Ch.15 Decision Analysis
Decision Analysis without Probabilities (Slide 3 of 10) *see excel spreadsheet
- In our PDC problem using the optimistic approach would lead us to the decision to build the large complex so decision d3 bc this decision offers a maximum payoff. In a situation where the demand is strong but we're optimistic (approach): we assume this will happen
Problem Formulation (Slide 3 of 10)
- In this problem, we want to MAXIMIZE the profit - We don't know what the demand is going to be - What we know so far is that there are 3 possible decisions alternatives for us to be denoted as: d1, d2, and d3
Decision Analysis without Probabilities (Slide 1 of 10)
- The decision analysis without probabilities although quite simple is is still a legit method and therefore, appropriate in situations - We usually use it when a simple best case worst case analysis is sufficient or when we have no or little confidence in our ability to actually somehow establish or estimate the probabilities.
Problem Formulation (Slide 7 of 10)
- The graphical representation of what happens in a table is called a decision tree. It represents the decision making process
Decision Analysis without Probabilities (Slide 4 of 10) *see excel file ch.15 example
- conservative approach is the opposite of optimistic approach - basically we're not at all optimistic, more pessimistic then eventually decide to go with the best of the worst possible outcomes or possible payoffs - with the conservative approach we analyze each decision alternative separately first then we determine the worst possible payoff for each decision alternatives Ex / 1st step in conservative approach: For the first decision alternative the smallest payoff is 7. For d2 the medium complex is 5 and for then -9 for the large complex 2nd step: Final step will be to choose the best of the worst payoffs - in our case they all happen when the weak demand happens. Just happened to be this way, it doesn't always have to be in 1 column always. - at times we can have more than 2 States of Nature so those worse payoffs per each decision alternative will be in different columns. The point is now to select the smallest one
Decision Analysis without Probabilities (Slide 2 of 10)
- for our example, where we maximize the profit (will be related to the largest payoff) but if we deal with the minimization problem where we for example minimize the cost then that best possible payoff will be the smallest one ^need to be flexible and careful
Equation (15.5) is a restatement of Bayes' theorem introduced in Chapter 4.
Equation (15.5) is known as Bayes' theorem, and it is used to compute posterior probabilities.
Computing Branch Probabilities with Bayes' Theorem (Slide 3 of 4)
Ex: we need to know the conditional probability of a favorable market research report given that the state of nature is strong demand for the PDC project. To carry out the probability calculations, we will need conditional probabilities for all sample outcomes given all states of nature, that is P(F | S1), P(F | S2), P (U | S1) and P(U | S2) ^These conditional probabilities are assessments of the accuracy of the market research; they are often estimated using historical performance of previous market research reports. Ex: P(F | S1) may be estimated via the historical frequency of strong demand being associated with a market research report that was favorable.
Decision Analysis without Probabilities (Slide 6 of 10)
The 3rd approach is called minimax regret approach. - we can treat that regret / think of it as an opportunity loss or the cost of a lost opportunity - for this approach we'll choose the decision alternative that minimizes the maximum state of regret that could occur over all possible states of nature
Introduction (Slide 2 of 2)
The decision alternative we'll consider
Decision Analysis with Probabilities (Slide 1 of 8)
The expected value approach for decision analysis is a method used in decision analysis for determining the best decision alternative when we have probability estimates for the possible states of nature. ^requires calculation
Risk profile:
The probability distribution of the possible payoffs associated with a decision alternative or decision strategy.
Risk analysis:
The study of the possible payoffs and probabilities associated with a decision alternative or a decision strategy in the face of uncertainty.
Decision Analysis techniques belong to what category in data analytics?
The third category, prescriptive analytics. So, we will have decision alternatives here and there will be some risk that will be related to EACH of those decision alternatives
The optimistic approach evaluates each
decision alternative in terms of the best payoff that can occur.
The conservative approach evaluates each
decision alternative in terms of the worst payoff that can occur.
Bayes' theorem can be used to compute branch probabilities for
decision trees.
Expected value (EV) of a decision alternative di
di = decision alternative I sj = state of nature J occuring P(sj) = probability of state of nature sj N = number of possible states of nature Vij = the payoff or value corresponding to making decision alternative di, and state of nature J occurs
One approach to sensitivity analysis is to select
different values for the probabilities of the states of nature and the payoffs and then resolve the decision analysis problem. If the recommended decision alternative changes, we know that the solution is sensitive to the changes made.
In many decision-making situations, we can obtain probability assessments for the states of nature. When such probabilities are available, we can use the
expected value approach (EVA) to identify the best decision alternative
Risk analysis helps the decision maker recognize the difference between the
expected value of a decision alternative and the payoff that may actually occur.
Risk analysis helps to provide the probability information about the
favorable as well as the unfavorable outcomes that may occur.
Risk profile
for a decision alternative shows the possible payoffs along with their associated probabilities.
For Optimistic Approach what are we interested in?
interested in the most largest possible payoff
Utility:
is a measure of the total worth or relative desirability of a particular outcome; it reflects the decision maker's attitude toward a collection of factors such as profit, loss, and risk.
Optimistic Approach is also called
maximax
Conservative Approach is also called
maximin - we're maximizing the minimum payoffs per each decision alternative
Utility is a
measure of the total worth of a consequence reflecting a decision maker's attitude toward considerations such as profit, loss, and risk.
As its name implies, under the minimax regret approach to decision analysis, one would choose the decision alternative that
minimizes the maximum state of regret that could occur over all possible states of nature.
We will in decision analysis consider problems that involve
some decision alternatives and few possible future events
Tree Diagrams help us
structure our decision analysis and chose between multiple decisional alternatives - box on far left is called a decision node - the branches that come out of the decision node represent the 3 different decision alternatives: we can chose to build 3 different size developments - the circles represent chance nodes, chance nodes include the states of nature - the states of nature of PDC are the demand can be strong or weak - on the far right are the payoffs. the payoffs use the notation VIJ. Example: V11 represents the payoff if we chose a small condo development and state of nature S1 occurs therefore, we have strong demand. Payoff of 8 million dollars
In words, the expected value of a decision alternative is the
sum of weighted payoffs for the decision alternative. The weight for a payoff is the probability of the associated state of nature and therefore the probability that the payoff will occur.
risk neutral:
the decision maker always prefers the decision with positive expected value payoff to one with a negative expected payoff value
conditional probability:
the probability of one event, given the known outcome of a (possibly) related event.
Sensitivity analysis:
the study of how changes in the probability assessments for the states of nature or changes in the payoffs affect the recommended decision alternative.
Researchers have found that as long as the monetary value of payoffs stays within a range that the decision maker considers reasonable, selecting the decision alternative with the best expected value usually leads to selection of the most preferred decision. However,
when the payoffs are extreme, decision makers are often unsatisfied or uneasy with the decision that simply provides the best expected value.
The question we ask in Decision Analysis is
which decision alternative do we choose in our situtation?
Utility for example incorporates the availability of a decision makers funds to absorb negative payoffs as well as the
willingness of the decision maker to risk small amounts for unlikely large payoffs
Utility Theory (Slide 4 of 18) Table 15.7: Payoff Table for Swofford, Inc.
So Swaford has conducted an extensive market analysis to determine the possible payoffs given a decision and a state of nature ^shown in table Label each of the possible payoffs using notation VIJ. Ex = V11 refers to the payoff if decision alternative D1 is chosen and the state of nature is S1 ^aka Swaford chooses to invest in real estate option A and prices go up. The payoff here is $30,000. The payoff for investing in real estate option A and prices remain stable is given as V12 which is $20,000. Similarly, V13 (V 1, 3 not V thirteen) is shown to be negative $50,000 which is what occurs if Swatford chooses investment alternative A and prices go down Investment Alternative B has payoff shown in the second row V21, V22, and V23 If Swatford chooses not to invest which is decision alternative D3 then the payoffs for each state of nature is simply 0 because Swaford does NOT risk anything and they have no possible return. These payoffs are indicted by V31, V 32, and V33 *Using these values we can then calculate the expected values of each decision alternative: EV(d1) = P(S1) * V11 + P(S2) * V12 + P(s3) * V13 EV(d1) = 0.3 x 30k + 0.5 x 20k + 0.2 x (-50k) = 9k ^ we have taken the probability of each state of nature occurring multiplied by the payoff of that decision when that state of nature occurs Therefore, the expected value of decision alternative 1 is $9k Similarly, the expected value of decision alternative 2 is -11k aka we expect to lose money if we chose decision alternative D2 Finally, the expected value of decision alternative D3 which is to NOT invest is simply 0. Based on these calculations, the optimal expected value (highest expected value) is achieved from decision alternative D1 of $9,000. Therefore, if Swaford is using a simple expected payoff value they should choose decision alternative 1 and INVEST in real estate option A. ^The prob here is there is a 20% chance Swaford will do this. Can possibilty lose 50k, which and deeply impact company if they don't have 50k so bankrupt. So, Swaford might not prefer this alternative just based on possibly losing 50k
To illustrate the concept of utility, we'll use the example of Swaford Real Estate Investment Firm
Swaford has 3 possible decisions to make: Can chose to invest in Real estate option A aka d1. Decision alternative two is to invest in real estate option B and D3 is to simply NOT invest. There are 3 possible states of nature that affects the payoffs related to these decisions: S1 = the state of nature when real estate prices go up S2 = the state of nature when real estate prices remain stable S3 = the state of nature when real estate prices go down
Bayes' theorem:
A theorem that enables the use of sample information to revise prior probabilities.
Problem Formulation (Slide 5 of 10)
- the table is called a Payoff Table for our problem - table is represented in millions - row labels: Decision Alternatives d1, d2, and d3. State of Nature columns are Strong Demand and Weak Demand - The table is called a Payoff Table bc that is what the numbers are. The numbers represent payoff which is the numerical outcomes resulting from a specific combination of a decision alternative in a state of nature - Example: If what happens is a strong demand and we end up building a small complex then we will earn 8 million dollars but if we decided to build a large condo complex then we'll go with decision d3 and the demand will be weak. We'll lose 9 million dollars. ^these #s are given, through analysis these payoffs are estimated and this is what we'll start with and then come to an optimal decision recommendation on which decision alternative to chose
Problem Formulation (Slide 4 of 10)
- the terminology we'll be using for the possible outcomes of a chance event is states of nature - states of nature must be mutually exclusive - In any given situation there is only 1 possible state of nature that will happen. That will occur - In our example the states of nature are related to the demand - the demand is our chance event. we don't know WHAT to expect but we know there are 2 possibilities called S1 and S2
Problem Formulation (Slide 2 of 10)
- we have 3 different sixes of the condominiums potentially to be built
15.3 Decision Analysis with Probabilities
1) Expected Value Approach 2) Risk Analysis 3) Sensitivity Analysis
Decision Analysis without Probabilities ^this is the simplistic technique w/o probabilities. we'll be discussing 3 different approaches
1) Optimistic Approach 2) Conservative Approach 3) Minimax Regret Approach
Problem Formulation
1) Payoff Tables 2) Decision Trees
15.6 Utility Theory
1) Utility and Decision Analysis 2) Utility Functions 3) Exponential Utility Function
Ex: A payoff of negative, -100,000 is exactly
100 times worse than a payoff of -$1,000 but clearly this isn't always true Ex: Lottery tickets almost always have a negative expected payoff value, BUT many people purchase lottery tickets. One of the reasons for this is what is known as the Utility Theory
Utility Theory (Slide 2 of 18) Swofford Inc. example
As an example of a situation in which utility can help in selecting the best decision alternative, let us consider the problem faced by Swofford, Inc., a relatively small real estate investment firm located in Atlanta, Georgia. Swofford currently has two investment opportunities that require approximately the same cash outlay. The cash requirements necessary prohibit Swofford from making more than one investment at this time. Consequently, three possible decision alternatives may be considered.
Problem Formulation (Slide 1 of 10)
As we formulate the problem, we'll always identify the decision alternatives and always will look at uncertain future events, etc
The expected value approach for decision analysis is best illustrated through an ex: PDC Condos - PDC is considering an investment in a new condo development - payoff from these condos is dependent on both the size of condo development that PDC chooses as well as demand for these condos - demand for these condos is outside of the control of PDC therefore, this is what is referred to as the states of nature - based on historical research and current market, we'll assume that PDC believes there is a 0.8 probability that demand for their condos will be strong and a 0.2 probability that demand will be weak.
Assume company PDC believes: Probability of strong demand for condos is 0.8 Probability of weak demand for condos is 0.2 ^will use to label our states of nature and the probability of these states of nature occuring.
15.5 Computing Branch Probabilities with Bayes' Theorem
Conditional probability is introduced in Chapter 4.
The PDC decision tree is shown again in Figure 15.10. Let:
F = favorable market research report U = unfavorable market research report S1 = strong demand (state of nature 1) S2 = weak demand Explanation: At chance node 2, we need to know the branch probabilities P(F) and P(U). At chance nodes 6, 7, and 8, we need to know the branch probabilities P(s1 | F) , which is read as "the probability of state of nature 1 given a favorable market research report," and P(s2 | F) , which is the probability of state of nature 2 given a favorable market research report. The notation | in P(s1 | F) and P(s2 | F) is read as "given" and indicates a conditional probability, because we are interested in the probability of a particular state of nature "conditioned" on the fact that we receive a favorable market report. At chance nodes 9, 10, and 11, we need to know the branch probabilities P(s1 | U) and P(s2 | U) ^note that these are also posterior probabilities, denoting the probabilities of the two states of nature given that the market research report is unfavorable At chance nodes 12, 13, and 14, we need the probabilities for the states of nature, P(s1) and P(s2) if the market research study is not undertaken.
Problem Formulation (Slide 8 of 10)
Figure 15.1: Decision Tree for the PDC Condominium Project ($ Millions) - for our example, in our decision tree we have nodes of different shapes. - 1st node is a square. Squares represent decision nodes. There are 3 branches coming out of the square node in small, medium, or large condomiumn option. These are the decision options. - 2, 3 and 4 are circles which depict chance event nodes. From each node branch we have a circle that represents a chance event. Our chance events are that the demand may be strong or the demand may be weak. *How we read the decision tree is similar to the table: ex: for a small condo complex, decision d1, in case we have a strong demand happening then our payoff is 8 million dollars AND for the weak demand in the case of a small condo complex then payoff is 7 million
Decision Analysis with Probabilities (Slide 2 of 8) Tree Diagram for PDC Condo example
Figure 15.2: PDC Decision Tree with State-of-Nature Branch Probabilities
Decision Analysis with Probabilities (Slide 3 of 8)
Figure 15.3: Applying the Expected Value Approach Using a Decision Tree for the PDC Condominium Project - We will calculate the possible value of each decision alternative using =SUMPRODUCT on Excel. NOT manually
Decision Analysis with Probabilities (Slide 6 of 8)
Figure 15.4: Risk Profile for the Large-Complex Decision Alternative for the PDC Condominium Project - Let us demonstrate risk analysis and the construction of a risk profile by returning to the PDC condominium construction project. Using the expected value approach, we identified the large condominium complex as the best decision alternative. The expected value of $14.2 million for is based on a 0.8 probability of obtaining a $20 million profit and a 0.2 probability of obtaining a $9 million loss. The 0.8 probability for the $20 million payoff and the 0.2 probability for the −$9 million payoff provide the risk profile for the large-complex decision alternative. This risk profile is shown graphically in Figure 15.4. NOTE: Sometimes a review of the risk profile associated with an optimal decision alternative may cause the decision maker to choose another decision alternative even though the expected value of the other decision alternative is not as good. ^example: the risk profile for the medium-complex decision alternative shows a 0.8 probability for a $14 million payoff and a 0.2 probability for a $5 million payoff. Bc no probability of a loss is associated with decision alternative , the medium-complex decision alternative would be judged less risky than the large-complex decision alternative. As a result, a decision maker might prefer the less risky medium-complex decision alternative even though it has an expected value of $2 million less than the large-complex decision alternative.
Decision Analysis with Probabilities (Slide 4 of 8)
However, often there are little, or no, historical data to guide the estimates of these probabilities. In these cases, we may have to rely on subjective estimates to determine the probabilities for the states of nature. When relying on subjective estimates, we often want to get more than one estimate because many studies have shown that even knowledgeable experts are often overly optimistic in their estimates. It is also particularly important when dealing with subjective probability estimates to perform risk analysis and sensitivity analysis, as we will explain.
In Section 15.4 the branch probabilities for the PDC decision tree chance nodes were provided in the problem description. No computations were required to determine these probabilities.
In this section, we show how Bayes' theorem can be used to compute branch probabilities for decision trees. The branch probabilities are the posterior probabilities for demand that have been updated based on the sample information of whether the market research report is favorable or unfavorable.
Problem Formulation (Slide 6 of 10)
Notation: - the different payoffs are denoted as Vij, the two indexes, i shows the different decision alternative and J shows the different state of nature - as we move through the payoff table, those different fees represent the payoffs
Computing Branch Probabilities with Bayes' Theorem (Slide 4 of 4)
Note that the preceding probability assessments provide a reasonable degree of confidence in the market research study. If the true state of nature is , the probability of a favorable market research report is 0.90, and the probability of an unfavorable market research report is 0.10. If the true state of nature is S2 the probability of a favorable market research report is 0.25, and the probability of an unfavorable market research report is 0.75. One reason for a 0.25 probability of a potentially misleading favorable market research report for state of nature, S2 is that when some potential buyers first hear about the new condominium project, their enthusiasm may lead them to overstate their real interest in it. A potential buyer's initial favorable response can change quickly to a "no-thank-you" when later faced with the reality of signing a purchase contract and making a down payment.
Extra effort and care should be taken to make sure the input value is as accurate as possible.
On the other hand, if a modest-to-large change in the value of one of the inputs does not cause a change in the recommended decision alternative, the solution to the decision analysis problem is not sensitive to that particular input. No extra time or effort would be needed to refine the estimated input value.
PDC Ex: There are 3 possible decisions that can be made by PDC condos
PDC can choose to build a : (decision alternative d1, d2, and d3) 1. Build small development (d1) ^PDC finds that the payoff from building a small development when there is a strong demand for condos is 8 million dollars and when a weak demand for condos is 7 million -V11 = 8 and V12 = 7 -from 1st payoff table 2. Build Medium development (d2) 3. Build Large development (d3) - Based on these 3 possibilities we can calculate the payoff given a decision alternative and the state of nature that occurs
Assume company PDC believes:
Probability of strong demand for condos is 0.8 ^assume s1 = strong demand for condos Probability of weak demand for condos is 0.2 ^assume s1 = weak demand for condos *we have defined both a state or nature for strong/weak demands of condos Now: Define P(s1) meaning the probability of strong demand P(s1) = 0.8 P(s2) = 0.2 Define P(s2) meaning the probability of weak demand
Problem Formulation (Slide 10 of 10)
Problem Formulation (Slide 10 of 10)
Sensitivity Analysis example: Suppose that in the PDC problem the probability for a strong demand is revised to 0.2 and the probability for a weak demand is revised to 0.8. Would the recommended decision alternative change? Using P(s1) = 0.2, P(s2) = 0.8, and equation (15.2), the revised expected values for the three decision alternatives are as follows: slide
With these probability assessments, the recommended decision alternative is to construct a small condominium complex , with an expected value of $7.2 million. The probability of strong demand is only 0.2, so constructing the large condominium complex is the least preferred alternative, with an expected value of −$3.2 million (a loss) Additional: Thus, when the probability of strong demand is large, PDC should build the large complex; when the probability of strong demand is small, PDC should build the small complex. Obviously, we could continue to modify the probabilities of the states of nature and learn even more about how changes in the probabilities affect the recommended decision alternative. Sensitivity analysis calculations can also be made for the values of the payoffs. We can easily change the payoff values and resolve the problem to see if the best decision changes.
Utility Theory:
allows us to adjust for different risk preferences and we can use the utility instead of the dollar amount as a measure of the payoffs
Sensitivity analysis can be used to determine how
changes in the probabilities for the states of nature or changes in the payoffs affect the recommended decision alternative.
The utility values allow us to consider
conditions such as Ev(d1)
The decision analysis situations presented so far in ch.15 have expressed outcomes (payoffs) in terms of
monetary value. With probability information available about the outcomes of the chance events, we defined the optimal decision alternative as the one that provides the best expected value However, in some situations the decision alternative with the best expected value may not be the preferred alternative. A decision maker may also wish to consider intangible factors such as risk, image, or other nonmonetary criteria in order to evaluate the decision alternatives. When monetary value does not necessarily lead to the most preferred decision, expressing the value (or worth) of a consequence in terms of its utility will permit the use of expected utility to identify the most desirable decision alternative. The discussion of utility and its application in decision analysis is presented in this section.
To find the expected value (EV) of a decision alternative we simply
multiply the probability of each state of nature occurring by the payoff associated with making a decision when that state of nature occurs and then adding these values
The drawback to the sensitivity analysis approach described in this section is the
numerous calculations required to evaluate the effect of several possible changes in the state-of-nature probabilities and/or payoff values. Many computer packages exist that can assist with the creation of decision trees and with the calculations required.
States of nature will always be things
outside of our control
In decision analysis, regret is the difference between the
payoff associated with a particular decision alternative and the payoff associated with the decision that would yield the most desirable payoff for a given state of nature. Thus, regret represents how much potential payoff one would forgo by selecting a particular decision alternative, given that a specific state of nature will occur. This is why regret is often referred to as opportunity loss.
P(s1 | F) and P(s2 | F) are referred to as
posterior probabilities because they are conditional probabilities based on the outcome of the sample information.
The minimax regret approach is neither
purely optimistic nor purely conservative
Methods such as the EVA for choosing the best decision alternative typically assume that the decision maker is what's known as
risk neutral
