Risk, Feasibility and Benefit/Cost Analysis Burns, Chapter 6
Recitation What are the components of any decision problem? Which of these have to be mutually exclusive? What is the optimist criterion? What is the pessimist criterion? What is the inbetweenist criterion? What decision environment do all of these assume?
RULE OF INSUFFICIENT REASON 1) For each row, add-up all of the payoffs in that row and record the result in a column to the right labeled ROW SUM. 2) Examine the column labeled ROW SUM to the right and pick the alternative with the largest payoff in that column.
REGRETTIST CRITERION 1) Form the regret table. 1) For each row in the regret table, find the largest regret number in the row and record that in a column to the right labeled ROW MAXIMUM. 2) Examine the column labeled ROW MAXIMUM to the right and pick the alternative with the smallest regret in that column.
Notes For any alternative, the expected value and expected regret numbers sum to the expected payoff of perfect information The expected value and expected regret criteria always select the same alternative, because when the former is maximized, the latter is minimized
Expected Payoff of Perfect Information, EPPI Calculated by finding the largest payoff in each column and then taking the products with the column probabilities and summing these products The EPPI is the best we could do if we had perfect information
Expected Payoff of Perfect Information, EPPI By definition, EPPI = ∑pi * max(Pij) The product of the state probability with the maximum payoff in that column, summing those products = $600,000 The EPPI is the payoff to us of the additional information The additional information is assumed to be ‘perfect’ here
Expected Value of Perfect Information, EVPI by definition, EVPI = EPPI - EV* EVPI = $600,000 - 0 = $600,000 The EVPI is the value to us of the additional information The value is the “best we could do with the additional information” minus the “best we could do without the additional information”
Joint, Marginal and Conditional Probabilities Consider two events A and B, occurring chronologically, temporally as event A and then event B The following relationships are true –P(A∙B) = P(B/A) ∙P(A) = P(A/B)∙P(B) –P(A), and P(B) are marginal probabilities –P(A∙B) = P(B∙A) are joint probabilities –P(A/B), P(B/A) are conditional probabilities
The Steps 1.Solution without additional information 2.Compute initial probabilities—P(S), P(F), P(PS/AS), P(PF/AS), P(PS/AF), P(PF/AF) 3.Do Bayesian Revision 4.Find solution assuming consultant predicts SUCCESS 5.Find solution assuming consultant predicts FAILURE 6.Using the selected alternative in Steps 4 and 5 calculate the EPSI and the EVSI
EPSI –Expected Payoff of Sample Information EPSI = ∑ Probj * highest payoff within each scenario Does NOT assume the information is ‘perfect’
EVSI = Expected Value of Sample Information EVSI = EPSI – EV* The best we could do with the additional information minus the best we could do without the additional information Does NOT assume ‘perfect’ information
Decision making without additional information Don’t do the project and realize a payoff of zero
Expected Payoff of Sample Information, EPSI Sample information is never perfect information By definition, EPSI =.25*$1,400,000 +.75*$0 = $350,000 Clearly, the payoff of sample information is a lot less than the payoff of perfect information
by definition, EVSI = EPSI - EV* EVSI = $350,000 - 0 = $350,000 The value is the “best we could do with the additional information” minus the “best we could do without the additional information” Expected Value of Sample Information, EVSI
Expected value and Expected regret will sum to the same number –what is that number Will always choose the same alternative?? Optimal expected regret is always equal to _____
Decision Trees The Payoff Table approach is useful for a single decision situation. Many real-world decision problems consist of a sequence of dependent decisions. Decision Trees are useful in analyzing multi-stage decision processes.
Characteristics of the Decision Tree –A Decision Tree is a chronological representation of the decision process – There are two types of nodes Decision nodes (represented by squares) State of nature nodes (represented by circles). –The root of the tree corresponds to the present time.
Characteristics of Decision Trees –The tree is constructed outward into the future with branches emanating from the nodes. A branch emanating from a decision node corresponds to a decision alternative. It includes a cost or benefit value. A branch emanating from a state of nature node corresponds to a particular state of nature, and includes the probability of this state of nature.
Probability Trees Describe the process for using these to do Bayesian revision
Probability Trees Construct backward-looking tree –Find joint probabilities at the end nodes by taking the product of all probabilities leading out to the end node Construct forward-looking tree –Move joint probabilities to their appropriate end nodes –Calculate marginal probabilities of indicator states –Calculate posterior conditional probabilities
Solving Decision Trees Are they solved backwards, forwards, sideways? What must every end-node have attached to it? What must every arc emanating from a chance node have attached to it? How are chance nodes handled? How are decision nodes handled?
The EVSI using Decision Trees What is the Definition for the EVSI? How do you calculate it using decision trees?
BILL GALLEN DEVELOPMENT COMPANY – B. G. D. plans to do a commercial development on a property. – Relevant data Asking price for the property is 300,000 dollars. Construction cost is 500,000 dollars. Selling price is approximated at 950,000 dollars. Variance application costs 30,000 dollars in fees and expenses –There is only 40% chance that the variance will be approved. –If B. G. D. purchases the property and the variance is denied, the property can be sold for a net return of 260,000 dollars. –A three month option on the property costs 20,000 dollars, which will allow B.G.D. to apply for the variance. A consultant can be hired for 5000 dollars. –P (Consultant predicts approval | approval granted) = 0.70 –P (Consultant predicts denial | approval denied) = 0.80
SOLUTION Construction of the Decision Tree –Initially the company faces a decision about hiring the consultant. –After this decision is made more decisions follow regarding Application for the variance. Purchasing the option. Purchasing the property.
Do not hire consultant Hire consultant -5000 0 1 Let us consider the decision to not hire a consultant 2 Do nothing 0 Buy land -300,000 Purchase option -20,000 0 3 11 4 Apply for variance -30,000 Apply for variance -30,000
5 Buy land and Apply for variance Approved Denied 0.4 0.6 67 BuildSell 950,000-500,000260,000 Sell 9 -70,000 10 120,000 8 12 Purchase option and Apply for variance Approved Denied 0.4 0.6 -300,000-500,000 950,000 1314 15 Buy land BuildSell 17 -50,000 100,000 16
This is where we are at this stage Let us consider the decision to hire a consultant
1 Do not hire consultant 0 2 Let us consider the decision to hire a consultant Hire consultant -5000 18 Predict Approval Predict Denial 0.4 0.6 19 35 Do Nothing Buy land -300,000 Purchase option -20,000 Do Nothing Buy land -300,000 Purchase option -20,000 -5000 21 28 44 37 36 20 Apply for variance -5000 -30,000
22 Approved Denied Consultant predicts approval The consultant serves as a source for additional information about denial or approval of the variance. Therefore, at this point we need to calculate the posterior probabilities for the approval and denial of the variance application ? ? Posterior Probability of approval | consultant predicts approval) = 0.70 Posterior Probability of denial | consultant predicts approval) = 0.30 0.30 0.70 2324 BuildSell 950,000-500,000260,000 Sell 26 -75,000 27 115,000 25
The rest of the Decision Tree is built in a similar manner. A complete picture can be obtained from WINQSB.
Determining the Optimal Strategy –Work backward from the end of each branch. –At a state of nature node, calculate the expected value of the node. –At a decision node, the branch that has the highest ending node value is the optimal decision. –The highest ending node value is the value for the decision node.
-75,000 115,000 -75,000 115,000 -75,000 115,000 -75,000 115,000 -75,000 22 115,000 - 75,000 Approved Denied (115,000)(0.7)=80500 (-75,000)(0.3)= -22500 -22500 80500 -22500 80500 -22500 80500 -22500 58,000 ? ? 0.30 0.70 2324 BuildSell 950,000-500,000 260,000 Sell 26 27 25 -75,000 115,000 With 58,000 as the chance node value, we continue backward to evaluate the previous nodes.
Decision Tree Evaluation and strategy determination Decision Tree Evaluation and strategy determination Hire the consultant (go to node 18)
If the consultant predicts an approval (indicated by node 19) then buy the land and apply for the variance. Wait for the results. If the variance is approved (indicated by node 23) …then build and sell. We proceed by the same manner and complete the strategy. We proceed by the same manner and complete the strategy.
Decision Making and Utility Introduction –The expected value criterion may not be appropriate if the decision is a one-time opportunity with substantial risks. –Decision makers do not always choose decisions based on the expected value criterion. A lottery ticket has a negative net expected return. Insurance policies cost more than the present value of the expected loss the insurance company pays to cover insured losses.
Example Suppose you play a coin-tossing game in which there is a.55 prob of heads, a.45 prob of tails. The stakes: you win $100,000 if the coin comes up heads; you lose $100,000 if the coin comes up tails The game has an expected return of $10,000 WOULD YOU PLAY???
Three types of Decision Makers –Risk Averse -Prefers a certain outcome to a chance outcome having the same expected value. –Risk Taking - Prefers a chance outcome to a certain outcome having the same expected value. –Risk Neutral - Is indifferent between a chance outcome and a certain outcome having the same expected value.
Risk Preferences Risk averse Risk neutral--most large firms are thought to be risk neutral –same result as Expected value or payoff--no need for utility conversion Risk seeking?? –Small firms do not get to be large firms without being willing to take on some very large, but calculated risks. –Would you say Microsoft’s market behavior is risk seeking? Yet this company turned $4.4 billion in profits in 1999 on just $14.4 billion revenues (ranked 284 of 500 firms) GM produced $3 billion in profits on $161 billion in revenues (ranked 1 out of 500 firms)