  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.
