Public goods are often either over-consumed in the absence of regulatory mechanisms, or remain completely unused, as in the case of 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  to efficiently allocate capacity constrained public goods. To design these mechanisms, we leverage the theory of Fisher markets, wherein each agent  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. The presence of additional physical constraints that arise in public goods allocation problems motivates the need for modifications to the Fisher market framework to account for these more general set of constraints. In our  mechanism, we specifically modify the Fisher market social optimization problem through a perturbation of the budgets of agents. The  perturbations are set based on the dual variables of the additional physical constraints such that the KKT conditions of the newly defined social and individual optimization problems are equivalent. While the perturbed social problem may not resemble the original societal objective, we show that the optimal allocation corresponding to the perturbed problem is closely related to that of the unperturbed problem . Finally, to compute the perturbation constants we present a fixed point procedure, establish its convergence and present numerical experiments to corroborate our convergence results . Thus, our mechanism, both theoretically and computationally, overcomes a fundamental limitation of Fisher markets, which only consider capacity and budget constraints.
Public goods are often either over-consumed in the absence of regulatory mechanisms, or remain completely unused, as in the case of 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. The presence of additional physical constraints that arise in public goods allocation problems motivates the need for modifications to the Fisher market framework to account for these more general set of constraints. In our market mechanism, we specifically modify the Fisher market social optimization problem through a perturbation of the budgets of agents. The budget perturbations are set based on the dual variables of the additional physical constraints such that the KKT conditions of the newly defined social and individual optimization problems are equivalent. This establishes that our mechanism sets market clearing prices such that each agent obtains their maximum possible utilities subject to their constraints . Finally, to compute the perturbation constants we present a fixed point scheme, establish the existence of a unique fixed point and present numerical experiments to illustrate convergence guarantees . Thus, our mechanism, both theoretically and computationally, overcomes a fundamental limitation of Fisher markets, which only consider capacity and budget constraints.
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designed
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in the economy
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market
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budget
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While the perturbed social problem may not resemble the original societal objective, we show that the optimal allocation corresponding to the perturbed problem is closely related to that of the unperturbed problem
This establishes that our mechanism sets market clearing prices such that each agent obtains their maximum possible utilities subject to their constraints
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procedure, establish its convergence
scheme, establish the existence of a unique fixed point
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corroborate our convergence results
illustrate convergence guarantees