Cardinal-Utility Matching Markets: The Quest for Envy-Freeness, Pareto-Optimality, and Efficient Computability

TL;DR

This paper investigates cardinal-utility matching markets, focusing on achieving envy-freeness and Pareto-optimality while remaining computationally feasible. It shows that computing an EF+PO lottery in one-sided markets is PPAD-hard (indeed PPAD-complete with CHR23) and that a $(2+\epsilon)$-approximately envy-free and exactly Pareto-optimal lottery can be obtained in polynomial time via Nash bargaining, also yielding $(2+\epsilon)$-IC. In two-sided markets, EF+PO allocations may not exist, but the authors establish the existence of rational justified-envy-free and weak Pareto-optimal lotteries, and provide a framework based on a DIP-like equilibrium to obtain JEF. The work thus delineates the computational limits of EF+PO mechanisms in cardinal-utility settings and highlights Nash bargaining as a practical alternative for one-sided markets, while identifying promising directions and open questions for two-sided scenarios.

Abstract

Unlike ordinal-utility matching markets, which are well-developed from the viewpoint of both theory and practice, recent insights from a computer science perspective have left cardinal-utility matching markets in a state of flux. The celebrated pricing-based mechanism for one-sided cardinal-utility matching markets due to Hylland and Zeckhauser, which had long eluded efficient algorithms, was finally shown to be intractable; the problem of computing an approximate equilibrium is PPAD-complete. This led us to ask the question: is there an alternative, polynomial time, mechanism for one-sided cardinal-utility matching markets which achieves the desirable properties of HZ, i.e. (ex-ante) envy-freeness (EF) and Pareto-optimality (PO)? We show that the problem of finding an EF+PO lottery in a one-sided cardinal-utility matching market is by itself already PPAD-complete. However, a $(2 + ε)$-approximately envy-free and (exactly) Pareto-optimal lottery can be found in polynomial time using the Nash-bargaining-based mechanism of Hosseini and Vazirani. Moreover, the mechanism is also $(2 + ε)$-approximately incentive compatible. We also present several results on two-sided cardinal-utility matching markets, including non-existence of EF+PO lotteries as well as existence of justified-envy-free and weak Pareto-optimal lotteries.

Figures (8)

• Figure 1: For each pair of agents $i$ and $i'$ (large red dots) we add interpolating agents (small black dots) to transition between the utility vector $u_i$ and $u_{i'}$ in small steps. This is done coordinate-wise and this figure depicts an example with only two goods $j$ and $j'$.
• Figure 2: Shown is an agent who is interested in goods $j_1$ to $j_4$ which are plotted by their bang per buck. If we change only the utility of good $j_1$ (red) and leave the rest the same, there are only three possible sets of maximum bang per buck goods: $\{j_4\}$, $\{j_1, j_4\}$, and $\{j_1\}$. So along any chain of interpolating agents where we change only the utility for $j_1$ (monotonically), there will be at most two times that the set of maximum bang per buck goods and with it the budget of the agent can change.
• Figure 3: Depicted is $\mathcal{H}$ and its relationship to optimal bundles. Each point represents a good or collection of goods with identical price and utility. Gray points are dominated and will never be part of an optimal bundle. Points on $\mathcal{H}$ can be part of an optimal bundle depending on the budget $t$. A typical case is shown in which $\theta_i(t)$ consists of the three red goods that lie on the edge of $\mathcal{H}$ which corresponds to the tight dual constraints at budget $t$.
• Figure 4: Shown are several convex hulls $\mathcal{H}$ (red) as the red good's utility is changed. Note that the structure of $\mathcal{H}$ only changes when we cross one of the bounding lines of $\mathcal{H}$ -- the convex hull without the red good.
• Figure 5: Shown is an example instance which demonstrates that 2-EF is tight for Nash bargaining. Dashed edges have utility 1, solid edges have utility 2, and missing edges have utility 0. Clearly both agents prefer $j$ to $j'$. A simple calculation shows that in the Nash bargaining solution, $i$ will get all of $j$ and thus $i'$ will envy $i$ by a factor of 2.

Theorems & Definitions (74)

• Theorem : Section \ref{['sec:ppad_hardness']}
• Theorem : Section \ref{['sec:one_sided_nash']}
• Theorem : Section \ref{['sec:justified_envy_freeness']}
• Theorem 1: B46, N53
• Definition 2
• Theorem 3: HZ79
• Definition 4
• Definition 5
• Lemma 6
• Theorem 7