Error rate control for classification rules in multiclass mixture models

In the context of finite mixture models one considers the problem of classifying as many observations as possible in the classes of interest while controlling the classification error rate in these same classes. Similar to what is done in the framework of statistical test theory, different type I and type II-like classification error rates can be defined, along with their associated optimal rules, where optimality is defined as minimizing type II error rate while controlling type I error rate at some nominal level. It is first shown that finding an optimal classification rule boils down to searching an optimal region in the observation space where to apply the classical Maximum A Posteriori (MAP) rule. Depending on the misclassification rate to be controlled, the shape of the optimal region is provided, along with a heuristic to compute the optimal classification rule in practice. In particular, a multiclass FDR-like optimal rule is defined and compared to the thresholded MAP rules that is used in most applications. It is shown on both simulated and real datasets that the FDR-like optimal rule may be significantly less conservative than the thresholded MAP rule.

Public Input Provision, Tax Base Mobility, and External Ownership

This article investigates the distortionary impacts of tax base mobility and external ownership on public input provision. Regional governments compete for mobile tax bases (e.g., business capital). The impact of regional public policy partially accrues to non-residents because immobile factors (e.g., business land) are subject to external ownership. This article derives an optimal rule for regional public input provision that illustrates how these two distortionary impacts depend on the nature of production functions. The impact of external ownership is particularly complex. To explore this impact in detail, the case of production functions with constant elasticity of substitution is examined. Public inputs with different productivity impacts yield fairly different implications of external ownership for inefficient public input provision.

Representative Committees of Peers

A population of voters must elect representatives among themselves to decide on a sequence of possibly unforeseen binary issues. Voters care only about the final decision, not the elected representatives. The disutility of a voter is proportional to the fraction of issues, where his preferences disagree with the decision. While an issue-by-issue vote by all voters would maximize social welfare, we are interested in how well the preferences of the population can be approximated by a small committee. We show that a k-sortition (a random committee of k voters with the majority vote within the committee) leads to an outcome within the factor 1+O(1/√ k) of the optimal social cost for any number of voters n, any number of issues m, and any preference profile. For a small number of issues m, the social cost can be made even closer to optimal by delegation procedures that weigh committee members according to their number of followers. However, for large m, we demonstrate that the k-sortition is the worst-case optimal rule within a broad family of committee-based rules that take into account metric information about the preference profile of the whole population.

On the Impact of Germs and Demons

Historical and experimental analyses suggest contradictory deviations from efficient reasoning. In some cases, people tend to oversimplify and ignore important factors like germs, while in others they seem to overcomplicate and consider non-existent factors like imaginary demons. The current study shows how this apparent contradiction can be the product of a tendency to rely on small samples of past experience. Simulations demonstrate that reliance on small samples triggers apparent over-simplicity when simple choice rules are counterproductive but better for most sets of samples; the opposite over-complexity bias emerges when the optimal rule is simple but fails in most samples. The descriptive value of this hypothesis is demonstrated in two preregistered studies with 300 Mechanical-Turk participants. Study 1 shows the qualitative pattern predicted by the reliance on the small-samples hypothesis. Study 2 compares alternative formulations of the sampling process and clarifies the importance of a distinction between choice and sampling processes.

Rules without Commitment: Reputation and Incentives

This article studies the optimal design of rules in a dynamic model when there is a time inconsistency problem and uncertainty about whether the policy maker can commit to follow the rule ex post. The policy maker can either be a commitment type, which can always commit to follow rules, or an optimizing type, which sequentially decides whether to follow rules or not. This type is unobservable to private agents, who learn about it through the actions of the policy maker. Higher beliefs that the policy maker is the commitment type (i.e. the policy maker’s reputation) help promote good behaviour by private agents. We show that in a large class of economies, preserving uncertainty about the policy maker’s type is preferable from an ex ante perspective. If the initial reputation is not too high, the optimal rule is the strictest one that is incentive compatible for the optimizing type. We show that reputational considerations imply that the optimal rule is more lenient than the one that would arise in a static environment. Moreover, opaque rules are preferable to transparent ones if reputation is high enough.

Secretary Problem with Possible Errors in Observation

The classical secretary problem models a situation in which the decision maker can select or reject in the sequential observation objects numbered by the relative ranks. In theoretical studies, it is known that the strategy is to reject the first 37% of objects and select the next relative best one. However, an empirical result for the problem is that people do not apply the optimal rule. In this article, we propose modeling doubts of decision maker by considering a modification of the secretary problem. We assume that the decision maker can not observe the relative ranks in a proper way. We calculate the optimal strategy in such a problem and the value of the problem. In special cases, we also combine this problem with the no-information best choice problem and a no-information second-best choice problem.