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

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.

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

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 2strategies aij = payoff from II to I if I selects row i and II selects column j

Neither player knows what strategy the other will followHow should you play this game?

-1 1

1 -1

More Interesting

Column player strategies 1 2 Row 1 player 2strategies 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)

10 -1

-1 0.5

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)

35 15 60

45 58 50

38 14 70

11.2: Dominated Strategies

Column player strategies 1 2 Row 1 player 2strategies 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 1If 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 -1

-1 -2

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 2strategies 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)

-1 1

1 -1

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)

(1/2, 0)

(0,1)

(0, -1)

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

10 -1

-1 0.5

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 -2

-1 -1

-3/2 0

1/2 0

0 1

Linear Programming

max z = vsubject to v <= 10x1 - x2

v <= -x1 + 0.5x2

x1+x2 = 1

ORmax vsubject to 10x1 - x2 -v >= 0

-x1 + 0.5x2 -v >= 0

x1+x2 = 1

end

Guess what the problem formulated for Player II is?(dual)!

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 changeAnalogies: global warming, arms race, Tour de France, chicken

(-5,-5) (0,-20)

(-20,0) (-1,-1)

Games Against Nature

So far we have assumed a rational opponentNature can be

– probabilistic (there is a probability distribution for its strategies)– no known distribution on it's strategies

Cranberry grower example, probabilistic 1 2 – 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 strategiesE(flood) = -1(.3) -1(.7) = -1E(no flood) = -20(.1) + 0(.9) = -2

What if there is no distribution?

-1 -1 -20 0

Pascal's Wager

An argument by Blaise Pascal that one should believe in GodYour strategies: believe in God, don't believe in GodPayoff 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