Game Theory

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Game Theory

• Jeremy Jimenez
• If its 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 S2 ... 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 s1?S1, s2?S2, ...
  sn?Sn

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Nashs Equilibrium

• This equilibrium occurs when each players
  strategy is optimal, knowing the strategy's of
  the other players.
• A players best strategy is that strategy that
  maximizes that players 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
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Nashs 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 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

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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(AB) P(BA)P(A)/P(B)
• To use Bayes Rule, you need to know a few
  things
• You need to know P(BA)
• 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
• Prisoners Dilemma
• Non-Dominant Strategy moves
• Mixing Moves
• Strategic Moves
• Bargaining
• Concealing and Revealing Information

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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 penny.
• -Otherwise, player 1 wins player 2s penny.

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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.

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Payoff Matrix for Prisoners Dilemma

• Ted
• Confess Not Confess

• 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

Payoffs
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Non-Dominant strategy games

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

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Non-Dominant strategy games

• Ted
• Confess Not Confess

• Confess
• Bill
• Not Confess

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Solution to Non-Dominant strategy games

• Ted Confesses Ted doesnt confess
• Bill Bill
• Confesses Not confess Confesses
  Not confess
• 7 years 9 years 6 years 5
  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 teams money to obtain the most
  wins.
• Correlation can be used to find the best way to
  allocate the teams 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
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
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, 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
  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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Acknowledgements

• I would like to thank Arne Kildegaard and
  Jong-Min Kim for making this presentation
  possible.

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Questions?

• Comments?