We're now ready to understand the Scientific Approach to Managerial Decisions Under Uncertainty. This approach helps to assess the value of a business idea. For example, Inkdome, discussed in a previous module considered the development of a search engine to find tattoo artists online. The starting point is to identify problems, scenarios, actions, values of actions in different scenarios. Scenarios are realizations of uncertain outcomes. Actions are implementations of solutions. These steps combine theory and tests with intuition and creativity. Definition of problems and scenarios, probability of scenarios, and the association of value to scenario-action pairs typically stem from logical propositions and tests; Creative solutions or test designs, or which data to use, depend instead to a greater extent on intuition and imagination. Scientific decision-makers first identify key problems. Theory helps to think logically on what are the key problems and why. There is no need to understand all problems. This is hard and confusing, leading to bad decisions. Solving few important problems often explains a lot of the value of business idea. Inkdome focused on choice of artists, online search, time of search, availability of information online. They ignored, for instance, means of payment, or the colors that customers like. Scenarios define testable hypothesis. Again, theory identifies the outcome of scenarios and the hypothesis. Inkdome defined two outcome per problem, leading to two to the power four or 16 scenarios. Actions are implementations of solutions such as Inkdome's search engine. We will see in a later session that there could be many actions and solutions. Here we focus for simplicity on one solution which defines two actions: Develop the search engine or not. Finally, theory associates values to scenario action pairs. Inkdome concluded that the search engine is not worth in all scenarios in which at least one hypothesis is false, it is only worth in the scenario in which they are all true. Suppose the decision-makers such as Inkdome set the value V̂ of an action equal to minus 1 or one under a bad or good scenario, the value of no action is 0. They also set prior probabilities of the scenarios using logical reasoning and experience. The next step is to collect data to test hypothesis. Inkdome tested whether users employ many versus one artist, search online, spent times to search, find all information they need online. Tests produce signals. In particular, if data support the prediction that all hypotheses are true, we get a positive signal because in this case, Inkdome's theory is that the value of the action is positive. If we label the signal V̂, then V̂ is equal to 1, vice-versa, if the data falsifies at least one hypothesis, the signal V̂ is equal to minus 1. Should we take the action? The answer depends on the update of the probability that V is equal to 1 after observing the signal. If, for example, the observed signal is V̂ equal to minus 1, the probability of V equal to 1 becomes probability of V equal to 1 given V̂ equal to minus 1, which is the posterior probability. We take the action if the expected value of V after observing the signal V̂ equal to minus 1 is positive. This expected value is equal to minus 1 times the probability of V equal to minus 1, given that the signal V̂ is minus 1 plus 1 times the probability of V̂ to 1, given that the signal is minus 1. The problem is that statistical tests do not produce the probability of V given V̂, but the probability of V̂ given V. However, using the Bayes theorem, we retrieve the probability of V given V̂ from the probability of V̂ given V as shown graphically. This expression also depends on prior probabilities which then play a critical role in the decision. We also retrieve the joint probability of V and V̂, which is equal to the formula shown graphically. Consider this table. The last row represents the prior probabilities of V equal to minus 1 and 1. The probabilities in the cells are the joint probabilities of V̂ and V in the corresponding rows and columns. They come from the statistical tests because they are the product of the probability of V̂ given V produced by the tests times the prior probabilities. The last column reports the marginal probabilities of observing the signal in the row. We compute the optimal threshold V*, such that the decision-makers takes the action, if V̂ is greater than V*. The optimal threshold is the minimum value of the signal, such that the expected value of V of the action condition the signal is positive. Suppose that the signal is V̂ equal to minus 1. The expected value of V, given V̂ equal to minus 1 is minus 0.5. If the signal is equal to 1, the expected value of V, given V̂ is 0.33. The optimal threshold is V* equal to 1 because if we set V* equal to minus 1, we take the action even if V̂ is equal to minus 1, which has a negative expected V. Consider now a higher prior of V equal to 1, leaving the conditional probabilities that generate the joint probabilities in the cells unaltered. For instance, given the signal V̂ they estimate, the decision-makers believe they face better opportunities, such as better technologies or higher demand. We obtain this new table where as shown by the last row, the prior of V equal to 1 is now 0.8. The optimal threshold is V* equal to minus 1, which you can show as an exercise. Intuitively, with higher priors you are more lenient. Of course, the optimal V* increases if you believe you face bad opportunities. By leaving the conditional probabilities unaltered, the statistical analysis or any other process that generates the signal, remains unaltered. For example, in terms of p values, which we discuss in later sessions, decision-makers estimate the same p values, but set a higher threshold for the p value for rejecting the null hypothesis. Since p values represent the probability of type 1 error, this is equivalent to a lower V*. Finally, note that we have highlighted two equivalent ways of making the decision. The first one sets prior probabilities, retrieves posterior probabilities from the tests, computes the expected value of V given V̂ from these probabilities acts if this expected value is higher than the expected value of the alternative action. In this case, you need to compute V* to make the decision. You need to set the prior probabilities which enable you to retrieve, together with the probabilities produced by the tests, the probabilities that you need to compute the expected value of V given V̂. You then check whether this expected value is greater than the expected value of the alternative action. The second tool simply observes the signal V̂ and sets a threshold V* higher or lower than 0 according to whether priors are positive or negative. You make a positive decision if V̂ is greater than V*. In this case, you do not need to set the prior probabilities. However, the two decision tools are equivalent. Raising V*, which makes it more difficult to take the action, is equivalent to setting lower priors of high values and vice versa. The second tool is more practical in that you retrieve V̂ from your statistical analysis or from any other analysis that provides you with a guess, estimate, or signal. You set it equal to the value of the alternative action, if you have no strong prior in one or the other direction. Otherwise, you raise or lower V* according to your priors. Managers can maneuver the threshold in many ways. Since these decisions are typically made by teams or committees, in the next session, we discussed how managers can set a consensus rule according to whether they want to set high or low priors.
