Lecture 4

Duality and game theory

Knapsack problem – duality illustration

• A thief robs a jewelery shop with a knapsack• He cannot carry too much weight• He can choose among well divisible objects (gold, silver,

diamond sand)• A thief wants to take the most valuable goods with him

The model• Parameters:

W – knapsack maximal weightN – number of goods in a shopwi – good i’s weight

vi – good i’s value

• Decision variables: xi – share of total amount of good i taken to the knapsack

• Objective function:Maximize the total value

• Constraints:(a) Cannot take more than available(b) Cannot take more than the knapsack capacity (c) Cannot take the negative (if he is a thief indeed)

The model

• Formulate as an LP:

Max

Problem of a thief (a primal problem)

• Substitute N=3, W=4, w=(2,3,4) i v=(5,20,3)gold, diamond sand and silver.

maxp.w.

A thief problem solution: (x1,x2,x3)=(0.5, 1, 0)Objective function value: 22.5

Analysis

• Only one good will be taken partially (gold). It is a general rule in all knapsack problems with N divisible goods.

• Intuition: – The optimal solution is unique. – In order to uniquely determine 3 unknowns, we need 3

independent linear equations. – So at least 3 constraints should be satisfied as equalities. – One constraint is the knapsack weight, but another two are

about goods quantities 0≤xi≤1. – Hence only one good may be taken in fractional amount in

the optimum.

Crime syndicate buys out the thief’s business

• The crime syndicate wants to buy out the goods from the thief together with his thief business (equipment etc. here: knapsack).

• They propose prices y1 for gold, y2 for diamond sand, y3 for silver and y4 for 1 kg knapsack capacity.

• But the thief may use 2 kg knapsack capacity and all the gold to generate 5 units of profit , so the price offered for gold 2y4+y1 should be at least 5. Similarly with other goods.

• The syndicate wants to minimize the amount it has to pay the thief y1+y2+y3+4y4

• The prices should not be negative, otherwise the thief will be insulted.

The syndicate problem (a dual problem)

• The syndicate problem may be formulated as follows:

min

p.w.

Syndicate problem solution: (y1,y2,y3,y4)=(0,12.5,0,2.5)Objective function value: 22.5

The thief problem

Is equivalent

Because e.g.

Transforming:

Because e.g.

It is equivalent to the sybdicate problem

Syndicate problem optimal solution: (y1,y2,y3,y4)=(0,12.5,0,2.5) dual pricesOptimal objectuve function value: 22.5

Thief problem optimal solution: (x1,x2,x3)=(0.5, 1, 0)Optimal objective function value: 22.5

Microsoft Excel 12.0 Raport wrażliwościArkusz: [knapsack.xlsx]primalRaport utworzony: 2013-03-27 16:27:40

Komórki decyzyjne Wartość Przyrost Współczynnik Dopuszczalny Dopuszczalny

Komórka Nazwa końcowa krańcowy funkcji celu wzrost spadek$B$2 x1 x 0,5 0 5 8,333333333 3,5$B$3 x2 x 1 0 20 1E+30 12,5$B$4 x3 x 0 -7 3 7 1E+30

Warunki ograniczające Wartość Cena Prawa strona Dopuszczalny Dopuszczalny

Komórka Nazwa końcowa dualna w. o. wzrost spadek$E$8 knapsack weight Ax 4 2,5 4 1 1$E$9 x1 Ax 0,5 0 1 1E+30 0,5$E$10 x2 Ax 1 12,5 1 0,333333333 0,333333333$E$11 x3 Ax 0 0 1 1E+30 1

Rozwiązanie problemu złodzieja: (x1,x2,x3)=(0.5, 1, 0)Optymalna wartość funkcji celu: 22.5

Rozwiązanie problemu syndyka: (y1,y2,y3,y4)=(0,12.5,0,2.5) ceny dualneOptymalna wartość funkcji celu: 22.5

Microsoft Excel 12.0 Raport wrażliwościArkusz: [knapsack.xlsx]dualnyRaport utworzony: 2013-03-27 16:27:15

Komórki decyzyjne Wartość Przyrost Współczynnik Dopuszczalny Dopuszczalny

Komórka Nazwa końcowa krańcowy funkcji celu wzrost spadek$B$2 y1 y 0 0,5 1 1E+30 0,5$B$3 y2 y 12,5 0 1 0,333333333 0,333333333$B$4 y3 y 0 1 1 1E+30 1$B$5 y4 y 2,5 0 4 0,999999999 1

Warunki ograniczające Wartość Cena Prawa strona Dopuszczalny Dopuszczalny

Komórka Nazwa końcowa dualna w. o. wzrost spadek$F$7 min price per gold A'y 5 0,5 5 8,333333333 3,5$F$8 min price per diamonds A'y 20 1 20 1E+30 12,5$F$9 min price per silver A'y 10 0 3 7 1E+30

Matching/assignment

x Gene Helen Irene RowsumsDavid 1 0 0 1Edward 0 1 0 1Fenix 0 0 1 1colsums 1 1 1

compatibility Gene Helen IreneDavid 1 0 0.5Edward 0.75 2 1Fenix 0.5 2.5 1.5

Objective fun 4.5

http://mathsite.math.berkeley.edu/smp/smp.html

Individual decision theory vs game theory

Zero-sum games

• In zero-sum games, payoffs in each cell sum up to zero

• Movement diagram

Zero-sum games• Minimax = maximin = value of the game

• The game may have multiple saddle points

Zero-sum games

• Or it may have no saddle points

• To find the value of such game, consider mixed strategies

Zero-sum games

• If there is more strategies, you don’t know which one will be part of optimal mixed strategy.

• Let Column mixed strategy be (x,1-x)• Then Raw will try to maximize

Zero-sum games• Column will try to choose x to minimize the upper envelope

Zero-sum games

• Tranform into Linear Programming

Fishing on Jamaica

• In the fifties, Davenport studied a village of 200 people on the south shore of Jamaica, whose inhabitants made their living by fishing.

• Twenty-six fishing crews in sailing, dugout canoes fish this area [fishing grounds extend outward from shore about 22 miles] by setting fish pots, which are drawn and reset, weather and sea permitting, on three regular fishing days each week … The fishing grounds are divided into inside and outside banks. The inside banks lie from 5-15 miles offshore, while the outside banks all lie beyond … Because of special underwater contours and the location of one prominent headland, very strong currents set across the outside banks at frequent intervals … These currents are not related in any apparent way to weather and sea conditions of the local region. The inside banks are almost fully protected from the currents. [Davenport 1960]

Jamaica on a map

Strategies

• There were 26 wooden canoes. The captains of the canoes might adopt 3 fishing strategies:– IN – put all pots on the inside banks – OUT – put all pots on the outside banks– IN-OUT) – put some pots on the inside banks,

some pots on the outside

Advantages and disadvantages of fishing in the open sea

Disadvantages

• It takes more time to reach, so fewers pots can be set

• When the current is running, it is harmful to outside pots – marks are dragged away – pots may be smashed while

moving– changes in temeperature

may kill fish inside the pots

Advanatages

• The outside banks produce higher quality fish both in variaties and in size. – If many outside fish are available,

they may drive the inside fish off the market.

• The OUT and IN-OUT strategies require better canoes. – Their captains dominate the

sport of canoe racing, which is prestigious and offers large rewards.

Collecting data

• Davenport collected the data concerning the fishermen average monthly profit depending on the fishing strategies they used to adopt.

Fishermen\Current FLOW NO FLOW

IN 17,3 11,5

OUT -4,4 20,6

IN-OUT 5,2 17,0

OUT Strategy

Zero-sum game? The current’s problem

• There is no saddle point• Mixed strategy:

– Assume that the current is vicious and plays strategy FLOW with probability p, and NO FLOW with probability 1-p

– Fishermen’s strategy: IN with prob. q1, OUT with prob. q2, IN-OUT with prob. q3

– For every p, fishermen choose q1,q2 and q3 that maximizes:

– And the vicious current chooses p, so that the fishermen get min

Graphical solution of the current’s problem

Mixed strategy of the current

Solution: p=0.31

5

7

9

11

13

15

17

19

21

IN

OUT

IN-OUT

The fishermen’s problem

• Similarly:– For every fishermen’s strategy q1,q2 and q3, the

vicious current chooses p so that the fishermen earn the least:

– The fishermen will try to choose q1,q2 and q3 to maximize their payoff:

Maximin and minimaxobjective function

Fishers' mixed strategy

q1 q2 q3Maximize 13,31 0,67 0,00 0,33

Expected payoff of the current whenFLOW 13,31 >= 13,31NO FLOW 13,31 >= 13,31probabilities 1,00 = 1,00

objective function

Mixed strategy of the current

p 1-pminimize 13,31 0,31 0,69

Expected payoff from strategy:IN 13,31 <= 13,31OUT 12,79 <= 13,31IN_OUT 13,31 <= 13,31probabilities 1,00 = 1,00

Optimal strategy for the fishermen

Optimal strategy for the current

Value of the game

Minimax sensitivity report

Microsoft Excel 12.0 Raport wrażliwości

Arkusz: [jamajka.xlsx]minimax

Raport utworzony: 2013-03-27 16:24:55

Komórki decyzyjne

    Wartość Przyrost Współczynnik Dopuszczalny Dopuszczalny

Komórka Nazwa końcowakrańcow

y funkcji celu wzrost spadek

$C$3objective function 13,31 0,00 1 1E+30 1

$D$3 p 0,31 0,00 0 11,8 5,8

$E$3 1-p 0,69 0,00 0 5,8 11,8

Warunki ograniczające

    Wartość Cena Prawa strona Dopuszczalny Dopuszczalny

Komórka Nazwa końcowa dualna w. o. wzrost spadek

$B$6 IN 13,31 -0,67 0 12,1 0,7

$B$7 OUT 12,79 0,00 0 1E+30 0,525

$B$8 IN-OUT 13,31 -0,33 0 0,3 12,1

$B$9 probabilities 1,00 13,31 1 1E+30 1

Microsoft Excel 12.0 Raport wrażliwościArkusz: [jamajka.xlsx]maximin

Raport utworzony: 2013-03-27 16:23:31

Komórki decyzyjne

    Wartość PrzyrostWspółczynni

kDopuszczaln

yDopuszczaln

yKomórk

a Nazwa końcowakrańcow

y funkcji celu wzrost spadek

$C$3objective function 13,31 0,00 1 1E+30 1

$D$3 q1 0,67 0,00 0 0,7 12,1$E$3 q2 0,00 -0,52 0 0,525 1E+30$F$3 q3 0,33 0,00 0 12,1 0,3

Warunki ograniczające

    Wartość Cena Prawa stronaDopuszczaln

yDopuszczaln

yKomórk

a Nazwa końcowa dualna w. o. wzrost spadek$B$6 FLOW 13,31 -0,31 0 5,8 11,8$B$7 NO FLOW 13,31 -0,69 0 11,8 5,8$B$8 probabilities 1,00 13,31 1 1E+30 1

Maximin sensitivity report

Forecast and observation

Game theory predicts• No fishermen risks fishing

outside• Strategy 67% IN, 33% IN-

OUT [Payoff: 13.31]• Optimal current’s strategy

31% FLOW, 69% NO FLOW

Observation shows• No fishermen risks fishing

outside • Strategy 69% IN, 31% IN-

OUT [Payoff: 13.38]• Current’s „strategy”: 25%

FLOW, 75% NO FLOW

The similarity is strikingDavenport’s finding went unchallenged for several yearsUntil …

Current is not vicious

• Kozelka 1969 and Read, Read 1970 pointed out a serious flaw:– The current is not a reasoning entity and cannot adjust to fishermen

changing their strategies.– Hence fishermen should use Expected Value principle:

• Expected payoff of the fishermen:– IN: 0.25 x 17.3 + 0.75 x 11.5 = 12.95– OUT: 0.25 x (-4.4) + 0.75 x 20.6 = 14.35– IN-OUT: 0.25 x 5.2 + 0.75 x 17.0 = 14.05

• Hence, all of the fishermen should fish OUTside.• Maybe, they are not well adapted after all

Fishermen\Current FLOW (25%) NO FLOW (75%)

IN 17,3 11,5

OUT -4,4 20,6

IN-OUT 5,2 17,0

Current may be vicious after all• The current does not reason, but it is very risky to fish outside.• Even if the current runs 25% of the time ON AVERAGE, it might

run considerably more or less in the short run of a year.• Suppose one year it ran 35% of the time. Expected payoffs:

– IN: 0.35 x 17.3 + 0.65 x 11.5 = 13.53– OUT: 0.35 x (-4.4) + 0.65 x 11.5 = 11.85– IN-OUT: 0.35 x 5.2 + 0.65 x 17.0 = 12.87.

• By treating the current as their opponent, fishermen GUARANTEE themselves payoff of at least 13.31.

• Fishermen pay 1.05 pounds as insurance premium

Actual (25%) Vicious (31%) 35%Optimal 13.3125 13.3125 13.3125Actual 13.291 13.31164 13.3254OUT 14.35 12.85 11.85

Decision making under uncertainty

Rybacy\Prąd Płynie Nie płynie

IN 0 9,1

OUT 21,7 0

IN-OUT 12,1 3,6

0,67 IN+0,33 IN-OUT 3,9875 7,2875

Fishermen\Current FLOW NO FLOW MAXIMIN MAXIMAX MINIMAX REGRET

IN 17,3 11,5 11,5 17,3 9,1

OUT -4,4 20,6 -4,4 20,6 21,7IN-OUT 5,2 17 5,2 17 12,10,67 IN+0,33 IN-OUT 13,3125 13,3125 13,3125 13,3125 7,2875

Decision making under uncertainty

Rybacy\Prąd Płynie Nie płynie

IN 0 9,1

OUT 21,7 0

IN-OUT 12,1 3,6

0,67 IN+0,33 IN-OUT 3,9875 7,2875

Fishermen\Current FLOW NO FLOW MAXIMIN MAXIMAX MINIMAX REGRET

IN 17,3 11,5 11,5 17,3 9,1

OUT -4,4 20,6 -4,4 20,6 21,7IN-OUT 5,2 17 5,2 17 12,10,67 IN+0,33 IN-OUT 13,3125 13,3125 13,3125 13,3125 7,2875

Decision making under uncertainty

Fishermen\Current FLOW NO FLOW

IN 0 9,1

OUT 21,7 0

IN-OUT 12,1 3,6

0,67 IN+0,33 IN-OUT 3,9875 7,2875

Fishermen\Current FLOW NO FLOW MAXIMIN MAXIMAX MINIMAX REGRET

IN 17,3 11,5 11,5 17,3 9,1

OUT -4,4 20,6 -4,4 20,6 21,7IN-OUT 5,2 17 5,2 17 12,10,67 IN+0,33 IN-OUT 13,3125 13,3125 13,3125 13,3125 7,2875

Regret matrix

Decision making under uncertainty

Fishermen\Current FLOW NO FLOW

IN 0 9,1

OUT 21,7 0

IN-OUT 12,1 3,6

0,67 IN+0,33 IN-OUT 3,9875 7,2875

Fishermen\Current FLOW NO FLOW MAXIMIN MAXIMAX MINIMAX REGRET

IN 17,3 11,5 11,5 17,3 9,1

OUT -4,4 20,6 -4,4 20,6 21,7IN-OUT 5,2 17 5,2 17 12,10,67 IN+0,33 IN-OUT 13,3125 13,3125 13,3125 13,3125 7,2875

Regret matrix

Decision making under uncertaintyFishermen\Current FLOW NO FLOW MAXIMIN MAXIMAX Hurwicz

optimism/pessimism index

IN 17,3 11,5 11,5 17,3 11,5α+17,3(1-α)

OUT -4,4 20,6 -4,4 20,6 -4,4α+20,6(1-α)IN-OUT 5,2 17 5,2 17 5,2α+17(1-α)0,67 IN+0,33 IN-OUT 13,3125 13,3125 13,3125 13,3125 13,3125

5

7

9

11

13

15

17

19

21

IN

OUT

IN-OUT