market designers

Abstract: We study the truthful and fair allocation of indivisible goods to $n$ strategic agents with additive valuations. Babaioff, Feige, and Manaker Morag [FOCS 2026] gave a randomized mechanism that uses only the agents' rankings of the goods, is truthful in expectation (TIE), and guarantees every agent $1/(H_{n-1}+2)=Θ(1/\log n)$ of her maximin share (MMS) in every realized allocation, where $H_{n-1}$ is the $(n-1)$th harmonic number; this is nearly the best possible with rankings alone. They conjectured that cardinal information allows TIE mechanisms to achieve a constant ex-post MMS guarantee. We confirm this conjecture: our TIE mechanism guarantees every agent at least $1/7$ of her MMS in every realized allocation; moreover, the mechanism is ex-ante envy-free and can be implemented in polynomial time. Our mechanism has two key technical ingredients, both of which may be of independent interest. The first is a truthful fractional allocation rule specifying each agent's probability of receiving each good: it favors each agent on her top $n-1$ goods and reduces her probability of receiving a good for each other agent who also ranks it among her top $n-1$ goods. The second is the balanced edge coloring: we decompose these probabilities into equally likely matchings from agents to high-value goods, those that alone meet an agent's guarantee, and balance these matchings in a fine-grained way without changing any marginal probability, so that every agent who receives no high-value good can obtain sufficient value from the remaining goods without over-allocating any good.

Abstract: We consider the fair division problem with indivisible goods and study the following decision problem: given a fair division instance, does there exist an allocation that is envy-free and efficient? We consider two efficiency criteria: Pareto-optimality and social welfare optimality. We provide a complete landscape on the computational complexity of this decision problem, with the number of agents ranging from $2$ to $\infty$, both additive valuations and general valuations, and the more restricted class of $k$-ary valuation functions (where an item's marginal value is restricted to $\{0,1,\ldots,k-1\}$ for some constant $k\geq2$). One interesting observation is that many versions of the above-mentioned decision problems fall into the ``gap'' between the first and the second levels of the polynomial hierarchy. Specifically, assuming the polynomial hierarchy does not collapse to the first level (i.e., assuming $\text{NP}\neq\text{coNP}$), these problems are in $(Σ_2^{\text{p}}\capΠ_2^{\text{p}})\setminus(\text{NP}\cup\text{coNP})$. In particular, we provide a fine-grained complexity analysis across different parameter regimes, including the number of agents and the choice of valuation models. Depending on different parameters, many problems admit different complexity classifications, ranging from the intermediate classes $Θ_2^{\text{p}}$ and $Δ_2^{\text{p}}$ between the two levels to $Σ_2^{\text{p}}$-completeness. Finally, De Keijzer et al. show the $Σ_2^{\text{p}}$-completeness of the decision problem when considering Pareto-optimality as the efficiency criterion with additive valuations. Our main results extend this result to more restricted settings, such as instances with a constant number of agents or $3$-ary valuation functions, which resolves the open problem given by Bouveret and Lang.

Abstract: We study gains from trade in matching markets with independent private values and costs, Bayesian incentive compatibility, interim individual rationality, and no expected budget deficit. A second-best guarantee for finite bilateral trade extends without loss to matching markets with independent Borel priors, arbitrary downward-closed feasibility, and finite expected first-best gains. For bounded buyers with monotone hazard rates and arbitrary bounded sellers, we determine the exact worst-case ratio of second-best to first-best gains, approximately $0.72490721$. For binary buyers and sellers with at most $m$ types, we determine the exact ratio for every $m$, including $8/9$ when $m=2$ and a limit of $4/5$ as $m$ grows. Both families of bounds are tight already in bilateral trade.

Abstract: Envy-freeness is a fundamental notion of fairness in the allocation of indivisible goods. In this paper, we study envy-free allocation under Leontief preferences, which model perfect complements. Although Leontief preferences have been extensively studied in the context of allocating divisible goods and market equilibria, they have received comparatively little attention for the allocation of indivisible goods. We show that, unlike additive valuations in cardinal preferences, an envy-free allocation always exists for Leontief preferences when there are at least two goods. In contrast, envy-free allocations may fail to exist when there is a single good, however it can be decided in polynomial time. We next study the problem of computing a welfare-maximizing envy-free allocation. We prove that this problem is NP-hard in general, whereas it is polynomial-time solvable when there is only a single good or agents have identical demands. Finally, we investigate the parameterized complexity of this problem.

Abstract: We study Bulow-Klemperer-style competition complexity in two-sided double auctions with $m$ unit-demand buyers drawn i.i.d. from $F_B$ and $n$ unit-supply sellers drawn i.i.d. from $F_S$. When $m \ge n$ and buyer valuations first-order stochastically dominate seller costs ($F_B \succeq_{\mathrm{FSD}} F_S$), we prove that recruiting just two additional sellers enables Seller Trade Reduction (STR), a prior-independent mechanism, to achieve expected Gains From Trade (GFT) at least the first-best GFT of the original market. When the buyer side is the smaller side of the market ($m \le n$), an analogous result holds for Buyer Trade Reduction with 2 additional buyers. This resolves open questions of Babaioff, Goldner, and Gonczarowski (SODA 2020) and Cai, Liaw, Mehta, and Zhao (STOC 2024). We complement our upper bound by showing that this uniform bound is optimal: already for $m = n = 1$, no prior-free mechanism (deterministic or randomized) that is dominant-strategy incentive-compatible, individually rational, and weakly budget-balanced can match the first-best GFT by recruiting only one additional seller.

Abstract: Matching mechanisms differ in how much of an agent's preference ranking must be determined and reported to obtain a particular object. A mechanism is singleton-attainable (SA) if every object that an agent can obtain through some report can also be obtained by reporting only that object as acceptable. With an SA mechanism, once an attainable object has been identified, the agent need not rank or report any other object. Singleton-attainability identifies a distinct dimension in matching theory and market design: transparent access to attainable outcomes, separate from incentives, stability, welfare or equity. We establish general conditions for SA and derive its strategic implications. Top-lift invariance and truncation invariance together imply SA, while strategyproofness and stability each imply SA. By contrast, no Pareto improvement over a strategyproof, individually rational, and non-wasteful mechanism is SA. In particular, every Pareto improvement over Deferred Acceptance violates SA. We introduce report width, which measures how many acceptable objects may have to be reported to obtain an object. SA mechanisms have report width one. Report width is unbounded for a large class of efficient mechanisms that Pareto-improve Deferred Acceptance. Stable selection with report-induced priorities has width one when priorities are monotone and maximal width under reverse priority dominance. Rank-welfare maximization has width one when the outside-option rank is fixed and maximal width when it is report-dependent. These results reveal a structural divide, which we call the width dichotomy: across all mechanisms and families in our classification and all structural classes we study, report width is either one or unbounded.

Abstract: Motivated by applications in fairness and foundational questions, we consider the problem of maximizing a monotone submodular function $f\colon 2^E \rightarrow \mathbb{R}_+$ over maximum cardinality sets in the intersection of two matroids on a common ground set $E$. An important special case is submodular perfect matching in bipartite graphs. Prior to this work, its approximability was poorly understood with only constant inapproximability known, despite not even a $\frac{1}{o(\sqrt{|E|})}$-approximation being known. Even when allowing to violate the cardinality constraint slightly, only a bicriteria approximation with a significant loss in the objective was known. Here, we obtain two results. First, we show that, within constant factors, the problem is approximation-equivalent to Submodular Orienteering in directed graphs. This yields an $Ω(1 / \log |E|)$-approximation in quasi-polynomial time together with an almost-matching hardness result. Second, we obtain an improved polynomial-time bicriteria approximation via a local search framework. More precisely, if $f(T^*)$ is the largest submodular value of a common independent set in both matroids of size at least $K$, we find a common independent set $T$ such that $|T| \geq (1 - ε) K$ and $f(T) \geq (1/2 - ε) f(T^*)$. In contrast, previous work only guarantees a value of $Ω(ε) f(T^*)$ while ensuring that $|T| \geq (1 - ε) K$.

Abstract: We investigate the asymptotic behavior of max-min fair allocations for indivisible goods under i.i.d. random utilities. For $N$ agents and $K$ goods with utilities ${\mU_{i,j}}$ drawn independently from a common distribution $F$, we derive asymptotic characterizations of the max-min value in the balanced case $K=N$ (and in $K=LN$ extensions) via distributional quantiles. We then study the efficiency impact of max-min fairness by comparing the resulting total welfare with the optimal sum welfare. For distributions with sufficiently light tails, we prove that the relative efficiency loss converges to zero as the market grows, implying that max-min fairness incurs negligible welfare loss in large random instances for a broad class of distributions.

Abstract: We study gains from trade (GFT) in two-sided matching markets with independent private types and arbitrary downward-closed feasibility constraints. The second-best benchmark is the maximum expected GFT achievable by a Bayesian incentive compatible, interim individually rational mechanism that is strongly budget balanced at every report profile. These constraints generally preclude attaining the first-best GFT and raise the question of how much efficiency must be lost. We prove that the second-best GFT is at least one half of the first-best GFT in every such matching market. This recovers and generalizes the recent $1/2$ guarantee for bilateral trade by Liu et al. (2026) to markets with multiple buyers and sellers and arbitrary downward-closed feasibility constraints. Together with their matching lower bound for bilateral trade, our result establishes a tight worst-case ratio of $1/2$ for this general class of matching markets. Our proof builds on the virtual-GFT framework of Brüstle et al. (EC 2017) to reduce the problem to a one-parameter Lagrangian. The main step is a geometric, edge-by-edge analysis based on first-best edge-selection regions, combined with a randomized contraction in rank space.

Abstract: Automated prediction markets require sponsors to prefund liquidity before observing order flow, creating a financing challenge at launch. We study whether nonnegative charges conditioned on observable payoff direction can improve recovery of this prefunded capital while limiting their effect on informed participation. We develop Seed Capital Flow (SCF), a direction-conditioned levy, in a stylized binary cost-function market with informed and liquidity-motivated traders. When order composition differs across directions, SCF concentrates the permitted fee burden on the direction with relatively more liquidity-motivated flow, whereas a uniform fee spreads it across both directions. Under a sufficiently tight common retention constraint, this allocation yields higher expected recovery capacity and can make additional liquidity choices financially viable. Synthetic numerical audits examine robustness to alternative flow patterns, stochastic arrivals, and label misspecification. The results characterize a mechanism-design tradeoff rather than an empirical prediction: the market remains prefunded, recovery is expected rather than guaranteed, and the analysis is limited to an opening-cohort setting.