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**Game Theory “ If it’s true that we are here to help others,**

then what exactly are the others here for? ” - George Carlin

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What is Game Theory? Game Theory: The study of situations involving competing interests, modeled in terms of the strategies, probabilities, actions, gains, and losses of 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 decision problem - Before the meal, every person agrees to split the bill evenly among them – a game

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**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 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"

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**Some Definitions for 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- 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 equilibrium Dominant Strategy -A strategy is dominant if, regardless of what any other players do, the 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.

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**Important Review Questions for Game Theory**

Strategy Who are the players? What strategies are available? What are the payoffs? What 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?

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**Five Assumptions Made to Understand Game Theory**

Each decision maker ("PLAYER“) has available to him two or more well-specified choices 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 payoffs 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 alternatives, will select the one that yields him the greater payoff.

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**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 modeled 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.

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**Interdependence of Player Strategies**

Sequential – Here the players move in sequence, knowing the other players’ previous moves. - To look ahead and reason Back 2) Simultaneous – Here the players act at the same time, not knowing the other players’ moves. - Use Nash Equilibrium to solve

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**Simultaneous-move Games of Complete Information**

A set of players (at least two players) S1 S Sn For each player, a set of strategies/actions {Player 1, S1, Player 2,S2 ... Player Sn} Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies ui(s1, s2, ...sn), for all s1S1, s2S2, ... snSn

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Nash’s Equilibrium This equilibrium occurs when each player’s strategy is optimal, knowing the strategy's of the other players. A player’s best strategy is that strategy that maximizes that player’s payoff (utility), knowing the strategy's of the other players. So when each player within a game follows their best strategy, a Nash equilibrium will occur. Logic Logic

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**Definition: Nash Equilibrium**

Given others’ choices, player i cannot be better-off if she deviates from si*

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**Nash’s Equilibrium cont.:**

Bayesian Nash Equilibrium The Nash Equilibrium of the imperfect-information game A 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 probability distribution on nature’s 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

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Bayes’ Rule A mathematical rule of logic explaining how you should change your beliefs in light of new information. Bayes’ Rule: P(A|B) = 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

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**Examples of Where Game Theory Can Be Applied**

Zero-Sum Games Prisoner’s Dilemma Non-Dominant Strategy moves Mixing Moves Strategic Moves Bargaining Concealing and Revealing Information

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**Zero-Sum Games Penny Matching: -1 , 1 1 , -1**

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 1’s penny. -Otherwise, player 1 wins player 2’s penny. Player 1 Player 2 Tail Head -1 , 1 1 , -1

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**Prisoner’s 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.

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**Payoff Matrix for Prisoner’s Dilemma**

Ted Confess Not Confess Both get 5 years 1 year for Bill 10 years for Ted 10 years for Bill 1 year for Ted Both get 3 years Confess Bill Not Confess

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**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 always the best option Players Bill Ted Strategies Not Confess Confess -5, -5 -1,-10 -10,-1 -3,-3 Payoffs

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**Non-Dominant strategy games**

There are many games when players do not have dominant strategies - A player’s strategy will sometimes depend on the other player's strategy - According to the definition of Dominant strategy, if a player depends on the other player’s strategy, he has no dominant strategy.

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**Non-Dominant strategy games**

Ted Confess Not Confess Confess Bill Not Confess 7 years for Bill 2 years for Ted 6 years for Bill 4 years for Ted 9 years for Bill 0 years for Ted 5 years for Bill 3 years for Ted

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**Solution to Non-Dominant strategy games**

Ted Confesses Ted doesn’t confess Bill Bill Confesses Not confess Confesses Not confess 7 years years years years Best Strategies There is not always a dominant strategy and sometimes your best strategy will depend on the other players move.

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**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 Bluffing in Poker

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**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 team’s money to obtain the most wins. Correlation can be used to find the best way to allocate the team’s money.

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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 long 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.

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**Salaries vs. Points scored/Allowed**

Position Correlation T-test RB .27 2.67 k .25 2.52 TE .17 1.74 OL .04 .34 QB .03 .32 WR -.03 -.30 Position Correlation T-test DE .25 2.52 CB .15 1.48 S .06 .61 LB .05 .52 DT .04 .34 P Running Backs edge out Kickers for best correlation of position spending to team points 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

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**Total Position spending vs. Wins**

Points Scored Points Allowed Total Position Correlation Spending K 0.27 0.17 CB 0.12 0.23 TE 0.16 0.2 0.15 OL 0.02 0.08 RB 0.11 -0.03 0.26 QB 0.1 0.04 DE -0.14 P 0.01 0.03 LB 0.05 -0.08 -0.02 S DT -0.01 -0.04 WR Note: Kicker has highest correlation also OL is ranked high also.

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What this means NFL teams are not very successful at delivering results for the big money spent on individual players. There's high risk in general, but more so at some positions over others in spending large chunks of your salary cap space.

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**Future Study Increase the Sample size. Cluster Analysis**

Correspondence analysis Exploratory Factor Analysis

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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.

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References Nasar, Sylvia (1998), A Beautiful Mind: A Biography of John Forbes Nash, Jr., Winner of the Nobel Prize in Economics, 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 Economists. 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 Applications, 2nd ed. (1998), Addison-Wesley Publishing Co. D. Blackwell and M. A. Girshick (1954) Theory of Games and Statistical Decisions, John Wiley & Sons, New York. NFL Official, 2004 NFL Record and Fact Book; Time Inc. Home Entertainment, New York, New York.

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