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**Introduction to Decision Analysis**

The field of decision analysis provides framework for making important decisions. Decision analysis allows us to select a decision from a set of possible decision alternatives when uncertainties regarding the future exist. The goal is to optimized the resulting payoff in terms of a decision criterion.

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**Components of a Decision Model**

A set of possible decisions discrete, continuous finite, infinite A set of possible events that could occur (states of nature) Payoffs (returns or costs) associated with each decision/state of nature pair

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**Payoff Table Analysis Payoff Tables**

Payoff Table analysis can be applied when - There is a finite set of discrete decision alternatives. The outcome of a decision is a function of a single future event. In a Payoff Table - The rows correspond to the possible decision alternatives. The columns correspond to the possible future events. Events (States of Nature) are mutually exclusive and collectively exhaustive. The body of the table contains the payoffs.

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**EXAMPLE TOM BROWN INVESTMENT DECISION**

Tom Brown has inherited $1000. He has decided to invest the money for one year. A broker has suggested five potential investments. Gold. (Junk) Bond. Growth Stock. Certificate of Deposit. Stock Option Hedge. Tom has to decide how much to invest in each investment.

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**Constructing a Payoff Table**

Determine the set of possible decision alternatives. for Tom this is the set of five investment opportunities. Defined the states of nature. Tom considers several stock market states (expressed by changes in the DJA) State of Nature DJA Correspondence S.1: A large rise in the stock market Increase over 1000 points S.2: A small rise in the stock market Increase between 300 and 1000 S.3: No change in the stock market Change between -300 and +300 S.4: A small fall in stock market Decrease between 300 and 800 S5: A large fall in the stock market Decrease of more than 800 The States of Nature are Mutually Exclusive and Collective Exhaustive

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**Determining Payoffs Our broker reasons:**

Stocks and bonds generally move in the same direction of the market Gold is an investment hedge that seems to move opposite to the market direction A C/D account pays the same interest irrespective of market conditions Specifically our broker’s analysis leads to the following payoff table:

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Payoff Table

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**Dominated Decision Alternatives**

The Stock Option hedge is dominated by the Bond Alternative, because the payoff for each state of nature for the stock option the payoff for the bond option. Thus the stock option hedge can be eliminated from any consideration.

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**Decision Making Criteria**

Classifying Decision Making Criteria Decision making under certainty The future state of nature is assumed known Decision making under risk There is some knowledge of the probability of the states of nature occurring. Decision making under uncertainty. There is no knowledge about the probability of the states of nature occurring.

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**Decision Making Under Certainty**

Given the “certain” state of nature, select the best possible decision for that state of nature

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**Decisions for Decision Making Under Certainty**

If we knew Large Small No Small Large this ===> Rise Rise Change Fall Fall Gold Bond Stock C/D Choose => Stock Stock Gold Gold C/D

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**Decision Making Under Uncertainty**

The decision criteria are based on the decision maker’s attitude toward life. These include an individual being pessimistic or optimistic, conservative or aggressive. Criteria Maximin Criterion - pessimistic or conservative approach. Minimax Regret Criterion - pessimistic or conservative approach. Maximax criterion - optimistic or aggressive approach. Principle of Insufficient Reasoning.

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**Decision Making Under Uncertainty**

The decision criteria are based on the decision maker’s attitude toward life. These include an individual being pessimistic or optimistic, conservative or aggressive. Criteria Maximax (Minimin), Maximin (Minimax), Minimax Regret, Principle of Insufficient Reasoning

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The Maximax Criterion This criterion is based on the best possible scenario. It fits both an optimistic and an aggressive decision maker. An optimistic decision maker believes that the best possible outcome will always take place regardless of the decision made. An aggressive decision maker looks for the decision with the highest payoff (when payoff is profit)

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**To find an optimal decision.**

Find the maximum payoff for each decision alternative. Select the decision alternative that has the maximum of the “maximum” payoff. TOM BROWN - continued The Optimal decision

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The Maximin Criterion This criterion is based on the worst-case scenario. It fits both a pessimistic and a conservative decision maker. A pessimistic decision maker believes that the worst possible result will always occur. A conservative decision maker wishes to ensure a guaranteed minimum possible payoff.

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**To find an optimal decision**

Record the minimum payoff across all states of nature for each decision. Identify the decision with the maximum “minimum payoff”. TOM BROWN - Continued The Optimal decision

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**The Minimax Regret Criterion (“I shoulda”)**

This criterion fits both a pessimistic and a conservative decision maker. The payoff table is based on “lost opportunity,” or “regret”. The decision maker incurs regret by failing to choose the “best” decision.

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**To find an optimal decision**

For each state of nature. Determine the best payoff over all decisions. Calculate the regret for each decision alternative as the difference between its payoff value and this best payoff value. For each decision find the maximum regret over all states of nature. Select the decision alternative that has the minimum of these “maximum regrets”.

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**TOM BROWN - continued 500 - (-100) = 600 500 -100 500 500 -100 -100**

= 600 500 TOM BROWN - continued -100 500 500 -100 -100 500 -100 -100 Investing in Gold incurs a regret when the market exhibits a large rise 500 500 The Optimal decision Let us build the Regret Table

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**The Principle of Insufficient Reason**

This criterion might appeal to a decision maker who is neither pessimistic nor optimistic. It assumes all the states of nature are equally likely to occur. The procedure to find an optimal decision. For each decision add all the payoffs. Select the decision with the largest sum (for profits).

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**TOM BROWN - continued Sum of Payoffs**

Gold $600 Bond $350 Stock $50 C./D $300 Based on this criterion the optimal decision alternative is to invest in gold.

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**Summary Maximax Decision -- Stock Maximin Decision -- C/D**

Minimax Regret -- Junk Bond Insufficient Reason -- Gold Isn’t there a better way?

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**Decision Making Under Risk**

The Expected Value Criterion If a probability estimate for the occurrence of each state of nature is available, payoff expected value can be calculated. For each decision calculate its expected payoff by Expected Payoff = S Based on past market performance the analyst predicts: P(large rise) = .2, P(small rise) = .3, P(No change) = .3, P(small fall) = .1, P(large fall) = .1 (Probability)(Payoff) Over States of Nature

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**TOM BROWN - continued The Optimal decision**

(0.2)(250) + (0.3)(200) + (0.3)(150) + (0.1)(-100) + (0.1)(-150) = 130

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**When to Use the Expected Value Approach**

The Expected Value Criterion is useful in cases where long run planning is appropriate, and decision situations repeat themselves. One problem with this criterion is that it does not consider attitude toward possible losses.

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**Expected Value of Perfect Information (EVPI)**

The Gain in Expected Return obtained from knowing with certainty the future state of nature is called: Expected Value of Perfect Information (EVPI) It is also the Smallest Expect Regret of any decision alternative. Therefore, the EVPI is the expected regret corresponding to the decision selected using the expected value criterion

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**TOM BROWN - continued Stock If it were known with certainty**

that there will be a “Large Rise” in the market Large rise -100 250 500 60 Stock ... the optimal decision would be to invest in... Similarly, Expected Return with Perfect information = 0.2(500)+0.3(250)+0.3(200)+0.1(300)+0.1(60) = $271 EVPI = ERPI - EV = $271 - $130 = $141

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**Bayesian Analysis - Decision Making with Imperfect Information**

Bayesian Statistic play a role in assessing additional information obtained from various sources. This additional information may assist in refining original probability estimates, and help improve decision making.

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**Should Tom purchase the Forecast ?**

TOM BROWN - continued Tom can purchase econometric forecast results for $50. The forecast predicts “negative” or “positive” econometric growth. Statistics regarding the forecast. Should Tom purchase the Forecast ? When the stock market showed a large rise the forecast was “positive growth” 80% of the time.

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SOLUTION Tom should determine his optimal decisions when the forecast is “positive” and “negative”. If his decisions change because of the forecast, he should compare the expected payoff with and without the forecast. If the expected gain resulting from the decisions made with the forecast exceeds $50, he should purchase the forecast.

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**Tom needs to know the following probabilities**

P(Large rise | The forecast predicted “Positive”) P(Small rise | The forecast predicted “Positive”) P(No change | The forecast predicted “Positive ”) P(Small fall | The forecast predicted “Positive”) P(Large Fall | The forecast predicted “Positive”) P(Large rise | The forecast predicted “Negative ”) P(Small rise | The forecast predicted “Negative”) P(No change | The forecast predicted “Negative”) P(Small fall | The forecast predicted “Negative”) P(Large Fall) | The forecast predicted “Negative”)

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**[ P(B | A 1)P(A 1)+ P(B | A 2)P(A 2)+…+ P(B | A n)P(A n) ]**

Bayes’ Theorem provides a procedure to calculate these probabilities P(B |A i)P(A i) [ P(B | A 1)P(A 1)+ P(B | A 2)P(A 2)+…+ P(B | A n)P(A n) ] P(A i | B) = Posterior Probabilities for “positive” economic forecast 0.16 0.56 X = 0.286 0.375 0.268 0.071 0.000 0.2 0.3 0.1 The probability that the stock market shows “Large Rise” given that the forecast predicted “Positive” The Probability that the forecast is “positive” and the stock market shows “Large Rise”. Observe the revision in the prior probabilities

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**Bayesian Prob. for “Negative” Forecast**

s P(s) P(Neg|s) P(Neg and s) P(s|Neg) Large Rise /.44= .091 Small Rise /.44= .205 No Change /.44= .341 Small Fall /.44= .136 Large Fall /.44= .227 .44

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**Expected Value of Sample Information**

The expected gain from making decisions based on Sample Information. With the forecast available, the Expected Value of Return is revised. Calculate Revised Expected Values for a given forecast as follows. EV(Invest in……. |“Positive” forecast) = =.286( )+.375( )+.268( )+.071( )+0( ) = EV(Invest in ……. | “Negative” forecast) = =.091( )+.205( )+.341( )+.136( )+.227( ) = Gold Bond -100 250 -100 100 200 100 150 200 200 -100 300 300 -150 $84 $180 -100 100 200 300 -100 Bond 100 200 300 250 -100 100 200 150 200 -100 300 -150 $ 65 $120

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**Should Tom purchase the Forecast ?**

The rest of the revised EV s are calculated in a similar manner. EREV = Expected Value Without Sampling Information = 130 Expected Value of Sample Information - Excel So, Should Tom purchase the Forecast ? Invest in Stock when the Forecast is “Positive” ERSI = Expected Return with sample Information = (0.56)(250) + (0.44)(120) = $193 Invest in Gold when the forecast is “Negative”

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**EVSI = Expected Value of Sampling Information**

= ERSI - EREV = = $63. Yes, Tom should purchase the Forecast. His expected return is greater than the Forecast cost. Efficiency = EVSI / EVPI = 63 / 141 = 0.45

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Decision Trees The Payoff Table approach is useful for a single decision situation. Many real-world decision problems consists of a sequence of dependent decisions. Decision Trees are useful in analyzing multi-stage decision processes.

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**Characteristics of the Decision Tree**

A Decision Tree is a chronological representation of the decision process There are two types of nodes Decision nodes (represented by squares) State of nature nodes (representing by circles). The root of the tree corresponds to the present time. The tree is constructed outward into the future with branches emanating from the nodes. A branch emanating from a decision node corresponds to a decision alternative. It includes a cost or benefit value. A branch emanating from a state of nature node corresponds to a particular state of nature, and includes the probability of this state of nature.

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**BILL GALLEN DEVELOPMENT COMPANY**

B. G. D. plans to do a commercial development on a property. Relevant data Asking price for the property is $300,000 Construction cost is $500,000 Selling price is approximated at $950,000 Variance application costs $30,000 in fees and expenses There is only 40% chance that the variance will be approved. If B. G. D. purchases the property and the variance is denied, the property can be sold for a net return of $260,000 A three month option on the property costs $20,000 which will allow B.G.D. to apply for the variance. A consultant can be hired for $5000 P (Consultant predicts approval | approval granted) = 0.70 P (Consultant predicts denial | approval denied) = 0.90

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**What should BGD do? Construction of the Decision Tree**

Initially the company faces a decision about hiring the consultant. After this decision is made more decisions follow regarding Application for the variance. Purchasing the option. Purchasing the property.

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**Let us consider the decision to not hire a consultant 11**

3 Do nothing 2 Buy land -300,000 4 Apply for variance -30,000 Do not hire consultant Hire consultant -5000 1 Purchase option -20,000 Let us consider the decision to not hire a consultant 11 Apply for variance -30,000

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Apply for variance Buy land and -70,000 10 120,000 8 6 7 Build Sell 950,000 -500,000 Approved Denied 0.4 0.6 5 260,000 Sell 9 -300,000 -500,000 950,000 13 14 15 Buy land Build Sell 17 -50,000 100,000 16 Approved Denied 0.4 0.6 12 Purchase option and Apply for variance

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**Let us consider the decision to hire a consultant**

This is where we are at this stage

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**Let us consider the decision to**

2 -5000 21 28 44 37 36 20 Apply for variance -30,000 Do not hire consultant Let us consider the decision to hire a consultant 19 35 Do Nothing Buy land -300,000 Purchase option -20,000 1 Hire consultant -5000 18 Predict Approval Denial 0.4 0.6

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**The consultant serves as a source for additional information **

-75,000 27 115,000 25 23 24 Build Sell 950,000 -500,000 22 Approved Denied 0.1765 0.8235 ? 260,000 Sell 26 Posterior Probability of approval | consultant predicts approval) = Posterior Probability of denial | consultant predicts approval) = Consultant predicts approval The consultant serves as a source for additional information about denial or approval of the variance. Therefore, at this point we need to calculate the posterior probabilities for the approval and denial of the variance application

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**Determining the Optimal Strategy**

The rest of the Decision Tree is built in a similar manner. Now, Work backward from the end of each branch. At a state of nature node, calculate the expected value of the node. At a decision node, the branch that has the highest ending node value is the optimal decision. The highest ending node value is the value for the decision node. Let us illustrate by evaluating one branch -- Hire consultant and consultant predicts approval

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**Thus if consultant recommends approval -- BUY LAND**

80500 -22500 115,000 -75,000 115,000 -75,000 115,000 -75,000 (115,000)(.8235)= (-75,000)(.1765)= 115,000 -75,000 115,000 -75,000 -75,000 115,000 -22500 80500 -75,000 115,000 Build Sell 80500 -22500 23 24 25 80500 - -500,000 950,000 Predict approval Approved ? 0.8235 22 Denied 81483 Sell 0.1765 26 27 260,000 $81,483 is the expected value if consultant predicts approval and land is bought In a similar manner, the expected value if consultant predicts approval and the option is purchased is $68,525 The expected value if the consultant predicts approval and BGD does nothing is -$5,000 (the consultant’s fee). Thus if consultant recommends approval -- BUY LAND

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**Evaluation of Other Branches**

DO NOT HIRE CONSULTANT Do nothing EV = $0 Buy land EV = $6,000 Purchase option EV = $10,000 <==BEST HIRE CONSULTANT -- PREDICTS DENIAL Do nothing EV = -$5,000 <===BEST Buy land EV = -$40,458 Purchase option EV = -$27,730

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**BEST COURSE OF ACTION OPTIMAL STRATEGY --**

EV(DO NOT HIRE CONSULTANT) = $10,000 EV(HIRE CONSULTANT) = .4(81,483) + .6(-5000) = $29,593.20 OPTIMAL STRATEGY -- HIRE CONSULTANT If consultant predicts approval -- buy the land If consultant predicts denial -- do nothing

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Game Theory Game theory can be used to determine optimal decision in face of other decision making players. All the players are seeking to maximize their return. The payoff is based on the actions taken by all the decision making players.

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**Classification of Games**

Number of Players Two players - Chess Multiplayer - More than two competitors (Poker) Total return Zero Sum - The amount won and amount lost by all competitors are equal (Poker among friends) Nonzero Sum -The amount won and the amount lost by all competitors are not equal (Poker In A Casino) Sequence of Moves Sequential - Each player gets a play in a given sequence. Simultaneous - All players play simultaneously.

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IGA SUPERMARKET The town of Gold Beach is served by two supermarkets: IGA and Sentry. Market share can be influenced by their advertising policies. The manager of each supermarket must decide weekly which area of operations to discount and emphasize in the store’s newspaper flyer.

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Data The weekly percentage gain in market share for IGA, as a function of advertising emphasis. A gain in market share to IGA results in equivalent loss for Sentry, and vice versa (i.e. a zero sum game)

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**IGA needs to determine an advertising emphasis that**

will maximize its expected change in market share regardless of Sentry’s action.

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**IGA’s Perspective - A Linear Programming model**

SOLUTION IGA’s Perspective - A Linear Programming model Decision variables X1 = the probability IGA’s advertising focus is on meat. X 2 = the probability IGA’s advertising focus is on produce. X 3 = the probability IGA’s advertising focus is on groceries. Objective Function For IGA Maximize expected market change (in its own favor) regardless of Sentry’s advertising policy. Define the actual change in market share as V.

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Constraints IGA’s market share increase for any given advertising focus selected by Sentry, must be at least V. The Model

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**Sentry’s Perspective - A Linear Programming model**

Decision variables Y1 = the probability that Sentry’s advertising focus is on meat. Y2 = the probability that Sentry’s advertising focus is on produce. Y3 = the probability that Sentry’s advertising focus is on groceries. Y4 = the probability that Sentry’s advertising focus is on bakery. Objective function Minimize changes in market share in favor of IGA Constraints Sentry’s market share decrease for any given advertising focus selected by IGA, must not exceed V.

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The Model

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**Optimal solutions For IGA For Sentry**

X1 = ; X2 = 0.5; X3 = 0.111 For Sentry Y1 = 0.6; Y2 = 0.2; Y3 = 0.2; Y4 = 0 For both players V =0 (a fair game).

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1 1 Slide Decision Theory Professor Ahmadi. 2 2 Slide Learning Objectives n Structuring the decision problem and decision trees n Types of decision making.

1 1 Slide Decision Theory Professor Ahmadi. 2 2 Slide Learning Objectives n Structuring the decision problem and decision trees n Types of decision making.

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