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Game Theory

Game Theory . Game theory was developed by John Von Neumann and Oscar Morgenstern in 1944 - Economists! One of the fundamental principles of game theory, the idea of equilibrium strategies was developed by John F. Nash, Jr. ( A Beautiful Mind ), a Bluefield, WV native.

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Game Theory

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  1. Game Theory • Game theory was developed by John Von Neumann and Oscar Morgenstern in 1944 - • Economists! • One of the fundamental principles of game theory, the idea of equilibrium strategies was developed by John F. Nash, Jr. (A Beautiful Mind), a Bluefield, WV native. • Game theory is a way of looking at a whole range of human behaviors as a game.

  2. Components of a Game • Games have the following characteristics: • Players • Rules • Payoffs • Strategies

  3. Types of Games • We classify games into several types. • By the number of players: • By the Rules: • By the Payoff Structure:

  4. Games as Defined by the Number of Players: • 1-person (or game against nature) • 2-person • n-person( 3-person & up)

  5. Games as Defined by the Rules: • These determine the number of options/alternatives in the play of the game. • The payoff matrix has a structure (independent of value) that is a function of the rules of the game. • Thus many games have a 2x2 structure due to 2 alternatives for each player.

  6. Games as Defined by the Payoff Structure: • Zero-sum • Non-zero sum • (and occasionally Constant sum) • Examples: • Zero-sum • Classic games: Chess, checkers, tennis, poker. • Political Games: Elections, War • Non-zero sum • Classic games: Football (?), D&D, Video games • Political Games: Policy Process

  7. Strategies • We also classify the strategies that we employ: • It is natural to suppose that one player will attempt to anticipate what the other player will do. Hence • Minimax - to minimize the maximum loss - a defensive strategy • Maximin - to maximize the minimum gain - an offensive strategy.

  8. Iterated Play • Games can also have sequential play which lends to more complex strategies. • (Tit-for-tat - always respond in kind. • Tat-for-tit - always respond conflictually to cooperation and cooperatively towards conflict.

  9. Game or Nash Equilibria • Games also often have solutions or equilibrium points. • These are outcomes which, owing to the selection of particular reasonable strategies will result in a determined outcomes. • An equilibrium is that point where it is not to either players advantage to unilaterally change his or her mind.

  10. Saddle points • The Nash equilibrium is also called a saddle point because of the two curves used to construct it: • an upward arching Maximin gain curve • and a downward arc for minimum loss. • Draw in 3-d, this has the general shape of a western saddle (or the shape of the universe; and if you prefer). .

  11. Some Simple Examples • Battle of the Bismark Sea • Prisoner’s Dilemma • Chicken

  12. The Battle of the Bismarck Sea • Simple 2x2 Game • US WWII Battle

  13. The Battle of the Bismarck Sea

  14. The Battle of the Bismarck Sea - examined • This is an excellent example of a two-person zero-sum game with an equilibrium point. • Each side has reason to employ a particular strategy • Maximin for US • Minimax for Japanese). • If both employ these strategies, then the outcome will be Sail North/Watch North.

  15. Decision Tree

  16. The Prisoners Dilemma • The Prisoner’s dilemma is also 2-person game but not a zero-sum game. • It also has an equilibrium point, and that is what makes it interesting. • The Prisoner's dilemma is best interpreted via a “story.”

  17. A Simple Prisoner’s Dilemma

  18. What makes a Game a Prisoner’s Dilemma? • We can characterize the set of choices in a PD as: • Temptation (desire to double-cross other player) • Reward (cooperate with other player) • Punishment (play it safe) • Sucker (the player who is double-crossed) • A game is a Prisoner’s Dilemma whenever: • T > R > P > S

  19. What is the Outcome of a PD? • The saddle point is where both Confess • This is the result of using a Minimax strategy. • Two aspects of the game can make a difference. • The game assumes no communication • The strategies can be altered if there is sufficient trust between the players.

  20. Iterated Play

  21. Solutions to PD? • The Reward option is the joint optimal payoff. • Can Prisoner’s reach this? • Minimax strategies make this impossible • Are there other strategies?

  22. The Theory of Metagames • Metagames step back from the game and look at the other players strategy • Strategic choice is based upon opponents choice. • For instance, we could adopt the following strategies: • Tit-for-tat • Tat-for-tit • Choose Confess regardless • Choose ~Confess regardless

  23. A Prisoner’s Dilemma Metagame

  24. The Full PD Metagame • Using the Metagame strategy, we get three possible equilibria • One the original both confess • The other two, both ~confess (a cooperative solution)

  25. Chicken • The game that we call chicken is widely played in everyday life • bicycles • Cars • Interpersonal relations

  26. The Game of Chicken

  27. Chicken is an Unstable game • There is no saddle point in the game. • No matter what one player chooses, the other player can unilaterally change for some advantage. • Chicken is therefore unstable. • We cannot predict the outcome

  28. Chicken is Nuclear Deterrence

  29. National Missile Defense • Let’s pick a current problem • National Missile Defense • Structure this as a game

  30. The Game of National Missile Defense

  31. Calculating Expected Utility of NMD • E(Build)=pA(B-C)+p~A(B-C) • E(~Build)=pA(B-C)+p~A(B-C) • E(Build)=pA(0-60)+p~A(0-60) • E(~Build)=pA(0-1000)+p~A(0-0) • Build NMD if E(Build)>E(~Build)

  32. Spreadsheet • Open Excel table

  33. Utility Curves p1

  34. Tragedy of the Commons • First observed during the British Enclosure movement • Describes the problem of the unregulated use of a public good • Take a commons – e.g. a common pasture for grazing of cattle in a village

  35. An example • Take a village with 10 families • Each family has 10 cows which just exactly provide the food they need. • The village commons has a carrying capacity of 100 cows

  36. Cow Carrying Capacity • Each cow produces 500 lbs of meat & dairy per year up to or at carrying capacity of the pasture. • 10 families X 10 Cows X 500 lbs = 50,000 lbs of food at carrying capacity • …and then Farmer Symthe’s wife has triplets…

  37. One more cow • So Farmer Smythe decides he really needs one more cow. • And there is no one to tell him no because the commons is an unregulated public good • Like • Air • Water • Security ?

  38. Reduced Capacity • With the overgrazing, each cow will now produce onl;y 490 lbs of food. • 10 families X 10 Cows X 490 lbs = 49,000 lbs of food at carrying capacity • Each family gets 4900 lbs of meat & dairy, instead of 5000. • Except Farmer Smythe, who gets 5390 lbs • Even with the reduced carrying capacity, it is still to his advantage to add the extra cow

  39. Look Familiar? • Look at the situation • N players • Equilibrium solution is to ~ cooperate • Joint optimal outcome is to cooperate, • This is an n-person Prisoner’s dilemma

  40. Thinking Strategically • Some simple concepts that are strategic in nature. • They are worth reviewing, as they all demonstrate some particular strategic choice.

  41. The Hot Hand • Are ‘hot hands’ just random sequences in a long series of trials? • Probabilistic analysis suggests that what sports observers claim as periods of exceptional performance are not statistically excessive • But hot hands may be masked by team responses. • Take tennis: • If your backhand is weak, your opponent will play to it. • Improve your backhand, and you get to use your better forehand more

  42. To Lead or not to Lead • Front-runner strategy • The leading sailboat copies the strategy of the trailing boat. • Doesn’t work in 3 boat races • Applicable to election? • When to go negative?

  43. Here I stand • Taking an irrevocable stand may change your opponent’s strategies. • A public statement makes a committement (and thereby changes payoffs) in ways that may dictate an outcome. • Opponent has to “take it or leave it. • May be costly next time!

  44. Belling the Cat • Is the individual willing to assume the risks of the group • This is a “Hostages Dilemma” • Note the reference to plane full of passengers powerless before a hijacker with a gun. • Is this likely to be the case after 9/11? • What does this say about this strategic decision?

  45. Mix your plays • Rely on you best strategy – strongest asset • But not exclusively • If you run the football every play, the defensive backfield will pull in and you will be less effective. • The pass sets up the run.

  46. Never give a sucker an even bet • When someone offers to bet you, they often know the odds – don’t bet. • Such as appliance warranties

  47. Game theory can be dangerous to your health • Check your bargaining position before you negotiate. • Do you negotiate first or afterwards? • Does your physical setting influence strategy?

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