Allocation with Weak Priorities and General Constraints

Abstract

With COVID 19 prevalent in the USA and the world, efficient social distance seating became an option for sports venues. The social distancing constraint requires six feet between individuals when the game has live audiences. Depending on the seats' dimensions, this would translate to a certain number of empty rows and empty seats in a row between the individuals. As a result, it is not possible to seat all ticket holders with safe social distancing. Hence, it necessitates reassigning spectators to games. An important feature of this problem is that season tickets are grouped by family, and only a safe distance between two different families needs to be maintained. Members of the same family can sit next to each other. Therefore, a large family needs fewer empty seats per person to maintain social distancing. A football season has about six home games. If priority is given to larger families for all the games, then many people can watch the live games, but the outcome will be highly unfair. Striking a good balance between efficiency and fairness is a nontrivial task. We model this as a resource allocation problem. Its novelty is the combination of three features: complex resource constraints, weak priority ranking over agents, and ordinal preferences over bundles of resources. We develop a mechanism based on a new concept called Competitive Stable Equilibrium (CSE). It has several attractive properties, unifies two different frameworks of one-sided and two-sided markets, and extends existing methods to richer environments. CSE is an extension of competitive equilibrium with endowed budgets that accommodates weak priorities. In particular, an agent only needs to pay for a resource if he belongs to the last tier among the agents currently consuming the resource. Furthermore, the price is positive only if the resource constraint binds: a market clearing condition as in a competitive equilibrium. Thus, if agents are endowed with equal budgets, then a CSE is a stable and envy-free outcome, which is both fair and Pareto optimal when the resource constraints are capacity constraints. Moreover, a CSE when agents are given a different budget corresponds to a tie-breaking rule among agents of the same tier. Tie-breaking rules can improve efficiency, especially when resource constraints are complex. Our framework also allows for an alternative and more flexible tie-breaking rule by giving agents different budgets. Furthermore, when agents consume a bundle of goods, it allows agents to distribute their tie-breaking budget over different resources. We illustrate this in the application of assigning seats in sports venues and compare our method with other tie-breaking alternatives. We empirically apply our mechanism to reassign season tickets to families in the presence of social distancing. Our simulation results show that our method outperforms the existing ones in both efficiency and fairness measures. However, CSE need not exist because of two different reasons: the income effect of a fixed budget and the complementarities of bundled preferences. To overcome them, we allow for an e-approximate solution of the budget as in Budish (Journal of Political Economy 2011), and a violation of resource constraints by accommodating a few extra agents as in Nguyen (Journal of Economic Theory 2016). The violation of the budget can be arbitrarily small, and the violation of resource constraints depends on the size of the largest bundle that an agent can consume. Our solution therefore, inherits the efficiency and fairness properties of existing methods for both one-sided and two-sided markets. Our mechanism is based on a nontrivial extension of Scarf's lemma to nonlinear constraints, which might be of independent interest. Because of the generality of our framework, it applies to settings beyond sport and entertainment events. For example, even in the context of social distancing, daycare facilities face a similar problem of reallocating slots to children and their siblings on different days of the week. Homeless shelters and refugee camps also need to reallocate families to limit the spread of the virus.