[Paper Review] Competitive Equilibria in Matching Markets with Budgets

Ning Chen, Xiaotie Deng+1 more|arXiv (Cornell University)|Apr 15, 2010

Auction Theory and Applications24 references4 citations

TL;DR

This paper presents a strongly polynomial-time algorithm to determine the existence of competitive equilibria and compute a minimum equilibrium in matching markets with indivisible goods, unit-demand buyers, and budget constraints. It establishes that competitive equilibria correspond exactly to strongly stable matchings under these constraints, resolving a fundamental discontinuity introduced by budgets that invalidates standard stability notions.

ABSTRACT

We study competitive equilibria in the classic Shapley-Shubik assignment model with indivisible goods and unit-demand buyers, with budget constraints: buyers can specify a maximum price they are willing to pay for each item, beyond which they cannot afford the item. This single discontinuity introduced by the budget constraint fundamentally changes the properties of equilibria: in the assignment model without budget constraints, a competitive equilibrium always exists, and corresponds exactly to a stable matching. With budgets, a competitive equilibrium need not always exist. In addition, there are now two distinct notions of stability, depending on whether both or only one of the buyer and seller can strictly benefit in a blocking pair, that no longer coincide due to the budget-induced discontinuity. We define weak and strong stability for the assignment model with transferable utilities, and show that competitive equilibria correspond exactly to strongly stable matchings. We consider the algorithmic question of efficiently computing competitive equilibria in an extension of the assignment model with budgets, where each buyer specifies his preferences over items using utility functions $u_{ij}$, where $u_{ij}(p_j)$ is the utility of buyer $i$ for item $j$ when its price is $p_j$. Our main result is a strongly polynomial time algorithm that decides whether or not a competitive equilibrium exists and if yes, computes a minimum one, for a general class of utility functions $u_{ij}$. This class of utility functions includes the standard quasi-linear utility model with a budget constraint, and in addition, allows modeling marketplaces where, for example, buyers only have a preference ranking amongst items subject to a maximum payment limit for each item, or where buyers want to optimize return on investment (ROI) instead of a quasi-linear utility and only know items' relative values.

Motivation & Objective

• To address the lack of existence and characterization of competitive equilibria in Shapley-Shubik assignment models when buyers face budget constraints.
• To resolve the breakdown of equivalence between competitive equilibria and stable matchings due to budget-induced discontinuities.
• To develop an efficient algorithm for computing competitive equilibria under general utility functions with budget constraints.
• To characterize the conditions under which truthful bidding is a Nash equilibrium in the minimum equilibrium mechanism.
• To extend the theoretical foundations of competitive equilibrium computation beyond quasi-linear utility models.

Proposed method

• The algorithm constructs a directed graph H based on preference and budget constraints to model feasible allocations and price adjustments.
• It uses a series of price reductions and matching checks to identify a minimum competitive equilibrium, leveraging a strongly polynomial-time procedure.
• The method relies on a key lemma showing that any competitive equilibrium must be representable within the constructed graph H.
• It applies a binary search-like approach over price ranges to find the minimal equilibrium prices that support a feasible matching.
• The algorithm is proven to terminate in strongly polynomial time, independent of input size in logarithmic terms.
• The approach is adapted to compute maximal equilibria by symmetric price-increasing procedures.

Experimental results

Research questions

• RQ1 Does a competitive equilibrium always exist in the Shapley-Shubik assignment model when buyers have budget constraints?
• RQ2 How do budget constraints alter the relationship between competitive equilibria and stable matchings?
• RQ3 Can a competitive equilibrium be computed efficiently in the presence of general utility functions with budget constraints?
• RQ4 What is the relationship between strong stability and competitive equilibria under budget constraints?
• RQ5 Is truthful bidding a Nash equilibrium in a mechanism that computes minimum competitive equilibria?

Key findings

• A competitive equilibrium does not always exist in the assignment model with budget constraints, unlike in the classical model.
• Competitive equilibria correspond exactly to strongly stable matchings, not weakly stable ones, due to the discontinuity introduced by budgets.
• The proposed algorithm computes a minimum competitive equilibrium in strongly polynomial time if one exists.
• The algorithm is correct and terminates in time bounded by a polynomial in the number of buyers and items, independent of the magnitude of budgets or values.
• Truthful bidding is a Nash equilibrium in the minimum equilibrium mechanism if a competitive equilibrium exists under true utilities.
• The mechanism is not dominant-strategy truthful, as the optimality of truthful bidding depends on the existence of an equilibrium, which is strategic and interdependent.