Thou Shalt Covet The Average Of Thy Neighbors' Cakes

We prove an $\Omega(n^2)$ lower bound on the query complexity of local proportionality in the Robertson-Webb cake-cutting model. Local proportionality requires that each agent prefer their allocation to the average of their neighbors' allocations in some undirected social network. It is a weaker fairness notion than envy-freeness, which also has query complexity $\Omega(n^2)$, and generally incomparable to proportionality, which has query complexity $\Theta(n \log n)$. This result separates the complexity of local proportionality from that of ordinary proportionality, confirming the intuition that finding a locally proportional allocation is a more difficult computational problem

Computational Topology and the Unique Games Conjecture

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\u27s 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\u27Donnell, FOCS \u2704; SICOMP \u2707) 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

Can Buyers Reveal for a Better Deal?

We study small-scale 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 where there is one buyer and one good, showing that such communication can simultaneously improve social welfare and ex ante buyer utility. 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 a surprising contrast to 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

Representation with Incomplete Votes

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

You can have your cake and redistrict it too

The design of algorithms for political redistricting generally takes one of two approaches: optimize an objective such as compactness or, drawing on fair division, construct a protocol whose outcomes guarantee partisan fairness. We aim to have the best of both worlds by optimizing an objective subject to a binary fairness constraint. As the fairness constraint we adopt the geometric target, which requires the number of seats won by each party to be at least the average (rounded down) of its outcomes under the worst and best partitions of the state; but we extend this notion to allow the two parties to compute their targets with respect to different election datasets. Our theoretical contribution is twofold: we introduce a new model of redistricting that closely mirrors the classic model of cake-cutting and we prove the feasibility of the geometric target in this model. Our empirical results, which use real election data and maps of six US states, demonstrate that the geometric target is feasible in practice and that imposing it as a fairness constraint comes at almost no cost to three well-studied optimization objectives.First author draf

Topological Universality of the Art Gallery Problem

We prove that any compact semi-algebraic set is homeomorphic to the solution space of some art gallery problem. Previous works have established similar universality theorems, but holding only up to homotopy equivalence, rather than homeomorphism, and prior to this work, the existence of art galleries even for simple spaces such as the M\"obius strip or the three-holed torus were unknown. Our construction relies on an elegant and versatile gadget to copy guard positions with minimal overhead, and is thus simpler than previous constructions, consisting of a single rectangular room with convex slits cut out from the edges. We additionally show that both the orientable and non-orientable surfaces of genus $n$ require galleries with only $O(n)$ vertices

Playing Divide-and-Choose Given Uncertain Preferences

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. The prior distributions on those values are independent and common knowledge. We characterize the structure of optimal divisions in the divide-and-choose game and show how to efficiently compute equilibria. We identify several striking differences between optimal strategies in the cases of known versus unknown preferences. Most notably, the divider has a compelling "diversification" incentive in creating the chooser's two options. This incentive, hereto unnoticed, leads to multiple goods being divided at equilibrium, quite contrary to the divider's optimal strategy when preferences are known. In many contexts, such as buy-and-sell provisions between partners, or in judging fairness, it is important to assess the relative expected utilities of the divider and chooser. Those utilities, we show, depend on the players' uncertainties about each other's values, the number of goods being divided, and whether the divider can offer multiple alternative divisions. We prove that, when values are independently and identically distributed across players and goods, the chooser is strictly better off for a small number of goods, while the divider is strictly better off for a large number of goods

Multiagent Evaluation Mechanisms

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

Health-status outcomes with invasive or conservative care in coronary disease

BACKGROUND In the ISCHEMIA trial, an invasive strategy with angiographic assessment and revascularization did not reduce clinical events among patients with stable ischemic heart disease and moderate or severe ischemia. A secondary objective of the trial was to assess angina-related health status among these patients. METHODS We assessed angina-related symptoms, function, and quality of life with the Seattle Angina Questionnaire (SAQ) at randomization, at months 1.5, 3, and 6, and every 6 months thereafter in participants who had been randomly assigned to an invasive treatment strategy (2295 participants) or a conservative strategy (2322). Mixed-effects cumulative probability models within a Bayesian framework were used to estimate differences between the treatment groups. The primary outcome of this health-status analysis was the SAQ summary score (scores range from 0 to 100, with higher scores indicating better health status). All analyses were performed in the overall population and according to baseline angina frequency. RESULTS At baseline, 35% of patients reported having no angina in the previous month. SAQ summary scores increased in both treatment groups, with increases at 3, 12, and 36 months that were 4.1 points (95% credible interval, 3.2 to 5.0), 4.2 points (95% credible interval, 3.3 to 5.1), and 2.9 points (95% credible interval, 2.2 to 3.7) higher with the invasive strategy than with the conservative strategy. Differences were larger among participants who had more frequent angina at baseline (8.5 vs. 0.1 points at 3 months and 5.3 vs. 1.2 points at 36 months among participants with daily or weekly angina as compared with no angina). CONCLUSIONS In the overall trial population with moderate or severe ischemia, which included 35% of participants without angina at baseline, patients randomly assigned to the invasive strategy had greater improvement in angina-related health status than those assigned to the conservative strategy. The modest mean differences favoring the invasive strategy in the overall group reflected minimal differences among asymptomatic patients and larger differences among patients who had had angina at baseline