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Public goods are often either over-consumed in the absence of regulatory mechanisms, or remain completely unused, as in the Covid-19 pandemic, where social distance constraints are enforced to limit the number of people who can share public spaces. In this work, we plug this gap through market based mechanisms designed to efficiently allocate capacity constrained public goods. To design these mechanisms, we leverage the theory of Fisher markets, wherein each agent in the economy is endowed with an artificial currency budget that they can spend to avail public goods. While Fisher markets provide a strong methodological backbone to model resource allocation problems, their applicability is limited to settings involving two types of constraints - budgets of individual buyers and capacities of goods. Thus, we introduce a modified Fisher market, where each individual may have additional physical constraints, characterize its solution properties and establish the existence of a market equilibrium. Furthermore, to account for additional constraints we introduce a social convex optimization problem where we perturb the budgets of agents such that the KKT conditions of the perturbed social problem establishes equilibrium prices. Finally, to compute the budget perturbations we present a fixed point scheme and illustrate convergence guarantees through numerical experiments. Thus, our mechanism, both theoretically and computationally, overcomes a fundamental limitation of classical Fisher markets, which only consider capacity and budget constraints. 

Abstract High‐speed video and electric field records of two positive cloud‐to‐ground (+CG) flashes were used to examine the effect of M‐components on needle activity after the return stroke onset. We observed enhancements of needle activity that were associated with the occurrence of M‐components identified by channel luminosity enhancements both at cloud altitudes and near the ground. Full‐fledged M‐components enhance needle activity via injection of negative charge into the bottom of grounded channel and reversing the direction of the radial electric field at the channel core, similar to +CG return strokes. Attempted M‐components, identified by channel luminosity enhancements at the cloud but not near the ground, did not enhance needle activity because of the absence of significant reflection from the ground, which causes electric field reversal at the core.