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Chapter 11 Game Theory

Chapter 11 Game Theory. Math 305 2008. Game Theory. What is it? a way to model conflict and competition one or more "players" make simultaneous decisions which affect the rewards accruing to each  Assumptions: 2 person (players) zero sum: what one wins the other loses

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Chapter 11 Game Theory

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  1. Chapter 11 Game Theory Math 305 2008

  2. Game Theory What is it? • a way to model conflict and competition • one or more "players" make simultaneous decisions which affect the rewards accruing to each  Assumptions: • 2 person (players) • zero sum: what one wins the other loses Strategies and payoffs represented by a matrix • player 1 has strategies 1-m • player 2 has strategies 1-m • aij = payoff from II to I if I selects row i and II selects column j. • [aij] = reward/payoff matrix.

  3. More Assumptions • each decision maker has two or more well-specified choices or sequences of choices • 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 • each decision maker has perfect knowledge of the game and of his opposition • all decision makers are rational; that is, each player will select the strategy that yields him the greater payoff

  4. Example: Odds/Evens Each player simultaneously holds 1 or 2 fingers. If the sum is odd, II (Even) pays I $1. If not, I (Odd) pays II $1. Payoff matrix: Column player strategies 1 2 Row 1 player 2 strategies aij = payoff from II to I if I selects row i and II selects column j Neither player knows what strategy the other will follow How should you play this game?

  5. More Interesting Column player strategies 1 2 Row 1 player 2 strategies aij = payoff from II to I if I selects row i and II selects column j These require a mixed strategy • select 1 x% of the time and 2, (1-x)% • player 1 strategy: (x1, 1- x1) • player 2 strategy: (y1, 1- y1)

  6. Constant Sum A generalization of zero sum: the sum of player winnings are a constant E.g. (p. 613) networks vying for audience of 100 million with strategies western, soap, and comedy. Payoff matrix is millions of viewers for network 1 W S C row min W 15 S 45 C 14 col max 45 58 70 Solve using minimax • max(row min) = min(col max) = 45 • saddlepoint at (2,1)

  7. 11.2:  Dominated Strategies Column player strategies 1 2 Row 1 player 2 strategies aij = payoff from II to I if I selects row i and II selects column j If you were player I, you would always pick strategy 1 If you were player II, you would always pick strategy 2 • equilibrium point at row 1, col 2 • value of the game = -1 Can also use saddle point condition • max(row minimum) = min(col maximum) • max (-1, -2) = min(10, -1) -=1  Does this work for odds/evens? max( -1,-1) != min(1,1)

  8. Example: Odds/Evens Not all games have a saddle point or dominated strategies leading to pure strategies for each player Back to this one: Column player strategies 1 2 Row 1 player 2 strategies aij = payoff from II to I if I selects row i and II selects column j Goal: probability distributions on the pure strategies (x1, 1- x1) for player I and (y1, 1- y1) for player II where xi = p(I holds i fingers) yi = P(II holds i fingers)

  9. Graphical Solution Payoff to I if II picks 1: -1(x1) + 1 (1-x1) = 1-2x1 Payoff to I if II picks 2: 1(x1) - 1 (1-x1) = -1 +2x1 payoff II to I x1 Note, we can ensure v=0 if x1= 1/2  with strategy (1/2, 1/2) Player II also has strategy (1/2, 1/2) (0,1) (1/2, 0) (0, -1)

  10. Graphical Solution Back to Player I: • payoff to I if II picks 1: 10(x1) - 1 (1-x1) = 11x1 -1 • payoff to I if II picks 2: -1(x1) + 0.5(1-x1) = -1.5x1 +0.5 • intersection at x1 = .12 • I strategy (.12, .88) • v = 10(.12) -1(.88) = .32 Player II: • if I selects strategy 1: 10y1 - (1-y1) = 11y1 -1 • if I selects strategy 2: -y1 + .5(1-y1) = -1.5y1+ .5 • intersection at y1= .12 • v = 11(.12) -1 =. 32 Mixed strategies will not always be the same for each player

  11. Graphical Solution Try Do p 619, table 13, eliminating dominated strategies first call fold PP PB BP BB Does graphing work for games with more than two strategies?

  12. Linear Programming max z = v subject to v <= 10x1 - x2 v <= -x1 + 0.5x2 x1+x2 = 1 OR max v subject to 10x1 - x2 -v >= 0 -x1 + 0.5x2 -v >= 0 x1+x2 = 1 end Guess what the problem formulated for Player II is? (dual)!

  13. 11.4 Two Person Nonconstant Games Prisoner's Dilemma: you and your partner in crime are being interrogated for a robbery in separate rooms confess don't confess don't Payoff (-x,-y) is x years for I and y years for II Dominated strategies leads to equilibrium point (-5, -5) Equilibrium point: neither player can benefit by a unilateral strategy change Analogies: global warming, arms race, Tour de France, chicken

  14. Games Against Nature So far we have assumed a rational opponent Nature can be • probabilistic (there is a probability distribution for its strategies) • no known distribution on it's strategies Cranberry grower example, probabilistic 12 • when there is a frost, one floods the bogs to protect the berries • it costs $$ to flood the bogs frost no frost • grower strategy: flood or not flood • nature strategy: freeze or not don't flood • the probability of a frost is .1 Approach: find the expected payoff for both strategies E(flood) = -1(.3) -1(.7) = -1 E(no flood) = -20(.1) + 0(.9) = -2 What if there is no distribution?

  15. Pascal's Wager An argument by Blaise Pascal that one should believe in God Your strategies: believe in God, don't believe in God Payoff matrix God exists God does not exist Beieve in God Don't believe Or

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