Decision Analysis: Uncertainty, Risk, and Expected Value of Perfect Information

Decision-Making Environments and Criteria Under Uncertainty

  • Three Environments of Decision Making

    • Decision Making Under Uncertainty: Scenarios or states of nature are known, but their probabilities of occurrence are completely unknown.
    • Decision Making Under Risk: States of nature are known, and explicit probabilities can be assigned to each state.
    • Decision Making Under Certainty: The exact state of nature that will occur is known with 100% confidence.

  • Objective Function and Payoff Definitions

    • Payoffs in decision tables represent net profit, which incorporates both revenues and expenses.
    • Because payoffs are modeled as profit, the general objective across rules under uncertainty is to maximize outcomes.
    • If the performance metric were strictly measured in terms of cost rather than profit, all decision rules would be inverted to minimize outcomes.

  • Decision Criteria Under Uncertainty

    • Maximax Criterion (Optimistic Perspective):
    • Assumes that the best possible state of nature will occur for whichever decision alternative is selected.
    • Procedure: Identify the maximum payoff for each alternative, then select the alternative corresponding to the maximum of those maximum payoffs.
    • Example Result: Selecting the alternative to construct a large plant, as its maximum potential outcome exceeds all other alternatives.
    • Maximin Criterion (Pessimistic Perspective):
    • Assumes that the worst-case scenario will materialize for each decision alternative.
    • Procedure: Identify the minimum payoff for each alternative, then choose the alternative that maximizes these minimum outcomes (the best of the worst).
    • Example Result: Selecting the alternative to do nothing. Given worst-case payoffs such as -\\text{\\$180,000}, -\\text{\\$20,000}, and \\text{\\$0}, the maximum of these worst outcomes is \\text{\\$0} (doing nothing).
    • Equally Likely / Laplace Criterion:
    • Assumes all potential states of nature have an equal probability of occurrence.
    • Procedure: Calculate the arithmetic mean outcome for each alternative across all states of nature, and choose the alternative with the highest expected average.

Decision Making Under Risk

  • Core Assumptions of Risk Models

    • Probabilities assigned to states of nature must sum exactly to 1:     sumj=1nP(sj)=1.0\\sum_{j=1}^{n} P(s_j) = 1.0
    • States of nature must be mutually exclusive (no overlap between distinct scenarios) and collectively exhaustive (encompassing every possible state that can occur).

  • Expected Monetary Value (EMV)

    • Definition: The weighted average payoff for an alternative, where weights are the probabilities of the respective states of nature.
    • General Mathematical Formula:     EMVi=sumj=1n(PayoffijtimesP(sj))EMV_i = \\sum_{j=1}^{n} (Payoff_{ij} \\times P(s_j))
    • Relationship to Equally Likely Criterion: The equally likely approach is a special case of EMVEMV where each scenario probability P(sj)=frac1nP(s_j) = \\frac{1}{n}.

  • Empirical Application: Plant Construction Decisions

    • State Probabilities:
    • Favorable Market Probability: P(Favorable)=0.6P(\text{Favorable}) = 0.6
    • Unfavorable Market Probability: P(Unfavorable)=0.4P(\text{Unfavorable}) = 0.4
    • Probability Validation: 0.6+0.4=1.00.6 + 0.4 = 1.0
    • Alternative Payoff Calculations:
    • Large Plant:       EMV_{\\text{Large}} = (0.6 \\times \\text{\\200,000}) + (0.4 \\times -\\text{\\48,000}) = \\text{\\120,000} - \\text{\\19,200} = \\text{\\$48,000}
    • Small Plant:       EMV_{\\text{Small}} = (0.6 \\times \\text{\\90,000}) + (0.4 \\times \\text{\\25,000}) = \\text{\\$52,000}
    • Do Nothing:       EMV_{\\text{Do Nothing}} = (0.6 \\times \\text{\\0}) + (0.4 \\times \\text{\\0}) = \\text{\\$0}
    • Optimal Decision Under Risk: The Small Plant alternative is selected because it yields the maximum EMVEMV of \\text{\\$52,000}.

Expected Value of Perfect Information (EVPI)

  • Conceptual Framework

    • Information regarding future states of nature typically carries a acquisition cost.
    • Expected Value of Perfect Information (EVPIEVPI) measures the maximum amount a decision-maker should logically pay to obtain perfect forecasts prior to choosing an alternative.
    • EVPIEVPI calculates the expected gain realized from eliminating all risk and uncertainty.

  • Mathematical Definition and Calculation

    • Expected Value with Perfect Information (EVwPI):
    • The expected return achieved if the decision-maker knows with certainty which state of nature will occur before choosing the optimal action for that state.
    • Formula:       EVwPI=sumj=1n(textBestPayoffinStatejtimesP(sj))EVwPI = \\sum_{j=1}^{n} (\\text{Best Payoff in State } j \\times P(s_j))
    • Expected Value of Perfect Information (EVPI):
    • Formula:       EVPI=EVwPI−max(EMV)EVPI = EVwPI - \\max(EMV)

  • Plant Construction EVPI Calculation

    • Best Payoff in Favorable Market: Construct Large Facility (\\text{\\$200,000})
    • Best Payoff in Unfavorable Market: Do Nothing (\\text{\\$0})
    • Expected Value with Perfect Information:     EVwPI = (0.6 \\times \\text{\\200,000}) + (0.4 \\times \\text{\\0}) = \\text{\\$120,000}
    • Expected Value of Perfect Information:     EVPI = \\text{\\120,000} - \\text{\\52,000} = \\text{\\$68,000}

  • Economic Decision Rules for Purchasing Information

    • Upper bound threshold: The absolute maximum price to pay for perfect information is \\text{\\$68,000}.
    • Information Cost Scenario A (\\text{\\$30,000}):
    • Decision: Purchase the information.
    • Justification: Information cost (\\text{\\$30,000}) is less than EVPIEVPI (\\text{\\$68,000}), yielding an expected net benefit of \\text{\\$38,000}.
    • Information Cost Scenario B (\\text{\\$80,000}):
    • Decision: Reject the information offer.
    • Justification: Information cost (\\text{\\$80,000}) exceeds EVPIEVPI (\\text{\\$68,000}), which would reduce overall expected monetary gain.

Comprehensive Application: Event Location Planning Under Weather Risk

  • Problem Definition and Parameters

    • Decision Alternatives: Outdoor Event, Indoor Event
    • States of Nature: Sunny Weather (P(Sunny)=0.7P(\text{Sunny}) = 0.7), Rainy Weather (P(Rainy)=0.3P(\text{Rainy}) = 0.3)
    • Probability Check: 0.7+0.3=1.00.7 + 0.3 = 1.0
    • Payoff Structure:
    • Outdoor Event under Sunny Weather: \\text{\\$10,000}
    • Outdoor Event under Rainy Weather: -\\text{\\$4,000}
    • Indoor Event under Sunny Weather: \\text{\\$6,000}
    • Indoor Event under Rainy Weather: \\text{\\$5,000}

  • Step-by-Step Expected Monetary Value (EMV) Calculations

    • EMVEMV for Outdoor Event:     EMV_{\\text{Outdoor}} = (0.7 \\times \\text{\\10,000}) + (0.3 \\times -\\text{\\4,000}) = \\text{\\7,000} - \\text{\\1,200} = \\text{\\$5,800}
    • EMVEMV for Indoor Event:     EMV_{\\text{Indoor}} = (0.7 \\times \\text{\\6,000}) + (0.3 \\times \\text{\\5,000}) = \\text{\\4,200} + \\text{\\1,500} = \\text{\\$5,700}
    • Base Choice under Risk: Select Outdoor Event (EMV = \\text{\\$5,800}).

  • Step-by-Step EVwPI and EVPI Calculations

    • Best Decision per State of Nature:
    • If Sunny: Select Outdoor Event (Payoff = \\text{\\$10,000})
    • If Rainy: Select Indoor Event (Payoff = \\text{\\$5,000})
    • Expected Value with Perfect Information:     EVwPI = (0.7 \\times \\text{\\10,000}) + (0.3 \\times \\text{\\5,000}) = \\text{\\7,000} + \\text{\\1,500} = \\text{\\$8,500}
    • Expected Value of Perfect Information:     EVPI = EVwPI - \\max(EMV) = \\text{\\8,500} - \\text{\\5,800} = \\text{\\$2,700}

  • Final Decision Analysis

    • Optimal base alternative under risk: Outdoor Event (EMV = \\text{\\$5,800}).
    • Maximum willingness to pay for a perfect weather forecast: \\text{\\$2,700}.