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

Uniform Folk Theorems in Repeated Anonymous Random Matching Games

We study infinitely repeated anonymous random matching games played by communities of players, who only observe the outcomes of their own matches. It is well known that cooperation can be sustained in equilibrium for the prisoner’s dilemma (PD) through grim trigger strategies. Little is known about games beyond the PD. We study a new equilibrium concept, strongly uniform equilibrium (SUE, which refines the notion of uniform equilibrium (UE) and has additional properties such as a strong version of (approximate) sequential rationality. We establish folk theorems for general games and arbitrary number of communities. Interestingly, the equilibrium strategies we construct are easy to play. We extend the results to a setting with imperfect private monitoring, for the case of two communities. We also show that it is possible for some players to get equilibrium payoffs that are outside the set of individually rational and feasible payoffs of the stage game. In particular, for the PD we derive a bound on the number of “free-riders” that can be sustained in society. A by-product of our analysis is an important result relating uniform equilibrium and strongly uniform equilibria: we show that, in general repeated games with finite players, actions, and signals, the set of UE and SUE payoffs coincide

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

Realistic theory of electromagnetically-induced transparency and slow light in a hot vapor of atoms undergoing collisions

We present a realistic theoretical treatment of a three-level $\Lambda$ system in a hot atomic vapor interacting with a coupling and a probe field of arbitrary strengths, leading to electromagnetically-induced transparency and slow light under the two-photon resonance condition. We take into account all the relevant decoherence processes including col5Blisions. Velocity-changing collisions (VCCs) are modeled in the strong collision limit effectively, which helps in achieving optical pumping by the coupling beam across the entire Doppler profile. The steady-state expressions for the atomic density-matrix elements are numerically evaluated to yield the experimentally measured response characteristics. The predictions, taking into account a dynamic rate of influx of atoms in the two lower levels of the $\Lambda$, are in excellent agreement with the reported experimental results for $^4$He*. The role played by the VCC parameter is seen to be distinct from that by the transit time or Raman coherence decay rate

Mobility and Conflict

We study the role of inter-group differences in the emergence of conflict. In our setting, society comprises two groups who compete in every period for political power, i.e. the right to allocate economic resources between the groups. Individuals can move from one group to another at a cost: this cost of mobility is the index of inter-group differences. Since mobility is costly, the group in power can keep a larger share for itself. The extent of such economic exclusion is limited by two constraints: excessive exclusion reduces the opposition’s opportunity cost of engaging in political conflict (conflict constraint) and, if a group keeps too much for itself, individuals switching from the other group will dilute the per capita share of resources (mobility constraint). In determining the optimal group size by attracting switchers, the incumbent faces a trade-off between low per capita surplus and high political strength. We characterize the resource allocations, group membership decisions and conflict decisions that arise in equilibrium. The two mechanisms of conflict and mobility act as constraints to expropriation, and the optimal sharing is dictated by which constraint binds. The extent of sharing turns out to be non-monotonic in the cost of mobility. We show that the limited commitment with respect to switching can lead to inefficient conflict in equilibrium. We also derive several testable predictions about when conflict will arise. Specifically, we show that conflict may arise when the cost of mobility is moderate, but may not necessarily emerge when the cost is high