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    1、Game Theory,Jeremy Jimenez“ If its true that we are here to help others,then what exactly are the others here for? ”- George Carlin,What is Game Theory?,Game Theory: The study of situations involving competing interests, modeled in terms of the strategies, probabilities, actions, gains, and losses o

    2、f opposing players in a game. A general theory of strategic behavior with a common feature of Interdependence.In other Words: The study of games to determine the probability of winning, given various strategies. Example: Six people go to a restaurant.- Each person pays for their own meal a simple de

    3、cision problem- Before the meal, every person agrees to split the bill evenly among them a game,A Little History on Game Theory,John von Neumann and Oskar Morgenstern - Theory of Games and Economic Behaviors John Nash- “Equilibrium points in N-Person Games“, 1950, Proceedings of NAS. “The Bargaining

    4、 Problem“, 1950, Econometrica. “Non-Cooperative Games“, 1951, Annals of Mathematics. Howard W. Kuhn Games with Imperfect information Reinhard Selten (1965) -“Sub-game Perfect Equilibrium“ (SPE) (i.e. elimination by backward induction) John C. Harsanyi - “Bayesian Nash Equilibrium“,Some Definitions f

    5、or Understanding Game theory,Players-Participants of a given game or games. Rules-Are the guidelines and restrictions of who can do what and when they can do it within a given game or games.Payoff-is the amount of utility (usually money) a player wins or loses at a specific stage of a game.Strategy-

    6、 A strategy defines a set of moves or actions a player will follow in a given game. A strategy must be complete, defining an action in every contingency, including those that may not be attainable in equilibriumDominant Strategy -A strategy is dominant if, regardless of what any other players do, th

    7、e strategy earns a player a larger payoff than any other. Hence, a strategy is dominant if it is always better than any other strategy, regardless of what opponents may do.,Important Review Questions for Game Theory,Strategy Who are the players? What strategies are available? What are the payoffs? W

    8、hat are the Rules of the game What is the time-frame for decisions? What is the nature of the conflict? What is the nature of interaction? What information is available?,Five Assumptions Made to Understand Game Theory,Each decision maker (“PLAYER“) has available to him two or more well-specified cho

    9、ices or sequences of choices (called “PLAYS“). Every possible combination of plays available to the players leads to a well-defined end-state (win, loss, or draw) that terminates the game. A specified payoff for each player is associated with each end-state (a ZERO-SUM game means that the sum of pay

    10、offs to all players is zero in each end-state). Each decision maker has perfect knowledge of the game and of his opposition; that is, he knows in full detail the rules of the game as well as the payoffs of all other players. All decision makers are rational; that is, each player, given two alternati

    11、ves, will select the one that yields him the greater payoff.,Cooperative Vs. Non-Cooperative,Cooperative Game theory has perfect communication and perfect contract enforcement.A non-cooperative game is one in which players are unable to make enforceable contracts outside of those specifically modele

    12、d in the game. Hence, it is not defined as games in which players do not cooperate, but as games in which any cooperation must be self-enforcing.,Interdependence of Player Strategies,Sequential Here the players move in sequence, knowing the other players previous moves.- To look ahead and reason Bac

    13、k 2) Simultaneous Here the players act at the same time, not knowing the other players moves.- Use Nash Equilibrium to solve,Simultaneous-move Games of Complete Information,Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations o

    14、f the strategiesui(s1, s2, .sn), for all s1S1, s2S2, . snSn,A set of players (at least two players)S1 S2 . SnFor each player, a set of strategies/actions Player 1, S1, Player 2,S2 . Player Sn,Nashs Equilibrium,This equilibrium occurs when each players strategy is optimal, knowing the strategys of th

    15、e other players.A players best strategy is that strategy that maximizes that players payoff (utility), knowing the strategys of the other players.So when each player within a game follows their best strategy, a Nash equilibrium will occur.,Logic,Logic,Definition: Nash Equilibrium,Nashs Equilibrium c

    16、ont.:,Bayesian Nash Equilibrium,The Nash Equilibrium of the imperfect-information gameA Bayesian Equilibrium is a set of strategies such that each player is playing a best response, given a particular set of beliefs about the move by nature.All players have the same prior beliefs about the probabili

    17、ty distribution on natures moves.So for example, all players think the odds of player 1 being of a particular type is p, and the probability of her being the other type is 1-p,A mathematical rule of logic explaining how you should change your beliefs in light of new information.Bayes Rule: P(A|B) =

    18、P(B|A)*P(A)/P(B)To use Bayes Rule, you need to know a few things: You need to know P(B|A) You also need to know the probabilities of A and B,Bayes Rule,Examples of Where Game Theory Can Be Applied,Zero-Sum Games Prisoners Dilemma Non-Dominant Strategy moves Mixing Moves Strategic Moves Bargaining Co

    19、ncealing and Revealing Information,Zero-Sum Games,Penny Matching: Each of the two players has a penny. Two players must simultaneously choose whether to show the Head or the Tail. Both players know the following rules: -If two pennies match (both heads or both tails) then player 2 wins player 1s pen

    20、ny. -Otherwise, player 1 wins player 2s penny.,Prisoners Dilemma,No communication:- Strategies must be undertaken without the full knowledge of what the other players (prisoners) will do.Players (prisoners) develop dominant strategies but are not necessarily the best one.,Payoff Matrix for Prisoners

    21、 Dilemma,ConfessBillNot Confess,Ted Confess Not Confess,Solving Prisoners Dilemma,Confess is the dominant strategy for both Bill and Ted.Dominated strategy -There exists another strategy which always does better regardless of other players choices -(Confess, Confess) is a Nash equilibrium but is not

    22、 always the best option,Payoffs,Non-Dominant strategy games,There are many games when players do not have dominant strategies- A players strategy will sometimes depend on the other players strategy- According to the definition of Dominant strategy, if a player depends on the other players strategy,

    23、he has no dominant strategy.,Non-Dominant strategy games,Confess BillNot Confess,TedConfess Not Confess,Solution to Non-Dominant strategy games,Ted Confesses Ted doesnt confess Bill BillConfesses Not confess Confesses Not confess7 years 9 years 6 years 5 yearsBest StrategiesThere is not always a dom

    24、inant strategy and sometimes your best strategy will depend on the other players move.,Examples of Where Game Theory Can Be Applied,Mixing Moves Examples in Sports (Football & Tennis) Strategic Moves War Cortes Burning His Own Ships Bargaining Splitting a Pie Concealing and Revealing Information Blu

    25、ffing in Poker,Applying Game Theory to NFL,Solving a problem within the Salary Cap. How should each team allocate their Salary cap. (Which position should get more money than the other) The Best strategy is the most effective allocation of the teams money to obtain the most wins. Correlation can be

    26、used to find the best way to allocate the teams money.,What is a correlation?,A correlation examines the relationship between two measured variables. - No manipulation by the experimenter/just observed. - E.g., Look at relationship between height and weight. You can correlate any two variables as lo

    27、ng as they are numerical (no nominal variables) Is there a relationship between the height and weight of the students in this room? - Of course! Taller students tend to weigh more.,Salaries vs. Points scored/Allowed,Running Backs edge out Kickers for best correlation of position spending to team poi

    28、nts scored. Tight Ends also show some modest relationship between spending and points. The Defensive Linemen are the top salary correlators, with cornerbacks in the second spot,Total Position spending vs. Wins,Note: Kicker has highest correlation also OL is ranked high also.,What this means,NFL team

    29、s are not very successful at delivering results for the big money spent on individual players. Theres high risk in general, but more so at some positions over others in spending large chunks of your salary cap space.,Future Study,Increase the Sample size. Cluster Analysis Correspondence analysis Exp

    30、loratory Factor Analysis,Conclusion,There are many advances to this theory to help describe and prescribe the right strategies in many different situations. Although the theory is not complete, it has helped and will continue to help many people, in solving strategic games.,References,Nasar, Sylvia

    31、(1998), A Beautiful Mind: A Biography of John Forbes Nash, Jr., Winner of the Nobel Prize in Economics, 1994. Simon and Schuster, New York. Rasmusen, Eric (2001), Games and Information: An Introduction to Game Theory, 3rd ed. Blackwell, Oxford. Gibbons, Robert (1992), Game Theory for Applied Economi

    32、sts. Princeton University Press, Princeton, NJ. Mehlmann, Alexander. The Games Afoot! Game Theory in Myth and Paradox. AMS, 2000. Wiens, Elmer G. Reduction of Games Using Dominant Strategies. Vancouver: UBC M.Sc. Thesis, 1969. H. Scott Bierman and Luis Fernandez (1993) Game Theory with Economic Appl

    33、ications, 2nd ed. (1998), Addison-Wesley Publishing Co. D. Blackwell and M. A. Girshick (1954) Theory of Games and Statistical Decisions, John Wiley Time Inc. Home Entertainment, New York, New York.,Acknowledgements,I would like to thank Arne Kildegaard and Jong-Min Kim for making this presentation possible.,Questions?,Comments?,

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