Utility and Game Theory

7/30/2019 Utility and Game Theory

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Chapter 5

Utility and Game Theory The Meaning of Utility

The Expected Utility Approach

Risk Avoiders versus Risk Takers Expected Monetary Value versus Expected Utility

Introduction to Game Theory

Pure Strategy Mixed Strategies

Dominated Strategies


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The Meaning of Utility Utilities are used when the decision criteria must be

based on more than just expected monetary values.

Utility is a measure of the total worth of a particularoutcome, reflecting the decision makers attitudetowards a collection of factors.

Some of these factors may be profit, loss, and risk.

This analysis is particularly appropriate in caseswhere payoffs can assume extremely high orextremely low values.


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DecisionAlternatives

State of NaturePrices Up, S1 Prices Stable, S2 Prices Down, S3

Investment A, d1 $30,000 $20,000 -50,000

Investment B, d2 50,000 -20,000 -30,000

Investment C, d3

0 0 0

P(S1) = 0.3 P(S2) = 0.5 P(S3) = 0.2

EV(d1) = $9,000 EV(d2) = -$1,000 EV(d3) = $0

Payoff table for Swofford, Inc.


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Steps for Determining the Utility of Money

Step 1:

Develop a payoff table using monetary values.

Step 2:

Identify the best and worst payoff values and assign

each a utility value, withU(best payoff) > U(worst payoff).

continued


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Step 3:

For every other monetary valueMin the payoff table:

3a: Define the lottery. The best payoff is obtainedwith probabilityp; the worst is obtained withprobability (1 p).

Lottery: Swofford obtains a payoff of $50,000 with probability of p and apayoff -$50,000 with probability of (1-p)

3b: Determine the value ofp such that the decisionmaker is indifferent between a guaranteed

payoff ofMand the lottery defined in step 3a.3c: Calculate the utility ofM:

U(M) =pU(best payoff) + (1 p)U(worst payoff)

continued


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Step 4:

Convert the payoff table from monetary values to utilityvalues.

Step 5:

Apply the expected utility approach to the utility tabledeveloped in step 4, and select the decision alternative

with the highest expected utility.


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Expected Utility Approach Once a utility function has been determined, the

optimal decision can be chosen using the expectedutility approach.

Here, for each decision alternative, the utilitycorresponding to each state of nature is multipliedby the probability for that state of nature.

The sum of these products for each decision

alternative represents the expected utility for thatalternative.

The decision alternative with the highest expectedutility is chosen.


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State of NaturePrices Up, S1 Prices Stable, S2 Prices Down, S3

Investment A, d1 $30,000 $20,000 -50,000

Investment B, d2 50,000 -20,000 -30,000

Investment C, d3

0 0 0

Payoff table for Swofford, Inc.


State of Nature

Prices Up, S1 Prices Stable, S2 Prices Down, S3Investment A, d1 9.5 9 0

Investment B, d2 10 5.5 4

Investment C, d3 7.5 7.5 7.5

Utility table for Swofford, Inc.

P(S1) = 0.3 P(S2) = 0.5 P(S3) = 0.2

EU(d1) = 7.35 EU(d2) = 6.55 EU(d3) = 7.5


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Sample Problem

Beth purchases strawberriesfor $3 a case and sells them for

$8 a case. The rather highmarkup reflects theperishability of the item andthe great risk of stocking it.The product has no value afterthe first day it is offered for

sale. Beth faces the problem ofhow many to order today fortomorrows business

Daily demand No. days

demanded

Probability of

each level

of demand

10 cases 18 0.2

11 cases 36 0.4

12 cases 27 0.3

13 cases 9 0.1

Total 90 days 1.0

What alternative would be chosen according to expected value?

What alternative would be chosen according to expected utility?


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Risk Avoiders vs. Risk Takers

A risk avoider will have a concave utility functionwhen utility is measured on the vertical axis andmonetary value is measured on the horizontal axis.Individuals purchasing insurance exhibit risk

avoidance behavior.

A risk taker, such as a gambler, pays a premium toobtain risk. His/her utility function is convex. Thisreflects the decision makers increasing marginalvalue of money.

A risk neutral decision maker has a linear utilityfunction. In this case, the expected value approach

can be used.


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Utility Example Graph of the Two Decision Makers Utility Curves

Decision Maker I

Decision Maker II

Monetary Value (in $1000s)

Utility

-60 -40 -20 0 20 40 60 80 100

100

60

40

20

80


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Utility Example Decision Maker I

Decision Maker I has a concave utility function.

He/she is a risk avoider.

Decision Maker II

Decision Maker II has convex utility function.

He/she is a risk taker.


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In decision analysis, a single decision maker seeks toselect an optimal alternative.

In game theory, there are two or more decision makers,

called players, who compete as adversaries against eachother.

It is assumed that each player has the same informationand will select the strategy that provides the best

possible outcome from his point of view. Each player selects a strategy independently without

knowing in advance the strategy of the other player(s).

continue


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The combination of the competing strategies provides

the value of the game to the players. Examples of competing players are teams, armies,

companies, political candidates, and contract bidders.


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Two-person means there are two competing players in

the game. Zero-sum means the gain (or loss) for one player is equal

to the corresponding loss (or gain) for the other player.

The gain and loss balance out so that there is a zero-sum

for the game. What one player wins, the other player loses.

Two-Person Zero-Sum Game


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Competing for Vehicle Sales

Suppose that there are only two vehicle dealer-ships in a small city. Each dealership is considering

three strategies that are designed to

take sales of new vehicles from

the other dealership over a

four-month period. The

strategies, assumed to be the

same for both dealerships, are onthe next slide.

Two-Person Zero-Sum Game Example


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Strategy Choices

Strategy 1: Offer a cash rebate

on a new vehicle.

Strategy 2: Offer free optional

equipment on anew vehicle.

Strategy 3: Offer a 0% loan

on a new vehicle.



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2 2 1

CashRebate

b1

0%Loan

b3

FreeOptions

b2

Dealership B

Payoff Table: Number of Vehicle Sales

Gained Per Week by Dealership A(or Lost Per Week by Dealership B)

-3 3 -1

3 -2 0

Cash Rebate a1Free Options a20% Loan a3

Dealership A



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Step 1: Identify the minimum payoff for each

row (for Player A).

Step 2: For Player A, select the strategy that provides

the maximum of the row minimums (called

the maximin).



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Identifying Maximin and Best Strategy

Row

Minimum

1

-3

-2

2 2 1

Cash

Rebate

b1

0%

Loan

b3

Free

Options

b2

Dealership B

-3 3 -1

3 -2 0

Cash Rebate a1Free Options a20% Loan a

3

Dealership A

Best StrategyFor Player A Maximin

Payoff



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Step 3: Identify the maximum payoff for each column

(for Player B).

Step 4: For Player B, select the strategy that provides

the minimum of the column maximums

(called the minimax).



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Identifying Minimax and Best Strategy

2 2 1

Cash

Rebate

b1

0%

Loan

b3

Free

Options

b2

Dealership B

-3 3 -1

3 -2 0


3

Dealership A

Column Maximum 3 3 1

Best StrategyFor Player B

Minimax

Payoff



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Pure Strategy

Whenever an optimal pure strategy exists:

the maximum of the row minimums equals theminimum of the column maximums (Player Asmaximin equals Player Bs minimax)

the game is said to have a saddle point (the

intersection of the optimal strategies) the value of the saddle point is the value of the game

neither player can improve his/her outcome bychanging strategies even if he/she learns in advance

the opponents strategy


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Row

Minimum1

-3

-2

Cash

Rebate

b1

0%

Loan

b3

Free

Options

b2

Dealership B

-3 3 -1

3 -2 0


3

Dealership A

Column Maximum 3 3 1

Pure Strategy Example

Saddle Point and Value of the Game

2 2 1

SaddlePoint

Value of thegame is 1


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Pure Strategy Example

Pure Strategy Summary

Player A should choose Strategya1 (offer a cashrebate).

Player A can expect a gain of at least 1 vehicle saleper week.

Player B should choose Strategyb3 (offer a 0%loan).

Player B can expect a loss of no more than 1 vehiclesale per week.


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Mixed Strategy

If the maximin value for Player A does not equal the

minimax value for Player B, then a pure strategy is notoptimal for the game.

In this case, a mixed strategy is best.

With a mixed strategy, each player employs more than

one strategy. Each player should use one strategy some of the time

and other strategies the rest of the time.

The optimal solution is the relative frequencies with

which each player should use his possible strategies.


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Mixed Strategy Example

b1 b2

Player B

11 5

a1a2

Player A

4 8

Consider the following two-person zero-sum game.

The maximin does not equal the minimax. There isnot an optimal pure strategy.

ColumnMaximum

11 8

RowMinimum

4

5

Maximin

Minimax


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p = the probability Player A selects strategya1(1 -p) = the probability Player A selects strategya2

If Player B selects b1:

EV = 4p + 11(1 p)If Player B selects b2:

EV = 8p + 5(1 p)


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4p + 11(1 p) = 8p + 5(1 p)

To solve for the optimal probabilities for Player A

we set the two expected values equal and solve forthe value ofp.

4p + 11 11p = 8p + 5 5p

11 7p = 5 + 3p

-10p = -6

p = .6

Player A should select:Strategya1 with a .6 probability andStrategya2 with a .4 probability.

Hence,(1 - p) = .4


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q = the probability Player B selects strategyb1(1 -q) = the probability Player B selects strategyb2

If Player A selects a1:

EV = 4q + 8(1 q)

If Player A selects a2:

EV = 11q + 5(1 q)


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4q + 8(1 q) = 11q + 5(1 q)

To solve for the optimal probabilities for Player B

we set the two expected values equal and solve forthe value ofq.

4q + 8 8q = 11q + 5 5q

8 4q = 5 + 6q

-10q = -3

q = .3

Hence,(1 - q) = .7

Player B should select:Strategyb1 with a .3 probability andStrategyb2 with a .7 probability.


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Value of the Game

For Player A:EV = 4p + 11(1 p) = 4(.6) + 11(.4) = 6.8

For Player B:

EV = 4q + 8(1 q) = 4(.3) + 8(.7) = 6.8

Expected gainper game

for Player A

Expected loss

per gamefor Player B


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Dominated Strategies Example

Row

Minimum

-2

0

-3

b1 b3b2

Player B

1 0 3

3 4 -3

a1a2a

3

Player A

ColumnMaximum 6 5 3

6 5 -2

Suppose that the payoff table for a two-person zero-

sum game is the following. Here there is no optimalpure strategy.

Maximin

Minimax


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b1 b3b2

Player B

1 0 3

Player A6 5 -2

If a game larger than 2 x 2 has a mixed strategy,

we first look for dominated strategies in order toreduce the size of the game.

3 4 -3

a1a2a3

Player As Strategya3 is dominated byStrategya1, so Strategya3 can be eliminated.


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b1 b3

Player B

Player A

a1a2

Player Bs Strategyb1 is dominated by

Strategyb2, so Strategyb1 can be eliminated.

b2

1 0 3

6 5 -2

We continue to look for dominated strategies in

order to reduce the size of the game.


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b1 b3

Player B

Player A

a1a2 1 3

6 -2

The 3 x 3 game has been reduced to a 2 x 2. It is

now possible to solve algebraically for the optimalmixed-strategy probabilities.


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Two-Person, Constant-Sum Games

(The sum of the payoffs is a constant other than zero.)

Variable-Sum Games

(The sum of the payoffs is variable.)

n-Person Games

(A game involves more than two players.)

Cooperative Games

(Players are allowed pre-play communications.)

Infinite-Strategies Games(An infinite number of strategies are available for theplayers.)

Other Game Theory Models

Utility and Game Theory

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