It is often argued that collusion is not possible in wholesale markets where suppliers and buyers privately bargain over discounts off list prices and sales quotas are unfeasible. In many wholesale markets, however, there are buyers of varying sizes and bargaining powers who compete for final consumers in the retail market. We study the role of public announcements of list prices in this wholesale–retail setting, whether suppliers collude or compete. Two effects are at work. When suppliers collude, public announcements of list prices further extend the possibility of collusion from small to large buyers (the multibuyer contact effect). When suppliers compete, these public announcements provide them with commitment to negotiate better terms with large buyers (the commitment effect).

Mindfulness-based Stress Reduction (MBSR) has shown good efficacy for improving wellbeing in employees experiencing occupational stress. However, comparisons with other interventions, longer-term follow-up, and data from varying sociocultural contexts are lacking. This three-arm, parallel randomised controlled trial (RCT) examined the effects of MBSR on psychological distress in non-physician health workers in direct contact with patients. One hundred and five participants were randomly allocated to either (1) MBSR (N = 35), (2) Stress Management Course (SMC; N = 34), or (3) wait-list (N = 36). Participants and those assessing outcomes were blinded to group assignment. Participants completed questionnaires pre- and post-intervention and four months after the intervention. Psychological distress was measured using the General Health Questionnaire (GHQ-12) and Outcome Questionnaire (OQ-45). Secondary outcomes included perceived stress, job satisfaction, mindfulness skills, and changes in salivary cortisol. Seventy-seven participants completed measures post-intervention and 52 at 4-month follow-up. MBSR showed a post-intervention effect in reducing GHQ-12 (β = −0.80 [SE = 1.58] p < 0.01) and OQ-45 (β = −0.72, [SE = 5.87] p < 0.05) psychological distress compared to SMC, and in reducing GHQ-12 (β = −1.30 [SE = 1.38] p < 0.001) and OQ-45 (β = −0.71, [SE = 5.58] p < 0.01) psychological distress compared to wait-list. In our secondary outcome, only MBSR was associated with a decrease in the cortisol awakening response by 23% (p < 0.05). At follow-up, only effects of MBSR on the psychological distress “social role” subscale (β = −0.76 [SE = 1.31] p < 0.05) remained significant compared to SMC. In conclusion, MBSR appears useful in reducing short-term psychological distress in healthcare workers, but these effects were not maintained at follow-up.

Inefficient contracts between firms and suppliers—contracts that do not maximize gains from trade—are ubiquitous. Research on bilateral oligopolies with inefficient contracts commonly employs the Nash-in-Nash bargaining equilibrium concept, but has been hampered by a lack of theoretical foundations for this equilibrium concept. We provide conditions for the existence and uniqueness of Nash-in-Nash equilibria in vertical relations games. These conditions imply that bargaining models are point identified, and can be verified in empirical work. We also provide an algorithm for fast equilibrium computation. These results advance our theoretical understanding of bilateral oligopolies and extend the toolset available to empirical researchers.

Random-restart heuristics are widely used in nonconvex optimization and equilibrium computation: practitioners run a local algorithm from many initial conditions and interpret repeated convergence to the same output as evidence that the result is robust, dominant, or even unique. Despite its widespread use, this reasoning is usually informal. We provide a probabilistic framework for interpreting restart evidence. We give broad, easy-to-verify sufficient conditions under which repeated runs of a solver can be treated as independent draws from a categorical distribution induced by random initial conditions. Within this framework, we develop Bayesian inference from repeated identical outputs. We derive posterior concentration rates for basin size and uniqueness. These rates demonstrate that uniqueness is inherently harder than learning basin size: posterior concentration for uniqueness is polynomial, whereas basin size concentrates exponentially fast. We also provide a verification protocol for checking whether a given problem fits our framework. We demonstrate the protocol on a widely used equilibrium solver for mixed-logit demand with multi-product firms, and complement the verification exercise with posterior tables that apply to any restart experiment satisfying the protocol. We conclude by delineating limits of restart-based inference, including failures induced by solver--problem mismatch and limited visibility of alternative outcomes.

I prove the existence and uniqueness of a Bertrand-Nash equilibrium in a mixed logit demand model with linear utility and multiproduct firms. The proof relies on a uniform convergence result showing that mixed logit profit functions, and their derivatives, converge to their logit counterparts as the variances of random coefficients shrink, combined with an existing uniqueness result for logit demand. This approach requires imposing variance constraints, which may bind in some applications. I propose a diagnostic for checking whether the constraints bind for a given model and parameter estimates.