players. Each player takes turn to make a move – Down (D) or Across (A). Player 1 moves first and Player 2 second. Each player gets to move no more than 5 times. Whenever a player chooses A, the game continues and the other player gets to move. However, whenever a player chooses D, the game is over and each player gets his payoff calculated as follows:
      • If Player 1 chooses D in his n -th move, he gets $4 n and Player 2 gets $4 n -1 .
      • If Player 2 chooses D in his m -th move, he gets $4 m  2 and Player 1 gets $4 m / 2.
    • What will happen ?
  • 12. Game Tree: Centipede Game The first number is Player 1’s payoff and the second is Player 2’s payoff D A 1 D A D A D A D A D A D A D A D A D A 2 1 2 1 2 1 2 1 2 4 1 2 8 16 4 8 32 64 16 32 128 256 64 128 512 1024 256 512 2048 4096 1024
  • 13. Split a Dollar Allan, Barbie, and Charles have $100 to split. Allan gets to offer first, and offers shares b and c to Barbie and Charles, keeping a for himself ( a + b + c = 100). If both accept, the game is over and the dollar is divided accordingly. If either Barbie or Charles rejects the offer, however, they come back the next day and start again, this time Barbie making the offer to Allan and Charles, and if this is rejected, on the third day Charles gets to make the offer. If this is rejected, they all get nothing. What will happen?
  • 14. Game Tree: Split a Dollar A Offer b, c Reject Accept ( a, b, c ) B Offer a , c Reject Accept ( a, b, c ) C Offer a, b Reject Accept ( a, b, c ) (0, 0, 0)
  • 15. Simultaneous Move
    • For simultaneous move games, we can often identify the best strategy by careful examination of the Payoff table of the game.
  • 16. Dominant Strategy
    • The strategy that results in the best payoff under all contingencies.
    • Rule: Use dominant strategy whenever there is one
  • 17. Prisoners’ Dilemma Game
    • The police have arrested two suspects of a crime. However, they lack sufficient evidence to convict either of them unless at least one of them confesses. The police hold the two suspects in separate cells and explain the consequences of their possible actions. If neither confesses, then both will be convicted of a minor offence and sentenced to one year in prison. If both confess, they will be sent to prison for six years. Finally, if only one of them confesses, then that prisoner will be released immediately while the other one will be sentenced to nine years in prison – six years for the crime and a further three years for obstructing the course of justice. What will happen ?
  • 18. Payoff Table: Prisoners’ Dilemma Game The first number is 1’s payoff and the second number is 2’s payoff -1, -1 -9, 0 Don’t Confess 0, -9 -6, -6 Confess Prisoner 1 Don’t Confess Confess Prisoner 2
  • 19. To Invest or Not to Invest?
    • Company A and B are duopoly in a market, each with 50% market share. Because of competition, both companies are considering whether to invest in IT technology. Presumably, investment in IT should help them improve their operational efficiencies and hence capture more market share. If only one company makes the investment, the one who makes the investment will be able to capture 25% more market share in addition to the 50% it originally enjoys. However, if both companies make investment, no one gains any advantage and hence they will still share the market equally. The cost of IT investment, in monetary term, is about equal to the monetary value for 5% of the market share. What will happen ?
  • 20. Payoff Table: To Invest or Not To Invest? The first number is A’s payoff and the second number is B’s payoff 50%, 50% 25%, 70% No 70%, 25% 45%, 45% Invest Company A No Invest Company B
  • 21. Indiana Jones and the Last Crusade
    • In the movie, Indiana Jones and the Last Crusade , Indiana Jones, his father, and the Nazis have all converged at the site of the Holy Grail. The two Joneses refuse to help the Nazis reach the last step. So the Nazis shoot Indiana’s dad. Only the healing power of the Holy Grail can save the senior Dr. Jones from his mortal wound. Suitably motivated, Indiana leads the way to the Holy Grail. But there is one final challenge. He must choose between literally scores of chalices, only one of which is the cup of Christ. While the right cup brings eternal life, the wrong choice is fatal. The Nazi leader impatiently chooses a beautiful golden chalice, drinks the holy water, and dies the sudden death that follows from a wrong choice. Indiana picks a wooden chalice, the cup of a carpenter. Exclaiming “There’s only one way to find out” he dips the chalice into the font and drinks what he hopes is the cup of life. Upon discovering that he has chosen wisely, Indiana brings the cup to his father and the water heals the mortal wound. Could Indiana Jones handle his situation in a better way ?
  • 22. Dominated Strategy
    • A strategy A is dominated by another strategy B if B always results in a better payoff than A regardless of the other players’ actions .
    • Rule: Never use a dominated strategy
  • 23. How would the players play this game? The first number is 1’s payoff and the second number is 2’s payoff 30, 0 0, 20 10, 30 Bottom 20, 30 10, 20 30, 25 Center Player 1 40, 20 10, 10 20, 0 Top Right Middle Left Player 2
  • 24. How would you play this game? Player 1 -3, 72 -2, -57 -1, 88 1, -13 -22, 0 E -1, 17 1, -12 -1, 39 -3, 43 1, -33 D 0, 4 4, -1 0, 2 95, -1 54, 1 C 2, 3 33, 0 2, 5 2, 2 32, 1 B -3, 19 -2, 45 -2, 0 28, -1 63, -1 A e d c b a Player 2
  • 25. An Armaments Game
    • A fighter command has four strategies, and its opponent bomber command has three counterstrategies. The diagram on the next slide shows the probability that the fighter destroys the bomber. What will happen ?
  • 26. Payoff Table: Armaments Game 0.10 0.16 0.21 Ramming 0.17 0.22 0.35 Toss-bombs 0.16 0.14 0.18 Rockets Fighter Command 0.15 0.25 0.30 Guns No Fire, High Speed Partial Fire, Med. Speed Full Fire, Low Speed Bomber Command
  • 27. Gulf War
    • The grid on the next slide shows the positions and the choices of the combatants. An Iraqi ship at the point I is about to fire a missile, intending to hit an American ship at A. The missile’s path is programmed at the launch; it can travel in a straight line, or make sharp right angled turns every 20 seconds. If the Iraqi missile flew in a straight line from I to A , American missile defenses could counter such a trajectory very easily. Therefore the Iraqis will try a path with some zigzags. All such paths that can reach A from I lie along the grid shown. Each length like IF equals the distance the missile can travel in 20 seconds.
    • The American ship’s radar will detect the launch of the incoming Iraqi missile, and the computer will instantly launch an antimissile. The antimissile travels at the same speed as the Iraqi missile, and can make similar 90-degree turns. So the antimissile’s path can also be set along the same grid starting at A . However, to allow for enough explosives to ensure a damaging open-air blast, the antimissile has only enough fuel to last one minute, so it can travel just three segments (e.g., A to B to C to F ).
    • If, before or at the end of the minute, US antimissile meets the incoming missile, it will explode and neutralize the threat. Otherwise Iraqi missile will go on to hit US ship. How should the trajectories of the two missiles be chosen?
  • 28. Gulf War A B C D E F G H I
  • 29. Payoff Table: Gulf War H: US missile hits Iraqi’s; O: US missile misses the target H H H O H H H H ADEB H H H O H H H O ADEF H H H O H H H O ADEH O O O H H O O O ADGH H H H H H H H O ABED H H H O H H H O ABEH H H H O H H H O ABEF H O O O O O O H ABCF US Strategies IHEF IHEB IHED IHGD IFEH IFED IFEB IFCB Iraqi Strategies
  • 30. Games without Dominated Strategies
    • Most games don’t have dominant or dominated strategies.
    • In such case, each player must consider “ What would my opponent do if he knew that I were doing this? Given what my opponent would do, should I do this? ” and must be fully aware of the fact that all other players can do the same reasoning.
    • If all the players are doing the same reasoning, the resulting outcome of a game is called Nash Equilibrium.
  • 31. Nash Equilibrium
    • Named after the renowned mathematician who first developed this concept – John Nash, the winner of 1994 Nobel Prize in Economics.
    • A solution concept used to predict the outcome of a game played by rational players .
    • A state in which no rational player has incentive to deviate from his or her strategy given the others’ strategy.
    • The solution found through iterated elimination of dominated strategies is a Nash Equilibrium.
  • 32. How would the players play this game? The first number is 1’s payoff and the second number is 2’s payoff 30, 0 0, 20 10, 30 Bottom 20, 20 10, 20 30, 25 Center Player 1 25, 20 10, 10 20, 0 Top Right Middle Left Player 2
  • 33. The Battle of the Sexes
    • Sometimes, a game may have more than one equilibrium, as the following example demonstrates.
    • A couple must decide what to do during the coming weekend. The husband would like to go to a soccer game while the wife prefers going to a concert. No matter what each other’s choice is, they both prefer doing things together than individually. Now, suppose that the husband and the wife cannot communicate when they make their choices, given the payoff on the next slide, what would they do ?
  • 34. Payoff Table: The Battle of the Sexes The first number is husband’s payoff and the second number is wife’s payoff 3, 10 0, 0 Concert 0, 0 10, 3 Soccer Husband Concert Soccer Wife
  • 35. Games with No Pure Strategy Equilibrium
    • Sometimes, a game may not have any pure strategy equilibrium. In such a case, they may play a mixed strategy equilibrium.
      • Pure Strategy Equilibrium : An equilibrium in which all players play their strategy for sure.
      • Mixed Strategy Equilibrium : An equilibrium in which some players play their strategies randomly .
  • 36. Stone-Scissors-Paper Game The first number is 1’s payoff and the second number is 2’s payoff 0, 0 -1, 1 1, -1 Paper 1, -1 0, 0 -1, 1 Scissors Player 1 -1, 1 1, -1 0, 0 Stone Paper Scissors Stone Player 2
  • 37. Applications
    • Picking Up Matches
    • The Three-way Duel
    • Sailors & Gold Coins
    • ZECK
    • The Voting Game
  • 38. Picking Up Matches
    • There are 30 matches on the table. Player 1 begins by picking up 1, 2, or 3 matches. Then Player 2 must pick up 1, 2, or 3 matches. Players continue in this fashion until the last match is picked up. The player who picks up the last match is the loser. How can Player 1 be sure of winning the game?
  • 39. Picking Up Matches
  • 40. The Three-way Duel
    • Three antagonists, Larry, Mo, and Curly, are engaged in a three-way duel. There are two rounds. In the first round, each player is given one shot: first Larry, then Mo, and then Curly. After the first round, any survivors are given a second shot, again beginning with Larry, then Mo, and then Curly.
    • For each duelist, the best outcome is to be the sole survivor. Next best is to be one of two survivors. In third place is the outcome in which no one gets killed. Dead last is that you get killed.
    • Larry is a poor shot, with only a 30% chance of hitting a person at whom he aims. Mo is a much better shot, achieving 80% accuracy. Curly is a perfect shot – he never misses.
    • What is Larry’s optimal strategy in the first round? Who has the greatest chance of survival in this problem?
  • 41.
    • If Larry shoots at Mo and hits, then he will die for sure (why?). If Larry shoots at Curly and hits, Mo will then shoot at Larry, in which case his chance of survival is only 20%. Larry’s best strategy is to fire up in the air in the first round. In this case, Mo will then shoot at Curly (why?), and if he misses, Curly will shoot and kill Mo (why?). Then it becomes the second round. Since only one other person remains, Larry has at least a 30% chance of survival.
    • Overall, Larry’s surviving probability is 0.412 , the sum of the following:
      • Mo killed Curly in the first round (80%) and Larry killed Mo in the second round (30%) = 0.24.
      • Mo killed Curly in the first round (80%) and Larry didn’t kill Mo in the second round (70%) and Mo misses in the second round (20%) = 0.112.
      • Mo didn’t kill Curly in the first round (20%) and Larry killed Curly in the second round (30%) = 0.06.
    • Curly can survive if he is not killed by Mo in the first round (20% chance) and by Larry in the second round (70% chance) . So, his survival probability is: 0.2 × 0.7 = 0.14.
    • Mo can only survive if he killed Curly in the first round and is not killed by Larry in the second round. So, his survival probability is: 0.8 × 0.7 = 0.56.
  • 42.
    • There is nothing weaker then water,
    • But none is superior to it in overcoming the hard,
    • For which there is no substitute,
    • That weakness overcomes strength,
    • And Gentleness overcomes rigidity,
    • No one does not know:
    • No one can put into practice.
    • Laotse, The Book of Tao
    • 天下莫柔弱於水,
    • 而攻堅強者莫之能勝,以其無以之。
    • 弱之勝強,
    • 柔之勝剛,
    • 天下莫不知;
    • 莫能行。
    • 老子道德經
  • 43.
    • Hardness and Stiffness are the companies of death,
    • And softness and gentleness are the companies of life.
    • Laotse, The Book of Tao
    • 堅強者死之徒,
    • 柔弱者生之徒。
    • 老子道德經
  • 44. Sailors & Gold Coins
    • In Joseph Conrad’s novel Typhoon , a number of sailors store gold coins in private boxes kept in the ship’s safe. The ship hits stormy weather, the boxes break open, and the coins are hopelessly mixed. Each sailor knows how many coins he started with, but nobody knows what anybody else started with. The captain’s problem is to return the correct number of coins to each sailor.
    • How should the captain solve the problem?
  • 45. ZECK
    • ZECK is a dot-game for two players. The object is to force your opponent to take the last dot. The game starts with dots arranged in any rectangular shape, for example a 7 × 4:
  • 46. ZECK
    • Each turn, a player removes a dot and with it all remaining dots to the northeast. If the first player chooses the fourth dot in the second row, this leaves his opponent with
    • Each period at least one dot must be removed. The person who is forced to take the last dot loses. Who will win this game?
  • 47. ZECK
  • 48. Player 1 will win. If Player 2 has a winning strategy, that means that for any opening move of the first player, the second has a response that puts him in a winning position. Imagine that the first player just takes the upper right-hand dot . Then, no matter how the second player responds, the board will be left in a configuration that the first player could have created in his first move. If this is truly a winning position, the first player should have and could have opened the game this way. There is nothing the second player can do to the first that the first cannot do unto him beforehand.
  • 49. The Customs Officers and Smugglers
    • Most of us have been to other countries and witnessed numerous officers hired at the Customs. We also notice that officers don’t usually check every passengers passing by their counters – they only do it occasionally.
    • On the other hand, even with so many officers on duty, smugglers can never really be deterred due to the innate nature of human greediness.
    • Suppose that officers are perfectly effective in catching smugglers when they do inspect. Would they inspect everyone? If they do, can they stop everyone from smuggling? If smuggling can be stopped, do we still need the Customs (assuming this is the only duty they do)?
    • Why don’t officers inspect every passenger and why can’t smuggling be stopped?
  • 50. The Voting Game
    • There are three job candidates A, B, and C, and three committee members V 1 , V 2 , and V 3 . Committee members vote to decide which candidate gets the job and they use a simple plurality rule – each member can only vote for one candidate and the candidate receiving the most vote wins the election. Since V 1 is the committee chair, if there is a tie, the candidate he votes for will get the job.
    • Suppose members rank the candidates as follows:
      • V 1 : A > B > C
      • V 2 : B > C > A
      • V 3 : C > A > B
    • If each member vote for his most preferred candidate, which candidate will get the job?
    • If all members vote strategically , which candidate will win? ( Voting Game.xls )
  • 51. The fact that you have power does not necessarily make you a powerful man.
  • 52. Exercises
    • Ultimatum Game
    • Repeated Prisoners’ Dilemma Game
  • 53. Ultimatum Game
    • In this exercise, you will be divided into two groups – one called Player 1 and the other Player 2. Each of you will get a number card and you will be matched with the person in the other group who has the same number card. You should not reveal your number card to anyone during this exercise.
    • The two matched players will then have to negotiate on how to divide 100 points. The game proceeds as follows. First, Player 1 makes a proposal for how to divide the 100 points. For example, he can say “I get 75 points and you get 25 points”. Then, Player 2 can say “ accept ” or “ reject ”. If 1’s proposal is accepted, 1 and 2 will get the points proposed; otherwise none of them gets anything.
    • We will play the game twice and you will have the chance to play both roles.
  • 54. Repeated Prisoners’ Dilemma Game
    • In this exercise, you will be divided into two groups – one called Player 1 and the other Player 2. Each of you will get a number card and you will be matched with the person in the other group who has the same number card. You should not reveal your number card to anyone during this exercise.
    • The two matched players will then play the Prisoners’ Dilemma Game repeatedly for 3 times.