Abstract
Decision-making can involve a single decision or multistage decisions. When the decision-maker has to make at least two decisions separated in time, such as a situation where a company must first decide if it makes sense to hire an expert to make an actual decision about an investment. As the decision-maker goes on making decisions, the uncertainty diminishes. The probabilities are determined in a reverse manner when compared to regular decision-tree used for decision-making. The resulting process is Baye’s rule or theorem. The company wants to find out if hiring an expert so that he or she can predict the best option that the company can take regarding the investment decision, is a worthwhile option or not. Since this problem is exactly the type of the problem that requires the application of Baye’s theorem, the company collects the data that the application needs and populate the decision-tree. Based on the resulting Expected Monetary Value (EMV) or the Payoff, we make a recommendation to the company about hiring the expert.
Keywords: Multistage decision-making, Baye’s theorem
Introduction
In the case of decision-making in uncertain situations, using decision trees is an important tool for helping us to arrive at a decision. In the case of situations where the decision-maker must make at least two decisions separated in time, then there alternating sets of decision nodes and chance nodes. In such as case, as the decision-maker makes decisions, the uncertainty at each node gets resolved. In such a case, the decision maker must use the Baye’s rule to obtain probabilities. Since the following the problem has multistage decision-making, it illustrates the use of Baye’s rule.
Background
Our company had to decide on investing in one of the three opportunities, after studying the market for 10 years. The first opportunity is real estate development, which is a highly risky venture with either a very high payoff if successful. The second option is a retail franchise for Just Hats, a boutique-type store selling fashion hats for men and women, which is slightly less risky than the previous option but also has a lower payoff if successful. The third option is a certainty, which is investing in High-Yield municipal bonds. Table 1 is the Payoff table.
Multistage Decision-making
The Baye’s theorem states the multistage decision-making formula as below:
Equation 1: Baye's Equation
Source:
Where P(Ai|B) = Posterior probabilities
and P(Ai) = the prior probabilities
General decision tree uses the conditional probability of occurrence when there is an occurrence of an earlier event. In the case of a situation using Baye’s theorem, we revise the probability of an earlier condition based on the occurrence of a later event. This enables a systematic approach to an optimal strategy.
The Baye’s rule of two outcomes, which results from the Baye’s theorem states thus:
Equation 2: Baye's rule for two outcomes
Source:
Where we are interested in calculating the probability of A given B. To do that, we assess the conditional probabilities in the opposite order, that is B given A or P(B|A) using the Baye’s rule in the decision tree.
The joint probabilities of the three investment choices when considering the expert's track record are in Table 3.
Considering the Real Estate option, we can make the following calculation.
Row 1 and 2 give the expert’s track record.
Row 3 is obtained by multiplying the probability of the favorable condition based on our 10-year research and the expert’s chance of being correct when the market is favorable and unfavorable. Adding these two provides the total probability.
Similarly, we obtain the row 4 values for unfavorable market conditions and favorable and unfavorable predictions by the expert in this condition as well as the total probability
We obtain row 6 and 7 values by dividing the probabilities with the corresponding total probabilities.
We repeat the above process for Just Hats also.
The Decision Tree and Recommendation
The decision tree (Figure 1), gives the following values.
In a scenario where A Favorable, B Favorable (A+, B+), the expert would suggest investing in real estate with a payoff of $6.125 million.
In a scenario where A Unfavorable, B Unfavorable (A-, B-), the expert would suggest investing in Just Hats with a payoff of $3.1 million.
In a scenario where A Favorable, B Unfavorable (A+, B-), the expert would suggest investing in real estate with a payoff of $6.125 million.
In a scenario where A Unfavorable, B Favorable (A-, B+), the expert would suggest investing in Just Hats with a payoff of $4.3 million.
The Expected Monetary Value (EMV) based on decision tree analysis is $4.75 million. However, moving towards the right in the chance node for using the consult expert branch and using the Baye’s rule, the pay-off becomes $5.275 million and the decision tree support hiring the expert as the difference the two payoffs is $575,000. Therefore, we can recommend to the company that unless the cost of hiring the expert is less than $575,000, it is not advisable to hire the expert.
Figure 1: Decision tree
References
Albright, S. C., Winston, W. L., & Broadie, M. (2015). Business analytics: data analysis and decision making (5th ed.). Stamford, CT: Cengage Learning.
Azad, K. (2017, January 24). An intuitive (and short) explanation of Bayes’ Theorem. Retrieved from betterexplained.com: https://betterexplained.com/articles/an-intuitive-and-short-explanation-of-bayes-theorem/
Mian, M. A. (2002). Project economics and decision analysis, volume II: Probabilistic models. Tulsa, OK: PennWell Corporation.
