1 15.053 Thursday, March 7 Duality 2 – The dual problem, in general – illustrating duality with 2-person 0-sum game theory Handouts: Lecture Notes.

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15.053 Thursday, March 7

• Duality 2 – The dual problem, in general – illustrating duality with 2-person 0-sum game theory

Handouts: Lecture Notes

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PRIMAL PROBLEM:

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1, x2, x3, x4 ≥ 0

maximize y1+ 3y2

subject to y1+ 2y2 ≥

y1+ 3y2 ≥

y1+ 4y2 ≥

y1+ 5y2 ≥

Observation 1.

The constraint matrix in theprimal is the transpose of theconstraint matrix in the dual.

Observation 2.

The RHS coefficients in theprimal become the costcoefficients in the dual.

DUSL PROBLEM:

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PRIMAL PROBLEM:

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1, x2, x3, x4 ≥ 0

DUSL PROBLEM:

maximize y1+ 3y2

subject to y1+ 2y2 ≥

y1+ 3y2 ≥

y1+ 4y2 ≥

y1+ 5y2 ≥

Observation 3. The cost coefficientsin the primal become the RHScoefficients in the dual.

Observation 4. The primal (in thiscase) is a max problem with equalityconstraints and non-negativevariables

The dual (in this case) is aminimization problem with ≥constraints and variablesunconstrained in sign.

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PRIMAL PROBLEM:

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1 2x1 + 3x2 +4x3 + 5x4 = 3 x1, x2, x3, x4 ≥ 0

DUSL PROBLEM:

maximize y1+ 3y2

subject to y1+ 2y2 ≥

y1+ 3y2 ≥

y1+ 4y2 ≥

y1+ 5y2 ≥

Question.

How does the dual changeif we have inequalityconstraints?

Recall that the optimaldual variables are theshadow prices for theprimal

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PRIMAL PROBLEM:

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 ≥

2x1 + 3x2 +4x3 + 5x4 ≤

x1, x2, x3, x4 ≥

Prices

Method. Recall that the optimal dualvariables are shadow prices.

Suppose we replace the 1 by 1 + ∆ Canthe optimal objective value go down?Can it go up?

Suppose we replace the 3 by 3 + ∆ Canthe optimal objective value go down?Can it go up?

Conclusion:y1 ≤0.

Conclusion:y2 ≥0 .

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PRIMAL PROBLEM:

maximize

subject toz = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1

2x1 + 3x2 +4x3 + 5x4 = 3

x1, x2, x3, x4 ≥ 0

DUSL PROBLEM:

maximize y1+ 3y2

subject to y1+ 2y2 ≥

y1+ 3y2 ≥

y1+ 4y2 ≥

y1+ 5y2 ≥

Now suppose that wepermit variables in theprimal problem to be either≤ 0 or unconstrained insign.

Recall: reduced costs arethe shadow prices of the “≥0” and “≤0” constraints.

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PRIMAL PROBLEM:

maximize

subject to

z = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 = 1

2x1 + 3x2 +4x3 + 5x4 = 3

x1 ≥ 0 x2 ≥ 0 x3 ≤ 0x4 uis

c1 = 3 – y1 – 2y2

Suppose we replace “x1 ≥ 0” by “x1 ≥ ∆”.Can the optimal objective value go down?Can it go up?

Conclusion:c1 ≤ 0, andthus y1 + 2y2 ≥ 3.

Prices

To do with your partner: figure out the sign on theshadow price for the constraint x3 ≤ 0. Also, whatdo we do with x4 uis?

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Summary for forming the dual of amaximization problem

PRIMAL Max Σj aijxj = bi Σj aijxj ≥ bi Σj aijxj ≤ bi xj ≥ 0 xj ≤ 0 xj u.i.s.

DUAL Min yi u.i.s. yi ≤0 yi ≥0Σj yiaij ≥cjΣj yiaij ≤cjΣj yiaij = cj

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Complementary Slackness

PRIMAL DUAL

Σj aijxj ≥ bi yi ≤ 0

Σj aijxj ≤ bi yi ≥ 0

xj ≥ 0 Σj yiaij ≥ cj

xj ≤ 0 Σj yiaij ≤ cj

Comp. Slackness

yi (Σj aijxj - bi ) = 0

yi (Σj aijxj - bi ) = 0

xj Σj yiaij - cj ) = 0

xj Σj yiaij - cj ) = 0

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Determine the dual

PRIMAL PROBLEM:

maximize

subject to

z = 3x1 + 4x2 +6x3 + 8x4

x1 + x2 + x3 + x4 ≥ 1

2x1 + 3x2 +4x3 + 5x4 = 3

x1 ≥ 0, x2 ≤ 0 x3 u.i.s.

Determine the dual of the above linearprogram. Then compare your result withthat of your partner.

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Duals of Minimization Problem

• The dual of a minimization problem is a maximization problem. • The shadow prices for the dual linear program form the optimal solution for the primal problem.

• The dual of the dual is the primal.

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2-person 0-sum game theory

Person R chooses a row: either 1, 2, or 3

Person C chooses a column: either 1, 2, or 3

This matrix is thepayoff matrix forplayer R. (And playerC gets the negative.)

e.g., R chooses row 3; C chooses column 1

R gets 1; C gets –1 (zero sum)

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Some more examples of payoffs

R chooses 2, C chooses 3

R gets 0; C gets 0 (zero sum)

R chooses row 3; C chooses column 3

R gets -2; C gets +2 (zero sum)

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Next: 2 volunteers

Player R puts out 1, 2 or 3 fingers

Player C simultaneously puts out 1, 2, or 3 fingers

We will runthe game for 5trials.

R tries to maximize his or her total

C tries to minimize R’s total.

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Next: Play the game with your partner(If you don’t have one, then watch)

Player R puts out 1, 2 or 3 fingers

Player C simultaneously puts out 1, 2, or 3 fingers

We will runthe game for 5trials.

R tries to maximize his or her total

C tries to minimize R’s total.

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Who has the advantage: R or C?

Suppose that R and C are both brilliantplayers and they play a VERY LONG TIME.

We will find alower and upperbound on thepayoff to R usinglinearprogramming.

Will R’s payoff be positive in the long run, orwill it be negative, or will it converge to 0?

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Computing a lower bound

Suppose that player R must announce his orher strategy in advance of C making a choice.

If R is forced toannounce a row,then what row willR select?

A strategy that consists of selecting the samerow over and over again is a “pure strategy.”R can guarantee a payoff of at least –1.

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Computing a lower bound on R’s payoff

Suppose we permit R to choose a randomstrategy.

Suppose R will flipa coin, and chooserow 1 if Heads, andchoose row 3 iftails.

The column player makes the choice afterhearing the strategy, but before seeing the flipof the coin.

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What is player’s C best response?

If C knows R’srandom strategy,then C candetermine theexpected payoff foreach columnchosen

Prob.

ExpectedPayoff

What would C’s best response be?

So, with a random strategy R can get at least -.5

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Suppose that R randomizes between row 1 and row 2.

Prob.

ExpectedPayoff

What would C’s best response be?

So, with a random strategy R can get at least 0.

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What is R’s best random strategy?

Prob.

C will choosethe column thatis the minimumof A, B, and C.

ExpectedPayoff

A: -2 x1 + 2 x2 + x3

B: x1 - x2

C: 2 x1 - 2 x3

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Finding the min of 2 numbers as anoptimization problem.

Let z = min (x, y)

Then z is the optimum solution value to the following LP:

maximize zsubject to z ≤ x z ≤ y

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R’s best strategy, as an LP

ExpectedPayoff

2-person0-sumgame

Maximize z (the payoff to x)

A: z ≤-2 x1 + 2 x2 + x3

C: z ≤2 x1 - 2 x3

B: z ≤x1 - x2

x1 + x2 + x3 = 1

x1 , x2 , x3 ≥0

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The Row Player’s LP, in general

ExpectedPayoff

Maximize z (the payoff to x)

Pj: z ≤a1j x1 + a2j x2 +… + anjxn for all j

x1 + x2 + … + xn = 1xj ≥0 for all j

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Here is the optimal random strategy for R.

This strategy is

a lower bound

on what R can

obtain if he or

she takes into

account what C

is doing.ExpectedPayoff

The optimal payoff to R is 1/9.

So, with a random strategy R guaranteesobtaining at least 1/9.

Prob.

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We can obtain a lower bound for C (and an upperbound for R) in the same manner.

Exp.payoff

If C chooses a

random

strategy, it will

give an upper

bound on what

R can obtain.

If C announced this random strategy, Rwould select 1 or 2.

So, with this random strategy C guaranteesthat R obtains at most 1/3.

Prob.

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The best random strategy for C is to minimize themax expected payoff.

The column playercan set up a linearprogram todetermine the bestrandom strategy

Note: C can use arandom strategy toensure that Rreceives at most 1/9on average.

Exp.payoff

So, with this random strategy R gets only 1/9.

Prob.

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The best random strategy for C is to minimize themax expected payoff.

So, the optimalaverage payoffto the game is1/9, assumingthat bothplayers play

optimally.

Exp.payoff

Prob.

For 2-person 0-sum games, the maximum payoff that R canguarantee by choosing a random strategy is the minimum payoffto R that C can guarantee by choosing a random strategy.

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2-person 0-sum games in general.

• Let x denote a random strategy for R, with value z(x) and let y denote a random strategy for C with value v(y).• z(x) ≤ v(y) for all x, y • The optimum x* can be obtained by solving an LP. So can the optimum y* .• z(x*) = v(y*)• The two linear programs are dual to each other.

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More on 2-person 0-sum games

• In principle, R can do as well with a random fixed strategy as by carefully varying a strategy over time

• Duality for game theory was discovered by Von Neumann and Morgenstern (predates LP duality).

• The idea of randomizing strategy permeates strategic gaming.