Competitive Equilibrium with Indivisible Goods and Generic Budgets

Abstract

Competitive equilibrium from equal incomes (CEEI) is a classic solution to the problem of fair and efficient allocation of goods (Foley 1967, Varian 1974). Every agent receives an equal budget of artificial currency with which to purchase goods, and prices match demand and supply. However, a CEEI is not guaranteed to exist when the goods are indivisible even in the simple two-agent, single-item market. Yet it is easy to see that, once the two budgets are slightly perturbed (made generic), a competitive equilibrium does exist. In this paper, we aim to extend this approach beyond the single-item case and study the existence of equilibria in markets with two agents and additive preferences over multiple items. We show that, for agents with equal budgets, making the budgets generic—by adding vanishingly small random perturbations—ensures the existence of equilibrium. We further consider agents with arbitrary nonequal budgets, representing nonequal entitlements for goods. We show that competitive equilibrium guarantees a new notion of fairness among nonequal agents and that it exists in cases of interest (such as when the agents have identical preferences) if budgets are perturbed. Our results open opportunities for future research on generic equilibrium existence and fair treatment of nonequals.

A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation

“…Recently, approaches based on maximin share fairness (a property weaker than proportionality) have been considered for computing fair allocation of indivisible goods to asymmetric agents [3,5,11] The acyclic consumption graph corresponding to the allocation in Table 1.…”

Learning Competitive Equilibria in Exchange Economies with Bandit Feedback

“…Since the seminal work of Varian [49], the fair division of multiple resource types has received significant attention in the game theory, economics, and computer systems literatures. One of the most common perspectives on this problem is as an exchange economy (or as a Fisher market, which is a special case of an exchange economy) [7,8,10,15,26,39,48,49]. In addition to this theoretical literature, fair-allocation mechanisms have been deployed in practical data-center resource-allocation tasks [55].…”

“…Under certain conditions, continually trading in this manner results in a competitive equilibrium, where the allocations have desirable Pareto-efficiency and fairness properties. EEs have attracted much research attention, historically since they are a tractable model to study human behavior and price determination in real-world economies, and more recently for designing multi-resource fair-division mechanisms [7,8,10,15,17,48].…”

Foundations of pseudomarkets: Walrasian equilibria for discrete resources

“…Our Second Welfare Theorem for environments without transfers is the first such result that allows for locally satiated agents. 9 Indeed, we think it is quite surprising that the insight of the Second Welfare Theorem holds true in the canonical no-transfer environment we study because the problems the received approaches to the Second Welfare Theorem run into in settings with locally satiated agents are well-known (Mas-Collel, Winston, and Green, 1995) and seem robust. The failure of local non-satiation implies that the Separating Hyperplane Theorem commonly used to prove the Second Welfare Theorem guarantees only the existence of a separating hyperplane that may have non-empty intersections with the set of Pareto-dominant aggregate assignments.…”

“…Mas-Collel et al (1995). 9 See the literature discussion at the end of the Introduction. Whether the Second Welfare Theorem obtains in settings without transfers and with possibly satiated agents was a puzzle except for deterministic assignments in single-unit demand settings, for which Abdulkadiroglu and Sonmez (1998) established a version of the Second Welfare Theorem.…”