DSC 3120 Generalized Modeling Techniques with Applications Part III. Decision Analysis Decision Analysis

Decision Analysis A Rational and Systematic Approach to Decision Making Decision Making: choose the “best” from several available alternative courses of action Key Element is Uncertainty of the outcome We, as decision maker, control the decision Outcome of the decision is uncertain to and uncontrolled by decision maker (controlled by nature) Decision Analysis

Example of Decision Analysis You have $10,000 for investing in one of the three options: Stock, Mutual Fund, and CD. What is the best choice? Question: Do you know the choices? Do you know the best choice? What is the uncertainty? How do you make your choice? Decision Analysis

Components of Decision Problem Alternative Actions -- Decisions There are several alternatives from which we want to choose the best States of Nature -- Outcomes There are several possible outcomes but which one will occur is uncertain to us Payoffs Numerical (monetary) value representing the consequence of a particular alternative action we choose and a state of nature that occurs later on Decision Analysis

Payoff Table Decision Analysis

An Example State of Nature Alternative Decision Analysis

Three Classes of Decision Models Decision Making Under Certainty Only one state of nature (or we know with 100% sure what will happen) Decision Making Under Uncertainty (ignorance) Several possible states of nature, but we have no idea about the likelihood of each possible state Decision Making Under Risk Several possible states of nature, and we have an estimate of the probability for each state Decision Analysis

Decision Making Under Uncertainty LaPlace (Assume Equal Likely States of Nature) Select alternative with best average payoff Maximax (Assume The Best State of Nature) Select alternative that will maximize the maximum payoff (expect the best outcome--optimistic) Maximin (Assume The Worst State of Nature) Select alternative that will maximize the minimum payoff (expect the worst situation--pessimistic) Minimax Regret (Don’t Want to Regret Too Much) Select alternative that will minimize the maximum regret Decision Analysis

Example: Newsboy Problem Payoff Table Decision Analysis

Example: Newsboy Problem LaPlace Criterion Decision Analysis

Example: Newsboy Problem Maximax Criterion Decision Analysis

Example: Newsboy Problem Maximin Criterion Decision Analysis

Minimax Regret Criterion: Step 1 Example: Newsboy Problem Minimax Regret Criterion: Step 1 Decision Analysis

Minimax Regret: Step 2 (Regret or Opportunity Loss Table) Example: Newsboy Problem Minimax Regret: Step 2 (Regret or Opportunity Loss Table) Decision Analysis

Decision Making Under Risk In this situation, we have more information about the uncertainty--probability Decision Analysis

Decision Making Under Risk Maximize Expected Return (ER) ERi =  (pj  rij) = p1ri1 + p2ri2 +…+ pmrim Where ERi = Expected return if choosing the ith alternative (Ai), (i = 1, 2, …, n) pj = The probability of state j (Sj) rij = The payoff if we choose alternative Ai and Sj state of nature occurs Decision Analysis

Expected Return & Variance Example: Newsboy Problem Expected Return & Variance Decision Analysis

Decision Making Under Risk High return is good, but on the other hand, low risk is also important Variance -- a measure of the risk Variancei =  pj  (rij - ERi)2 Where pj = The probability of state j (Sj) rij = The payoff if choose Ai and Sj occurs ERi= Expected return for alternative Ai Decision Analysis

Expected Value of Perfect Information EVPI measures the maximum worth (value) of the “Perfect Information” that we should pay for in order to improve our decisions EVPI = ER w/ perfect info. - ER w/o perfect info. ER w/ perfect info. =  pj  max(rij) ER w/o perfect info. = max(ERi) = max( pj  rij) Decision Analysis

Example: Newsboy Problem Calculate EVPI ER w/o PI ER w/ PI EVPI Decision Analysis

Expected Opportunity Loss (EOL) We can also use EOL to choose the best alternative Minimizing EOL = Maximizing ER both criteria yield the same best alternative EOLi =  pj  OLij where pj = The probability of state j (Sj) OLij = The opportunity loss if choose Ai and Sj occurs min(EOLi) = EVPI Decision Analysis

Expected Opportunity Loss Example: Newsboy Problem Expected Opportunity Loss EVPI Decision Analysis

Decision Making with Utilities Problem with Monetary Payoffs People do not always just look at the highest expected monetary return to make decisions; they often evaluate the risk Example: A company wants to decide to develop a new product or not Decision Analysis

Decision Making with Utilities Utility -- combines monetary return with people’s attitude toward risk Utility Function -- a mathematical function that transforms monetary values into utility values Three general types of utility functions (1) Risk-Averse (2) Risk-Neutral (3) Risk-Seeking MV Utility MV Utility MV Utility Decision Analysis

Risk-Averse Utility Function 0.910 0.850 0.775 0.680 0.524 100 200 300 400 500 Dollars Properties of Risk-averse Utility Function non-decreasing: more money is always better concave: utility increase for unit ($100, e.g.) increase of money is decreasing (extra money is less attractive) Decision Analysis

How to Create Utility Function Method I. Equivalent Lottery Start with two endpoints A (the worst possible payoff) and B (the best possible payoff) and assign U(A) = 0 and U(B) = 1 Then to find the utility for a possible payoff z between A and B, select the probability p (=U(z)) such that you are indifferent between the following two alternatives receive a payoff of z for sure receive a payoff of B with probability p or a payoff of A with probability 1 - p Decision Analysis

How to Create Utility Function Method II. Exponential Utility Function where x is the monetary value, r>0 is an adjustable parameter called risk tolerance First, the value of r can be estimated such that we are indifferent between the following choices a payoff of zero a payoff of r dollars or a loss of r/2 dollars with 50-50 chance Then the utility for a particular monetary value x can be found using the above assumed exponential utility function Decision Analysis

Example: Newsboy Problem Expected Utility Decision Analysis