We conclude our discussion of the optimal decision rule by highlighting some practical ways to organize the decision-making process. We have seen that different circumstances, or different priors about the true value, imply that - given the prediction - we choose a more stringent or a more lenient prediction. We have also anticipated that the best way to think about these decision processes is not as decisions made by individuals. They are made by teams and the final decision, in particular which threshold to set, is the outcome of a mediation within this discussion. Consider the following situation. Three managers have to make a decision, and the probability that individually they approve a project is p. They can set different consensus rules. In particular, they can decide to go ahead with the project if all three say yes, at least two say yes, at least one says yes. If so, the probabilities of launching the project are, respectively, p to the power three; three times p to the power two times one minus p, plus p to the power three; or one minus one minus p to the power three. In the first case, we need all three decision makers to approve. In the second case, the project is approved if any two decision makers approve out of the three combinations of any two of them. And the third probability is the complement to one of the cases in which no one approves. The first probability is the smallest probability of approval, while the last probability is the highest. This suggests that the higher the consensus rule, the less likely a project is approved, which implies fewer type I errors, but more type II errors. The third probability is the highest. It implies more type I errors, but fewer type II errors. The consensus rule is a way to maneuver a threshold. Suppose that a company or a team of decision makers wants to be more conservative. For example, their priors are such that they believe that the project is unlikely to be successful, or they have other projects that can be cannibalized by the one they are considering. In this case, they would launch the project only if very successful. By setting a high consensus rule, for example all three have to agree, the company sets de facto a high threshold V*. Vice versa, if they want to be lenient and lower the threshold, they could set a consensus rule of one. For example, in periods of high opportunities you want a low consensus rule. Two extreme forms of the organization of decision making are hierarchies or polyarchies. In a hierarchy, if any decision maker approves the project, the project moves to his boss who has to decide and so on. If anyone says no at any stage, the project stops. Thus, in a hierarchy, a project is approved if everyone says yes. In a polyarchy, the project is approved if at least one person says yes. Therefore, a hierarchy corresponds to a high threshold, and makes relatively more type II errors than type I errors. Vice versa, for the polyarchy. Also, if the probability that an individual approves the project is p, in a hierarchy with n different layers, the probability that the project is approved is p to the power n. In a polyarchy with n decision makers, it is one minus one minus p to the power n. As n increases, the hierarchy becomes less likely to approve a project, while a polyarchy becomes more likely to approve a project. Companies that organize the decision process as hierarchies can then increase the layers of approval if they want to be more stringent, and vice versa, if they want to be more lenient. Companies that organize the decision process as polyarchies can, instead, increase the number of decision makers if they want to be less stringent. These are practical ways to maneuver the threshold. Finally, consider committees. Committees are teams of n decision makers. The consensus rule is that a project is approved if k smaller or equal than n decision makers approve the project. Committees are an intermediate form of organization between hierarchies and polyarchies. In committees, a higher threshold corresponds to a higher consensus rule k. A higher k reduces the number of projects approved and increases type II relatively to type I errors. Vice versa, if we reduce k. Thus, companies can maneuver the consensus rule k to obtain more stringent or more lenient thresholds and reduce relatively more type I versus type II errors. In the extreme case of k equal to one, we have the polyarchy, while k equal to n is equivalent to the hierarchy. Interestingly, all this also suggests that in business committees, the majority rule is not necessarily optimal. You may want minority ruling if you believe that there are opportunities to be exploited. Or, you want very strong majorities if you believe that opportunities are limited, and similarly, if you feel that decision-makers are too overconfident or optimists, or too underconfident or pessimists.
