Adrian Vetta

Margarida Carvalho, Alison Caulfield, Yi Lin et al.

A kidney exchange program, also called a kidney paired donation program, can be viewed as a repeated, dynamic trading and allocation mechanism. This suggests that a dynamic algorithm for transplant exchange selection may have superior performance in comparison to the repeated use of a static algorithm. We confirm this hypothesis using a full scale simulation of the Canadian Kidney Paired Donation Program: learning algorithms, that attempt to learn optimal patient-donor weights in advance via dynamic simulations, do lead to improved outcomes. Specifically, our learning algorithms, designed with the objective of fairness (that is, equity in terms of transplant accessibility across cPRA groups), also lead to an increased number of transplants and shorter average waiting times. Indeed, our highest performing learning algorithm improves egalitarian fairness by 10% whilst also increasing the number of transplants by 6% and decreasing waiting times by 24%. However, our main result is much more surprising. We find that the most critical factor in determining the performance of a kidney exchange program is not the judicious assignment of positive weights (rewards) to patient-donor pairs. Rather, the key factor in increasing the number of transplants, decreasing waiting times and improving group fairness is the judicious assignment of a negative weight (penalty) to the small number of non-directed donors in the kidney exchange program.

Dimitris Avramidis, Alexandra Lassota, Ulrike Schmidt-Kraepelin et al.

Approval-based committee voting has received significant attention in the social choice community. Among the studied rules, Thiele rules, and especially Proportional Approval Voting (PAV), stand out for desirable properties such as proportional representation, Pareto optimality, and support monotonicity. Their main drawback is that computing a Thiele outcome is NP-hard in general. A glimpse of hope comes from the fact that Thiele rules are better behaved under structured preferences. On the candidate interval (CI) domain, they are computable in polynomial time via a linear program (LP) that has a totally unimodular constraint matrix. Surprisingly, this approach fails for the related voter interval (VI) domain, and the complexity of the problem has repeatedly been posed as an open question. Our main result resolves this question: although the relevant matrix is not totally unimodular, the ``standard'' LP still admits at least one optimal integral solution, and we provide a fast algorithm for finding it. Our technique naturally extends to the voter-candidate interval (VCI) domain, also known as the 1-dimensional voter-candidate range (1D-VCR) domain, and to the linearly consistent (LC) domain, both of which generalize the candidate and voter interval domains. Although both the VCI and LC domains have been studied in social choice, their relationship was unknown. We show, through connections to graph theory, that LC strictly contains VCI. We also provide an alternative definition of LC that is closer in spirit to VCI and has a natural interpretation in approval elections; this equivalence may be of independent interest. Finally, we study an alternative tree-based generalization of VCI and show that Thiele rules become NP-hard to compute on this domain.

Frank Connor, Max Dupré la Tour, Louis-Roy Langevin et al.

We analyze the house allocation problem, in which a set of agents must be matched to a set of objects for which they have cardinal utilities. A central mechanism for this problem is random serial dictatorship (RSD), which has long served as a canonical subject of study due to its simplicity and the existence of exact characterizations by its properties. Despite this extensive understanding, a basic quantitative question about the fairness of this mechanism remains unresolved. Although RSD is often viewed as fair ex ante, surprisingly, it is not envy-free in expectation. We quantify its deviation from envy-freeness via the envy-ratio, which is the maximum over all instances and pairs of agents of the ratio between an agent's expected utility for another agent's random object and for its own random object. Prior work shows a factor-$\sqrt{2}\approx 1.414$ lower bound on the envy-ratio of RSD. Our headline result is a matching upper bound, showing that RSD is $\sqrt{2}$-envy-free in the house allocation problem. We further analyze the two natural extensions of RSD (the randomized round-robin mechanism and the iterated-RSD mechanism) to settings with unequal numbers of agents and objects and more general valuation classes. For additive valuations, this ratio increases to at least $1.5$ and at most $1.707$ for randomized round-robin, but remains exactly $\sqrt{2}$ for iterated-RSD. For submodular valuations, we prove constant-factor upper and lower bounds for both mechanisms, leaving only a small constant gap in both cases. For the more general classes of XOS and subadditive valuations, we present a tight analysis for both mechanisms, showing that the envy-ratio is unbounded in the number of agents. These results provide the first tight or nearly tight quantitative guarantees on the extent to which random serial dictatorship and its natural generalizations approximate envy-freeness.