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 – a ij = payoff from II to I if I selects row i and II selects column j. – [a ij ] = 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 a ij = 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? 1 1

5. More Interesting Column player strategies 1 2 Row 1 player 2 strategies a ij = 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: (x 1, 1- x 1 ) – player 2 strategy: (y 1, 1- y 1 ) 10 0.5

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 a ij = 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) 10 -2

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 a ij = payoff from II to I if I selects row i and II selects column j Goal: probability distributions on the pure strategies (x 1, 1- x 1 ) for player I and (y 1, 1- y 1 ) for player II where x i = p(I holds i fingers) y i = P(II holds i fingers) 1 1

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

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

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? -2 0 4 -3/2 0 1/2 0 0 1

12. Linear Programming max z = v subject to v <= 10x 1 - x 2 v <= -x 1 + 0.5x 2 x 1 +x 2 = 1 OR max v subject to 10x 1 - x 2 -v >= 0 -x 1 + 0.5x 2 -v >= 0 x 1 +x 2 = 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 (-5,-5) (0,-20) (-20,0) (-1,-1)

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 1 212 – 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? -20 0

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 ∞ 0 -∞ 100 a religious life and an eternity of happiness a religious life a life of poisonous pleasures of the flesh and an eternity of suffering a life of poisonous pleasures of the flesh