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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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Steps for Determining the Utility of Money
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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Steps for Determining the Utility of Money
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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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
Payoff table for Swofford, Inc.
DecisionAlternatives
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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Introduction to Game Theory
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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Introduction to Game Theory
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.
Two-Person Zero-Sum Game Example
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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
Two-Person Zero-Sum Game Example
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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).
Two-Person Zero-Sum Game Example
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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
Two-Person Zero-Sum Game Example
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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).
Two-Person Zero-Sum Game Example
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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
Cash Rebate a1Free Options a20% Loan a
3
Dealership A
Column Maximum 3 3 1
Best StrategyFor Player B
Minimax
Payoff
Two-Person Zero-Sum Game Example
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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
Cash Rebate a1Free Options a20% Loan a
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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Mixed Strategy Example
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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Mixed Strategy Example
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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Mixed Strategy Example
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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Mixed Strategy Example
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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Mixed Strategy Example
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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Dominated Strategies Example
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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Dominated Strategies Example
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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Dominated Strategies Example
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
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