Decision Analysis

Decision Analysis

• Alternatives and States of

Nature

• Good Decisions vs. Good

Outcomes

• Payoff Matrix

• Decision Trees

• Utility Functions

• Decisions under Uncertainty

• Decisions under Risk

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Decision Analysis - Payoff Tables

Case Problem - (A) p. 38

As director of a home health agency, you are in the process of evaluating the possible addition of physical therapy services for your clients. Three options are under consideration:

Option A: The home health agency would contract with an independent practitioner to provide

PT services on a fee-for-service basis.

Costs: The PT will be paid $60 per home visit.

Option B: The home health agency would hire a staff physical therapist, provide the therapist with an automobile, and cover medical supplies and gasoline.

Costs: Monthly PT salary of $4000

Monthly automobile lease payment of $400

Medical supplies and gasoline allowance of $7 per visit

Option C: The home health agency would utilize an independent contractor to provide PT services, provide the contractor with an automobile and fringe benefits, and pay a medical supply and gasoline allowance.

Costs: PT services of $35 per visit

Monthly automobile lease payment of $400

Fringe benefits (including insurance) of $200 per month

Medical supplies and gasoline allowance of $7 per visit

Under all three alternatives the average payment for a PT home visit is $75 per visit.

The home health agency is trying to decide for which of these three options the maximum profit would be realized. In searching for the best option, the agency realizes that the demand for services influences the optimum choice. The agency would like to better understand the relationship between demand and optimum choice. Specific questions that the agency has posed include the following:

1. With no knowledge of demand for PT services, which option should be chosen?

2. For a given probability distribution of demand for PT services, which option provides maximum profit?

3. How sensitive is the choice of best option to the probability distribution of demand?

4. How much value should be placed on a system that can forecast future demand for PT services?

Assume that, using a combination of marketing analysis, analysis of competitors' volume data, and some careful assumptions, it has been determined that there are only four possible values of monthly demand for PT services: 30, 90, 140, or 150 visits. Assume further that the relative frequency with which each of these demands occurs is also available from health facilities utilization data compiled in a recent statewide study. These relative frequencies provide reasonable estimates of the probabilities associated with each value of monthly demand, which for the case problem are assumed to equal 0.1, 0.4, 0.2, and 0.3, respectively.

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Decisions under Uncertainty

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Decisions under Risk

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Decision Analysis - Utility Theory

• Utility theory provides a way to incorporate the decision maker’s attitudes and preferences toward risk and return in the decision analysis process so that the most desirable decision alternative is identified.

• A utility function translates each of the possible payoffs in a decision problem into a nonmonetary measure known as a utility.

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Utility

1.00

0.75

0.50

0.2

5

0 risk averse risk neutral risk seeking

Payoff

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• The utility of a payoff represents the total worth, value, or desirability of the outcome of a decision alternative to the decision maker.

• A risk averse decision maker assigns the largest relative utility to any payoff but has a diminishing marginal utility for increased payoffs.

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• A risk seeking decision maker assigns the smallest utility to any payoff but has an increasing marginal utility for increased payoffs.

• A risk neutral decision maker falls in between these two extremes and has a constant marginal utility for increased payoffs.

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Constructing Utility Functions

•

Step 1 - Assign a utility value of 0 to the worst outcome (W) in a decision problem and a utility value of 1 to the best outcome (B).

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• Step 2 - For any other outcome x , find the probability p at which the decision maker is indifferent between the following two alternatives:

– Receive x with certainty or

– Receive B with probability p or W with probability 1p

The value of p is the utility that the decision maker assigns to the outcome x .

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For example, let’s compute the utility for the $450 entry that corresponds to alternative

A and state of nature N=30. The problem consists on finding the value of p that makes the following two options equally attractive for the decision maker:

– Receive $450 with certainty

– Play a game in which the decision maker can make $5,800 with probability p or lose $2,360 with probability 1p

Let’s assume that the value of p that makes these two choices equally attractive to the decision maker is 0.7. Then the utility that the decision maker assigns to the $450 is

0.7.

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The Exponential Utility Function

• A sensible value for R is the maximum value of Y for which the decision maker is willing to participate in a game of chance with the following possible outcomes:

• Win $Y with probability 0.5

– Lose $Y/2 with probability 0.5

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The Exponential Utility Function

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Decision Analysis

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