On the Use of Cauchy Prior Distributions for Bayesian Logistic Regression

In logistic regression, separation occurs when a linear combination of the predictors can perfectly classify part or all of the observations in the sample, and as a result, finite maximum likelihood estimates of the regression coefficients do not exist. Gelman et al. (2008) recommended independent Cauchy distributions as default priors for the regression coefficients in logistic regression, even in the case of separation, and reported posterior modes in their analyses. As the mean does not exist for the Cauchy prior, a natural question is whether the posterior means of the regression coefficients exist under separation. We prove theorems that provide necessary and sufficient conditions for the existence of posterior means under independent Cauchy priors for the logit link and a general family of link functions, including the probit link. We also study the existence of posterior means under multivariate Cauchy priors. For full Bayesian inference, we develop a Gibbs sampler based on Polya-Gamma data augmentation to sample from the posterior distribution under independent Student-t priors including Cauchy priors, and provide a companion R package in the supplement. We demonstrate empirically that even when the posterior means of the regression coefficients exist under separation, the magnitude of the posterior samples for Cauchy priors may be unusually large, and the corresponding Gibbs sampler shows extremely slow mixing. While alternative algorithms such as the No-U-Turn Sampler in Stan can greatly improve mixing, in order to resolve the issue of extremely heavy tailed posteriors for Cauchy priors under separation, one would need to consider lighter tailed priors such as normal priors or Student-t priors with degrees of freedom larger than one

Mobility and Conflict

We study the role of inter-group differences in the emergence of conflict. In our setting, two groups compete for the right to allocate societys resources, and we allow for costly intergroup mobility. The winning group offers an allocation, that the opposition can either accept, or reject and wage conflict. Expropriating a large share of resources increases political strength by attracting opposition members, but such economic exclusion implies lower per capita shares and higher risk of conflict. In equilibrium, allocations are non-monotonic in the cost of mobility. Moreover, limited commitment with respect to mobility gives rise to inefficient conflict in equilibrium.conflict, inter-group mobility, political competition, resource allocation

Reputation Building Under Uncertain Monitoring

We study a canonical model of reputation between a long-run player and a sequence of short-run opponents, in which the long-run player is privately informed about an uncertain state that determines the monitoring structure in the reputation game. The long-run player plays a stage-game repeatedly against a sequence of short-run opponents. We present necessary and sufficient conditions (on the monitoring structure and the type space) to obtain reputation building in this setting. Specifically, in contrast to the previous literature, with only stationary commitment types, reputation building is generally not possible and highly sensitive to the inclusion of other commitment types. However, with the inclusion of appropriate dynamic commitment types, reputation building can again be sustained while maintaining robustness to the inclusion of other arbitrary types

Reputation with Opportunities for Coasting

Reputation concerns can discipline agents to take costly effort and generate good outcomes. But what if outcomes are not always observed? We consider a model of reputation with shifting observability, and ask how this affects agents’ incentives. We identify a novel and intuitive mechanism by which infrequent observation or inattention can actually strengthen reputation incentives and encourage effort. If an agent anticipates that outcomes may not be observed in the future, the benefits from effort today are enhanced due to a “coasting” effect. By investing effort when outcomes are more likely observed, the agent can improve her reputation, and when the audience is inattentive in the future, she can coast on this reputation without additional effort. We show that future opportunities to rest on one’s laurels can lead to greater overall effort and higher efficiency than constant observation. This has implications for the design of review systems or performance feedback systems in organizations. We provide a characterization of the optimal observability structure to maximize efficient effort in our setting

Prepare-and-measure based QKD protocol under free-space losses

In this study, we have theoretically presented a prepare-and-measure-based SARG04 protocol over free space. It has shown that the highest secret key rate is possible even under free-space losses with a maximum tolerance of noise.Comment: 4 pages, 1 figur

Reputation for a Servant of Two Masters

Classic models of reputation consider an agent taking costly actions to affect a single, homogeneous audience’s beliefs about his ability, preferences or other characteristic. However, in many economic settings, agents must maintain a reputation with multiple parties with diverse interests. In this paper we study reputation incentives for an agent who faces two audiences with opposed preferences. We ask if the existence of multiple audiences per se changes reputation incentives. Further, should the agent deal with the different audiences commonly or separately? Our analysis yields some new qualitative insights. Specifically, the presences of heterogeneous audiences is more likely to lead the agent towards “pooling” equilibria in which he takes an intermediate compromise action. Instead, dealing with only one audience leads the agent to cater towards that audience’s preferences, giving rise to a “separating” outcome or pooling on some extreme action. We analyze the welfare implications, and show that the agent most prefers that both audiences commonly observe all the actions that he takes. In our setting, reputation acts as an informal contract that enforces desirable behavior through future continuation payoffs. Our analysis highlights that the presence of multiple heterogeneous audiences can, naturally, lead these rewards to be non-monotonic in an agent’s reputation. We show different ways that this non-monotonicity arises. In an infinite horizon setting, it can emerge through endogenous interactions between the audiences, through equilibrium expectations of the agent’s choice of action. It can also arise, perhaps more trivially, through direct payoff interactions