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 pricessuch 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.
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.
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Thus, we introduce a modified Fisher market, where each individual may have
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that arise in public goods allocation problems motivates the need for modifications to the Fisher market framework
, characterize its solution properties and establish the existence of a market equilibrium. Furthermore,
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these more general set of constraints . In our market mechanism, we specifically modify the Fisher market social optimization problem through a perturbation of
additional constraints we introduce a social convex optimization problem where we perturb
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. The budget perturbations are set based on the dual variables of the additional physical constraints
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newly defined social and individual optimization problems are equivalent. This establishes that our mechanism sets market clearing pricessuch that each agent obtains their maximum possible utilities subject to their constraints
perturbed social problem establishes equilibrium prices
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perturbation constants
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, establish the existence of a unique fixed point and present numerical experiments to
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through numerical experiments
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