Decision Theory Professor Ahmadi
Learning Objectives • Structuring the decision problem and decision trees • Types of decision making environments: • Decision making under uncertainty when probabilities are not known • Decision making under risk when probabilities are known • Expected Value of Perfect Information • Decision Analysis with Sample Information • Developing a Decision Strategy • Expected Value of Sample Information
Types Of Decision Making Environments • Type 1: Decision Making under Certainty. Decision maker know for sure (that is, with certainty) outcome or consequence of every decision alternative. • Type 2: Decision Making under Uncertainty. Decision maker has no information at all about various outcomes or states of nature. • Type 3: Decision Making under Risk. • Decision maker has some knowledge regarding probability of occurrence of each outcome or state of nature.
Decision Trees • A decision tree is a chronological representation of the decision problem. • Each decision tree has two types of nodes; round nodes correspond to the states of nature while square nodes correspond to the decision alternatives. • The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives. • At the end of each limb of a tree are the payoffs attained from the series of branches making up that limb.
Decision Making Under Uncertainty • If the decision maker does not know with certainty which state of nature will occur, then he/she is said to be making decision under uncertainty. • The five commonly used criteria for decision making under uncertainty are: • the optimistic approach (Maximax) • the conservative approach (Maximin) • the minimax regret approach (Minimax regret) • Equally likely (Laplace criterion) • Criterion of realism with (Hurwicz criterion)
Optimistic Approach • The optimistic approach would be used by an optimistic decision maker. • The decision with the largest possible payoff is chosen. • If the payoff table was in terms of costs, the decision with the lowest cost would be chosen.
Conservative Approach • The conservative approach would be used by a conservative decision maker. • For each decision the minimum payoff is listed and then the decision corresponding to the maximum of these minimum payoffs is selected. (Hence, the minimum possible payoff is maximized.) • If the payoff was in terms of costs, the maximum costs would be determined for each decision and then the decision corresponding to the minimum of these maximum costs is selected. (Hence, the maximum possible cost is minimized.)
Minimax Regret Approach • The minimax regret approach requires the construction of a regret table or an opportunity loss table. • This is done by calculating for each state of nature the difference between each payoff and the largest payoff for that state of nature. • Then, using this regret table, the maximum regret for each possible decision is listed. • The decision chosen is the one corresponding to the minimum of the maximum regrets.
Example: Marketing Strategy Consider the following problem with two decision alternatives (d1 & d2) and two states of nature S1 (Market Receptive) and S2 (Market Unfavorable) with the following payoff table representing profits ( $1000): States of Nature s1s3 d1 20 6 Decisions d2 25 3
Example: Optimistic Approach An optimistic decision maker would use the optimistic approach. All we really need to do is to choose the decision that has the largest single value in the payoff table. This largest value is 25, and hence the optimal decision is d2. Maximum DecisionPayoff d1 20 choose d2d2 25 maximum
Example: Conservative Approach A conservative decision maker would use the conservative approach. List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs. Minimum DecisionPayoff choose d1d1 6 maximum d2 3
Example: Minimax Regret Approach For the minimax regret approach, first compute a regret table by subtracting each payoff in a column from the largest payoff in that column. The resulting regret table is: s1s2 Maximum d1 5 0 5 d2 0 3 3 minimum Then, select the decision with minimum regret.
Example: Equally Likely (Laplace) Criterion Equally likely, also called Laplace, criterion finds decision alternative with highest average payoff. • First calculate average payoff for every alternative. • Then pick alternative with maximum average payoff. Average for d1 = (20 + 6)/2 = 13 Average for d2 = (25 + 3)/2 = 14 Thus, d2 is selected
Example: Criterion of Realism (Hurwicz) • Often called weighted average, the criterion of realism (or Hurwicz) decision criterion is a compromise between optimistic and a pessimistic decision. • First, select coefficient of realism, a, with a value between 0 and 1. When a is close to 1, decision maker is optimistic about future, and when a is close to 0, decision maker is pessimistic about future. • Payoff = a x (maximum payoff) + (1-a) x (minimum payoff) In our example let = 0.8 Payoff for d1 = 0.8*20+0.2*6=17.2 Payoff for d2 = 0.8*25+0.2*3=20.6 Thus, select d2
Decision Making with Probabilities • Expected Value Approach • If probabilistic information regarding the states of nature is available, one may use the expected Monetary value (EMV) approach (also known as Expected Value or EV). • Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring. • The decision yielding the best expected return is chosen.
Expected Value of a Decision Alternative • The expected value of a decision alternative is the sum of weighted payoffs for the decision alternative. • The expected value (EV) of decision alternative di is defined as: where: N = the number of states of nature P(sj) = the probability of state of nature sj Vij = the payoff corresponding to decision alternative di and state of nature sj
Example: Marketing Strategy • Expected Value Approach Refer to the previous problem. Assume the probability of the market being receptive is known to be 0.75. Use the expected monetary value criterion to determine the optimal decision.
Expected Value of Perfect Information • Frequently information is available that can improve the probability estimates for the states of nature. • The expected value of perfect information (EVPI) is the increase in the expected profit that would result if one knew with certainty which state of nature would occur. • The EVPI provides an upper bound on the expected value of any sample or survey information.
Expected Value of Perfect Information • EVPI Calculation • Step 1: Determine the optimal return corresponding to each state of nature. • Step 2: Compute the expected value of these optimal returns. • Step 3: Subtract the EV of the optimal decision from the amount determined in step (2).
Example: Marketing Strategy • Expected Value of Perfect Information Calculate the expected value for the best action for each state of nature and subtract the EV of the optimal decision. EVPI= .75(25,000) + .25(6,000) - 19,500 = $750
Decision Analysis With Sample Information • Knowledge of sample or survey information can be used to revise the probability estimates for the states of nature. • Prior to obtaining this information, the probability estimates for the states of nature are called prior probabilities. • With knowledge of conditional probabilities for the outcomes or indicators of the sample or survey information, these prior probabilities can be revised by employing Bayes' Theorem. • The outcomes of this analysis are called posterior probabilities.
Posterior Probabilities • Posterior Probabilities Calculation • Step 1: For each state of nature, multiply the prior probability by its conditional probability for the indicator -- this gives the joint probabilities for the states and indicator. • Step 2: Sum these joint probabilities over all states -- this gives the marginal probability for the indicator. • Step 3: For each state, divide its joint probability by the marginal probability for the indicator -- this gives the posterior probability distribution.
Expected Value of Sample Information • The expected value of sample information (EVSI) is the additional expected profit possible through knowledge of the sample or survey information. • EVSI Calculation • Step 1: Determine the optimal decision and its expected return for the possible outcomes of the sample using the posterior probabilities for the states of nature. • Step 2: Compute the expected value of these optimal returns. • Step 3: Subtract the EV of the optimal decision obtained without using the sample information from the amount determined in step (2).
Efficiency of Sample Information • Efficiency of sample information is the ratio of EVSI to EVPI. • As the EVPI provides an upper bound for the EVSI, efficiency is always a number between 0 and 1.
Refer to the Marketing Strategy Example • It is known from past experience that of all the cases when the market was receptive, a research company predicted it in 90 percent of the cases. (In the other 10 percent, they predicted an unfavorable market). Also, of all the cases when the market proved to be unfavorable, the research company predicted it correctly in 85 percent of the cases. (In the other 15 percent of the cases, they predicted it incorrectly.) Answer the following questions based on the above information.
Example: Marketing Strategy 1. Draw a complete probability tree. 2. Find the posterior probabilities of all states of nature. 3. Using the posterior probabilities, which plan would you recommend? 4. How much should one be willing to pay (maximum) for the research survey? That is, compute the expected value of sample information (EVSI).