Apportionment is the act of distributing the seats of a legislature among political parties (or states) in proportion to their vote shares (or populations). A famous impossibility by Balinski and Young (2001) shows that no apportionment method can be proportional up to one seat (quota) while also responding monotonically to changes in the votes (population monotonicity). Grimmett (2004) proposed to overcome this impossibility by randomizing the apportionment, which can achieve quota as well as perfect proportionality and monotonicity --- at least in terms of the expected number of seats awarded to each party. Still, the correlations between the seats awarded to different parties may exhibit bizarre non-monotonicities. When parties or voters care about joint events, such as whether a coalition of parties reaches a majority, these non-monotonicities can cause paradoxes, including incentives for strategic voting.

Motivated by the difficulty of specifying complete ordinal preferences over a large set of m candidates, we study voting rules that are computable by querying voters about t < m candidates. Generalizing prior works that focused on specific instances of this problem, our paper fully characterizes the set of positional scoring rules that can be computed for any 1 ≤ t < m, which, notably, does not include plurality. We then extend this to show a similar impossibility result for single transferable vote (elimination voting). These negative results are information-theoretic and agnostic to the number of queries. Finally, for scoring rules that are computable with limited-sized queries, we give parameterized upper and lower bounds on the number of such queries a deterministic or randomized algorithm must make to determine the score-maximizing candidate. While there is no gap between our bounds for deterministic algorithms, identifying the exact query complexity for randomized algorithms is a challenging open problem, of which we solve one special case.

We introduce and study the problem of designing an equitable school redistricting map, which we formalize as that of assigning n students to school attendance zones in a way that is fair to various demographic groups. Drawing on methodology from fair division, we consider the demographic groups as players and seats in schools as homogeneous goods. Due to geographic constraints, not every school can be assigned to every student. This raises new obstacles, rendering some classic fairness criteria infeasible. Nevertheless, we show that it is always possible to find an almost proportional allocation among g demographic groups if we are allowed to add O(g log g) extra seats. For any fixed g, we show that such an allocation can be found in polynomial time, obtaining a runtime of O(n2 log n) in the special (but practical) case where g ≤ 3.

We prove that a polynomial fraction of the set of k-component forests in the m × n grid graph have equal numbers of vertices in each component, for any constant k. This resolves a conjecture of Charikar, Liu, Liu, and Vuong, and establishes the first provably polynomial-time algorithm for (exactly or approximately) sampling balanced grid graph partitions according to the spanning tree distribution, which weights each k-partition according to the product, across its k pieces, of the number of spanning trees of each piece. Our result follows from a careful analysis of the probability a uniformly random spanning tree of the grid can be cut into balanced pieces. Beyond grids, we show that for a broad family of lattice-like graphs, we achieve balance up to any multiplicative (1 ± ε) constant with constant probability. More generally, we show that, with constant probability, components derived from uniform spanning trees can approximate any given partition of a planar region specified by Jordan curves. This implies polynomial-time algorithms for sampling approximately balanced tree-weighted partitions for lattice-like graphs. Our results have applications to understanding political districtings, where there is an underlying graph of indivisible geographic units that must be partitioned into k population-balanced connected subgraphs. In this setting, tree-weighted partitions have interesting geometric properties, and this has stimulated significant effort to develop methods to sample them.

We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we prove lower bounds against the accuracy of FPC-interpretations that map Unique Games instances (encoded as relational structures) to rational numbers giving the approximate fraction of constraints that can be satisfied. We prove two new FPC-inexpressibility results for Unique Games: the existence of a $(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within any constant factor. Previous recent work has established similar FPC-inapproximability results for a small handful of other problems. Our construction builds upon some of these ideas, but contains a novel technique. While most FPC-inexpressibility results are based on variants of the CFI-construction, ours is significantly different. We start with a graph of very large girth and label the edges with random affine vector spaces over $\mathbb{F}_2$ that determine the constraints in the two structures. Duplicator's strategy involves maintaining a partial isomorphism over a minimal tree that spans the pebbled vertices of the graph.

Platforms for online civic participation rely heavily on methods for condensing thousands of comments into a relevant handful, based on whether participants agree or disagree with them. These methods should guarantee fair representation of the participants, as their outcomes may affect the health of the conversation and inform impactful downstream decisions. To that end, we draw on the literature on approval-based committee elections. Our setting is novel in that the approval votes are incomplete since participants will typically not vote on all comments. We prove that this complication renders non-adaptive algorithms impractical in terms of the amount of information they must gather. Therefore, we develop an adaptive algorithm that uses information more efficiently by presenting incoming participants with statements that appear promising based on votes by previous participants. We prove that this method satisfies commonly used notions of fair representation, even when participants only vote on a small fraction of comments. Finally, an empirical evaluation using real data shows that the proposed algorithm provides representative outcomes in practice.

We study pseudorandomness and pseudorandom generators from the perspective of logical definability. Building on results from ordinary derandomization and finite model theory, we show that it is possible to deterministically construct, in polynomial time, graphs and relational structures that are statistically indistinguishable from random structures by any sentence of first order or least fixed point logics. This raises the question of whether such constructions can be implemented via logical transductions from simpler structures with less entropy. In other words, can logical formulas be pseudorandom generators? We provide a complete classification of when this is possible for first order logic, fixed point logic, and fixed point logic with parity, and provide partial results and conjectures for first order logic with parity.

Mutually Fair Redistricting Even When Parties Disagree Congressional redistricting is the process of partitioning a state into districts, each of which elects a representative to Congress. Several recent high-profile redistricting efforts aim to increase the political power of a party. This raises the question of whether “fair” redistricting plans exist. In “You Can Have Your Cake and Redistrict It Too,” Benadè, Procaccia, and Tucker-Foltz propose a new theoretical model for redistricting inspired by classical cake-cutting models. In this model, it shown that is always possible to find redistricting plans that satisfy a particular notion of fairness, called the geometric target, simultaneously for both parties, even when the parties disagree about voter preferences. On real-world data, they find that this fairness constraint can be satisfied in all instances evaluated; moreover, requiring fairness comes at little cost in terms of traditional redistricting objectives. This suggests it is possible and practical to guarantee mutual fairness even in a climate of extreme partisanship.

We study the classic divide-and-choose method for equitably allocating divisible goods between two players who are rational, self-interested Bayesian agents. The players have additive private values for the goods, gD1, gD2, ..., gDn for the divider, and gC1, gC2, ..., gCn for the chooser. The prior distributions GDi and gCi, from which gDi and gCi are respectively drawn, are independent and common knowledge. After observing each gDi, the divider allocates each good fractionally between Pile 1 and Pile 2, then the chooser decides which player gets which pile.

We study market interactions in which buyers are allowed to credibly reveal partial information about their types to the seller. Previous recent work has studied the special case of one buyer and one good, showing that such communication can simultaneously improve social welfare and ex ante buyer utility. However, with multiple buyers, we find that the buyer-optimal signalling schemes from the one-buyer case are actually harmful to buyer welfare. Moreover, we prove several impossibility results showing that, with either multiple i.i.d. buyers or multiple i.i.d. goods, maximizing buyer utility can be at odds with social efficiency, which is surprising in contrast with the one-buyer, one-good case. Finally, we investigate the computational tractability of implementing desirable equilibrium outcomes. We find that, even with one buyer and one good, optimizing buyer utility is generally NP-hard but tractable in a practical restricted setting.

In the design and analysis of political redistricting maps, it is often useful to be able to sample from the space of all partitions of the graph of census blocks into connected subgraphs of equal population. There are influential Markov chain Monte Carlo methods for doing so that are based on sampling and splitting random spanning trees. Empirical evidence suggests that the distributions such algorithms sample from place higher weight on more “compact” redistricting plans, which is a practically useful and desirable property. In this paper, we confirm these observations analytically, establishing an inverse exponential relationship between the total length of the boundaries separating districts and the probability that such a map will be sampled. This result provides theoretical underpinnings for algorithms that are already making a significant real-world impact.

We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). We prove two new FPC- inexpressibility results for Unique Games: the existence of a ( \frac12,\frac13 + δ )-inapproximability gap, and inapproximability to within any constant factor. Previous recent work has established similar FPC-inapproximability results for a small handful of other problems. Our construction builds upon some of these ideas, but contains a novel technique. While most FPC-inexpressibility results are based on variants of the CFI-construction, ours is significantly different.

We consider settings where agents are evaluated based on observed features, and assume they seek to achieve feature values that bring about good evaluations. Our goal is to craft evaluation mechanisms that incentivize the agents to invest effort in desirable actions; a notable application is the design of course grading schemes. Previous work has studied this problem in the case of a single agent. By contrast, we investigate the general, multi-agent model, and provide a complete characterization of its computational complexity.

Covering spaces of graphs have long been useful for studying expanders (as "graph lifts") and unique games (as the "label-extended graph"). In this paper we advocate for the thesis that there is a much deeper relationship between computational topology and the Unique Games Conjecture. Our starting point is Linial's 2005 observation that the only known problems whose inapproximability is equivalent to the Unique Games Conjecture - Unique Games and Max-2Lin - are instances of Maximum Section of a Covering Space on graphs. We then observe that the reduction between these two problems (Khot-Kindler-Mossel-O'Donnell, FOCS '04; SICOMP '07) gives a well-defined map of covering spaces. We further prove that inapproximability for Maximum Section of a Covering Space on (cell decompositions of) closed 2-manifolds is also equivalent to the Unique Games Conjecture. This gives the first new "Unique Games-complete" problem in over a decade. Our results partially settle an open question of Chen and Freedman (SODA, 2010; Disc. Comput. Geom., 2011) from computational topology, by showing that their question is almost equivalent to the Unique Games Conjecture. (The main difference is that they ask for inapproximability over Z_2, and we show Unique Games-completeness over Z_k for large k.) This equivalence comes from the fact that when the structure group G of the covering space is Abelian - or more generally for principal G-bundles - Maximum Section of a G-Covering Space is the same as the well-studied problem of 1-Homology Localization. Although our most technically demanding result is an application of Unique Games to computational topology, we hope that our observations on the topological nature of the Unique Games Conjecture will lead to applications of algebraic topology to the Unique Games Conjecture in the future.