Decision Analysis A method for determining optimal strategies when faced with several decision alternatives and an uncertain pattern of future events. The Decision Analysis Approach Identify the decision alternatives - di Identify possible future events - sj mutually exclusive - only one state can occur exhaustive - one of the states must occur Determine the payoff associated with each decision and each state of nature - Vij Apply a decision criterion Types of Decision Making Situations Decision making under certainty state of nature is known decision is to choose the alternative with the best payoff Types of Decision Making Situations Decision making under uncertainty The decision maker is unable or unwilling to estimate probabilities Apply a common sense criterion Decision Making Under Uncertainty Maximin Criterion (for profits) - pessimistic list minimum payoff for each alternative choose alternative with the largest minimum payoff Decision Making Under Uncertainty Maximax Criterion (for profits) - optimistic list maximum payoff for each alternative choose alternative with the largest maximum payoff Decision Making Under Uncertainty Minimax Regret Criterion calculate the regret for each alternative and each state list the maximum regret for each alternative choose the alternative with the smallest maximum regret Decision Making Under Uncertainty Minimax Regret Criterion Regret - amount of loss due to making an incorrect decision - opportunity cost Rij | V * j Vij | Types of Decision Making Situations Decision making under risk Expected Value Criterion compute expected value for each decision alternative select alternative with “best” expected value Computing Expected Value Let: P(sj)=probability of occurrence for state sj and N=the total number of states Computing Expected Value Since the states are mutually exclusive and exhaustive N P( s ) P( s ) P( s ) P( s ) 1 j 1 j 1 and P( sj ) 0 for all j 2 N Types of Decision Making Situations Then the expected value of any decision di is N EV (di ) P(sj )Vij j 1 Decision Trees A graphical representation of a decision situation Most useful for sequential decisions Decision Making Under Risk: Another Criterion Expected Regret Criterion Compute the regret table Compute the expected regret for each alternative Choose the alternative with the smallest expected regret The expected regret criterion will always yield the same decision as the expected value criterion. Expected Regret Criterion The expected regret for the preferred decision is equal to the Expected Value of Perfect Information - EVPI EVPI is the expected value of knowing which state will occur. EVPI – Alternative to Expected Regret EVPI – Expected Value of Perfect Information EVwPI – Expected Value with Perfect Information about the States of Nature EVwoPI – Expected Value without Perfect Information about the States of Nature EVPI=|EVwPI-EVwoPI| Bayes Law P( A | B) P( B) P( B | A) P( A | B) P( B) P( A | B ) P( B ) P( A) P( A | B) P( B) P( A | B ) P( B ) In this equation, P(B) is called the prior probability of B and P(B|A) is called the posterior, or sometimes the revised probability of B. The idea here is that we have some initial estimate of the probability of B, we get some additional information about whether A happens or not, and then we use Bayes Law to compute this revised probability of B. Expected Value of Sample Information – EVSI EVSI – Expected Value of Sample Information EVwSI – Expected Value with Sample Information about the States of Nature EVwoSI – Expected Value without Sample Information about the States of Nature EVSI=|EVwSI-EVwoSI| Efficiency of Sample Information – E Perfect Information has an efficiency rating of 100%, the efficiency rating E for sample information is computed as follows: EVSI E 100 EVPI Note: Low efficiency ratings for sample information might lead the decision maker to look for other types of information Accounting for Risk in Decision Analysis Mean-Variance m Vari (rij ERi ) pj 2 j 1 Accounting for Risk in Decision Analysis Utility Theory replacing the payoffs with a unitless scale that accounts for both the value of the payoff and the decision makers risk attitude Risk Aversion A decision maker is risk averse if he/she would prefer a certain x dollars to a risky alternative with ER=x dollars. Accounting for Risk in Decision Analysis Direct assessment of utility Utility functions For example U ( x) 1 e x / r where x is the amount of the payoff and r is the risk aversion parameter.