Chapter 23 Decision Analysis
Decision Analysis… • In decision analysis: • • We deal with the problem of selecting one alternative from a list of several possible decisions. • • There may be no statistical data, or if there are data, the decision may depend only partly on them, and • • Profits and losses are directly involved. • We will draw upon concepts from probability theory, • Bayes’ Law, and expected value.
Decision Problem – Terminology… • Choices, decisions, possibilities, alternative courses of action, decision alternatives… these are referred to as acts (ai). • E.g.: a1: Invest in a GIC, or • a2: Invest in the stock market. • Acts are controllable; they reflect our choices.
Decision Problem – Terminology… • What actually comes to pass in the future, an outcome, is referred to as a state of nature (si): • E.g.: s1: Interest rates increase, or • s2: Interest rates stay steady or decline. • States of nature are uncontrollable. When enumerating states of nature, we should define all possible outcomes, (i.e. the list should be mutually exclusive and collectively exhaustive)
Decision Problem – Terminology… • For each combination of an act and a state of nature, the amount of profit is calculated. All this data: acts, states of nature, and profits are summarized into a payoff table: E.g. This is the payoff we will receive IF we take action a2AND s1 comes to pass in the future…
Decision Problem – Terminology… • An opportunity loss is the difference between • what the decision maker’s profit for an act is • and what the profit could have been • had the best decision been made. • Opportunity loss is calculated row-wise, by taking the combination of act & state of nature with the highest value and then subtracting this maximum value from all the payoffs in the row. If done correctly, we will be left with one zero (where the maximum payoff was located) and positive numbers for all other act / state of nature combinations.
Example 23.1… • Let’s put some numbers to these concepts… • A man wants to invest $1 million for 1 year. He has three possible alternatives (i.e. acts) • a1: Invest in a GIC paying 10%. • a2: Invest in a bond at 8%. • a3: Invest in stocks.
Example 23.1… • Our investor concludes that there are three possible outcomes that can happen during the year his money is invested (i.e. states of nature) • s1: Interest rates increase. • s2: Interest rates stay the same. • s3: Interest rates decrease.
Example 23.1… • Our investor determines the amount of profit he will make for each possible combination of an act and a state of nature and we create a profit table… E.g. We expect to lose $50,000IF we invest in an 8% bond AND the interest rate goes up in the future…
Example 23.1… Opportunity loss is calculated row-wise, by taking the combination of act & state of nature with the highest value and then subtracting this maximum value from all the payoffs in the row… If done correctly, we will be left with one zero (where the maximum payoff was located) and positive numbers for all other act / state of nature combinations…
Decision Trees… • Often a decision maker must choose between sequences of acts. Here a payoff table will not suffice to determine the best alternative; instead, we require a decision tree. • Though similar to a probability tree, a decision tree represents acts and states of nature sequentially (chronologically). By convention: a square node is used to denote a point where a decision (act) is made, and • a circular node represents a state of nature…
decision state of nature $100,000* a1 s1 –$50,000 a2 s2 $80,000 s3 $180,000 a3 s1 $150,000 s2 $90,000 s3 $40,000 Decision Tree… start with a decision node *A GIC (a1) always pays $100,000 regardless of (si) t
Expected Monetary Value Decision… • Often it is possible to assign probabilities to the states of nature, that is some values P(si). • Our investor believes that future interest rates are most likely to remain essentially the same as they are today and that (of the remaining two states of nature) rates are more likely to decrease than to increase. His probabilities might be: • P(s1) = .2 • P(s2) = .5 • P(s3) = .3
Expected Monetary Value Decision… • Since we’ve determined probabilities for the states of nature, we can compute the expected monetary value (EMV) of each act… • EMV(aj) = P(s1)xPayoff1,j + P(s2)xPayoff2,j + … + P(si)xPayoffi,j • Hence in our example: • EMV(a1) = .2(100,000) + .5(100,000) + .3(100,000) = $100,000 • EMV(a2) = .2(-50,000) + .5(80,000) + .3(180.000) = $84,000 • EMV(a3) = .2(150,000) + .5(90,000) + .3(40,000) = $87,000 • Hence we choose the act (a1) with the highest expected monetary value; i.e. EMV* = $100,000
Expected Monetary Value… • We can use our payoff table for this purpose as well… • EMV* = $100,000
Expected Opportunity Loss (EOL) Decision • We can also calculate the expected opportunity loss (EOL) of each act by using the opportunity loss table and our probabilities for states of nature…
Rollback Technique for Decision Trees… • j Add the probabilities for the states of nature to the branches on our probability tree… P(s)=1.00 $100,000 a1 P(s1)=.20 –$50,000 a2 P(s2)=.50 $80,000 P(s3)=.30 $180,000 a3 P(s1)=.20 $150,000 P(s2)=.50 $90,000 P(s3)=.30 $40,000
Rollback Technique for Decision Trees… • k Calculate the expected monetary value (EMV) for each round node… P(s)=1.00 EMV= (1)(100,000) = $100,000 $100,000 a1 P(s1)=.20 EMV= (.20)(-50,000) + (.50)(80,000) + (.30)(180,000) = $84,000 –$50,000 a2 P(s2)=.50 $80,000 P(s3)=.30 $180,000 a3 P(s1)=.20 EMV= (.20)(150,000) + (.50)(90,000) + (.30)(40,000) = $87,000 $150,000 P(s2)=.50 $90,000 P(s3)=.30 $40,000
Rollback Technique for Decision Trees… • k Calculate the expected monetary value (EMV) for each round node… $100,000 $100,000 a1 –$50,000 $84,000 a2 $80,000 $180,000 a3 $150,000 $87,000 $90,000 $40,000
Rollback Technique for Decision Trees… • l At each square node, we make a decision by choosing the branch with the largest EMV. $100,000 $100,000 a1 –$50,000 $84,000 a2 $80,000 $180,000 a3 $150,000 $87,000 $90,000 $40,000
Additional Information… • We can acquire useful information from consultants, surveys, or other experiments in an effort to improve our decision process, but this additional information comes at a cost to us. • What’s the maximum price we’d be willing to pay for such a survey? • This leads us to the concept of expected payoff with perfect information (EPPI).
Expected Payoff With Perfect Information • If we knew with certainty which state of nature would come to pass, we would would make our decisions accordingly. • If our investor knew for sure that state of nature #1(s1 – interest rates rising) were to come to pass, he would invest in stocks in order to earn $150,000 (instead of investing in GICs [return = $100,000] which we calculated when we didn’t have perfect information). • Back to our payoff table – what is the highest payoff for each state of nature (each row)?
Expected Payoff With Perfect Information • We calculate our Expected Payoff With Perfect Information (EPPI) as any other expected value (i.e. use P(si)’s) • EPPI = .2(150,000) + .5(100,000) + .3(180,000) = $134,000
Expected Value of Perfect Information… • Without perfect information, our investor could put his money into a GIC and earn EMV* = $100,000 expected profit. • Subtracting EMV* from EPPI leaves us with a quantity known as EVPI: the Expected Value of Perfect Information • EVPI = EPPI – EMV* = $134,000 – $100,000 = $34,000
Expected Value of Perfect Information… • Expected Value of Perfect Information • EVPI = EPPI – EMV* = $134,000 – $100,000 = $34,000 • The expected value of perfect information is $34,000 more than what our investor could earn without perfect information. • Thus EVPI provides an upper bound for survey costs to improve the decision making information on hand.
Decision Making w/ Additional Information • Suppose our investor wants to improve his decision-making capabilities by hiring a consultant. IMC (the consultant), will, for a $5,000 fee, analyze the economic conditions and forecast the behavior of interest rates over the next 12 months. • IMC details of their past successes forecasting interest rates. These are stated as conditional probabilities (also known as likelihood probabilities). • Should we pay the $5,000 fee? Is it worth it to buy this data?
Likelihood Probabilities… • I1: IMC predicts the interest rates will increase • I2: IMC predicts the interest rates will stay the same • I3: IMC predicts the interest rates will decrease E.g. when interest rates rose in the past, IMC correctly predicted this happening 60% of the time…
Likelihood Probabilities… • P(I1|s1) = .60 • When the interest rates actually increased (s1) in the past, IMC correctly predicted this happening 60% of the time. • That means 40% of the time they were wrong. • 30% of the time they predicted stable rates and 10% of the time they predicted falling rates, hence the other likelihood probabilities: P(I3|s1) = .10 (say) • We refer to Ii as an “indicator variable”.
Terminology… • Our original probabilities, P(si) are called prior probabilities — they were determined prior to any new information. • We have likelihood probabilities, P(Ii | sj) from the company providing the information. • What we’re interested in finding are posterior probabilities(a.k.a. revised probabilities), which are of the form P(sj | Ii) – what is the probability of a state of nature occurring, given that it was indicated by additional information. • That is: what’s the probability the consultant will be right!
Posterior Probabilities… • We can compute posterior probabilities for each indicator variable in turn with this approach: • j P(sj) – prior probabilities; known. • k P(Ii | sj) – likelihood probabilities; known. • l P(sj and Ii) = P(sj) x P(Ii | sj) – from: • m P(Ii) = sum of all P(sj and Ii)’s • n P(sj | Ii) = P(sj and Ii) / P(Ii) – again, from • Thus we have our posterior probability! (You may recall this as Bayes’ Law from earlier)
Calculating Posterior Probabilities…(I1) • j and k are given, we use to get l
Calculating Posterior Probabilities…(I1) • Add up column l to get the marginal probability for I1 – m
Calculating Posterior Probabilities…(I1) • Again, using divide the values in l by m to get n *repeat this process for indicators I2 and I3…
Posterior Probabilities… • After the probabilities have been revised (i.e. made into posterior probabilities) we can use them in exactly the same way we used the prior probabilities — to calculate the expected monetary value of each act: • EMV(a1) = .60(100,000) + .25(100,000) + .15(100,000)= $100,000 • EMV(a2) = .60(-50,000) + .25(80,000) + .15(180,000)= $17,000 • EMV(a3) = .60(150,000) + .25(90,000) + .15(40,000)= $118,500 • Thus, if IMC forecasts an increase in interest rates, the best course of action is act a3 (since it has the highest EMV). *repeat this process for indicators I2 and I3…
Summary of the EVM Calculations… • The action with the highest expected monetary value will change depending on what the consultant predicts…
Preposterior Analysis… • But we still haven’t answered the question – is it worth $5,000 to buy the consultant’s information or go it alone? • If IMC predicts s1, choose a3, payoff = $118,500 • If IMC predicts s2, choose a1, payoff = $100,500 • If IMC predicts s3, choose a2, payoff = $145,770 • We also know the marginal probabilities… • P(I1) = .20, P(I2) = .52, P(I3) = .28
Expected Monetary Value… • We can calculate an expected monetary value with additional information as the weighted average of: • the expected monetary values, and • P(I1), P(I2), and P(I3) as weights. • Hence: • EMV' =.20(118,500) + .52(100,000) + .28(145,770)= $116,516
Expected Monetary Value… • The value of the consultant’s forecast is the delta between • the expected monetary value with additional information (EMV') • and the expected monetary value without additional information (EMV*). • This difference is called the expected value of sample information and is denoted EVSI. Thus, • EVSI = EMV' - EMV* = $116,516 - $100,000 = $16,516
Expected Value of Sample Information • Since the expected value of sample information is greater than the cost of obtaining the information… • $16,516 > $5,000 • …we should advise our investor to hire IMC consultants and obtain an interest rate forecast before investing.