We prove axiomatic characterizations of several important multiwinner rules within the class of approval-based committee choice rules. These are voting rules that return a set of (fixed-size) committees. In particular, we provide axiomatic characterizations of Proportional Approval Voting, the Chamberlin--Courant rule, and other Thiele methods. These rules share the important property that they satisfy an axiom called consistency, which is crucial in our characterizations.
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The class of Basic Feasible Functionals BFF$_2$ is the type-2 counterpart of the class FP of type-1 functions computable in polynomial time. Several characterizations have been suggested in the literature, but none of these present a programming language with a type system guaranteeing this complexity bound. We give a characterization of BFF$_2$ based on an imperative language with oracle calls using a tier-based type system whose inference is decidable. Such a characterization should make it possible to link higher-order complexity with programming theory. The low complexity (cubic in the size of the program) of the type inference algorithm contrasts with the intractability of the aforementioned methods and does not overly constrain the expressive power of the language.
While the stable marriage problem and its variants model a vast range of matching markets, they fail to capture complex agent relationships, such as the affiliation of applicants and employers in an interview marketplace. To model this problem, the existing literature on matching with externalities permits agents to provide complete and total rankings over matchings based off of both their own and their affiliates' matches. This complete ordering restriction is unrealistic, and further the model may have an empty core. To address this, we introduce the Dichotomous Affiliate Stable Matching (DASM) Problem, where agents' preferences indicate dichotomous acceptance or rejection of another agent in the marketplace, both for themselves and their affiliates. We also assume the agent's preferences over entire matchings are determined by a general weighted valuation function of their (and their affiliates') matches. Our results are threefold: (1) we use a human study to show that real-world matching rankings follow our assumed valuation function; (2) we prove that there always exists a stable solution by providing an efficient, easily-implementable algorithm that finds such a solution; and (3) we experimentally validate the efficiency of our algorithm versus a linear-programming-based approach.
Online meal delivery is undergoing explosive growth, as this service is becoming increasingly popular. A meal delivery platform aims to provide excellent and stable services for customers and restaurants. However, in reality, several hundred thousand orders are canceled per day in the Meituan meal delivery platform since they are not accepted by the crowd soucing drivers. The cancellation of the orders is incredibly detrimental to the customer's repurchase rate and the reputation of the Meituan meal delivery platform. To solve this problem, a certain amount of specific funds is provided by Meituan's business managers to encourage the crowdsourcing drivers to accept more orders. To make better use of the funds, in this work, we propose a framework to deal with the multi-stage bonus allocation problem for a meal delivery platform. The objective of this framework is to maximize the number of accepted orders within a limited bonus budget. This framework consists of a semi-black-box acceptance probability model, a Lagrangian dual-based dynamic programming algorithm, and an online allocation algorithm. The semi-black-box acceptance probability model is employed to forecast the relationship between the bonus allocated to order and its acceptance probability, the Lagrangian dual-based dynamic programming algorithm aims to calculate the empirical Lagrangian multiplier for each allocation stage offline based on the historical data set, and the online allocation algorithm uses the results attained in the offline part to calculate a proper delivery bonus for each order. To verify the effectiveness and efficiency of our framework, both offline experiments on a real-world data set and online A/B tests on the Meituan meal delivery platform are conducted. Our results show that using the proposed framework, the total order cancellations can be decreased by more than 25\% in reality.
Multifidelity approximation is an important technique in scientific computation and simulation. In this paper, we introduce a bandit-learning approach for leveraging data of varying fidelities to achieve precise estimates of the parameters of interest. Under a linear model assumption, we formulate a multifidelity approximation as a modified stochastic bandit, and analyze the loss for a class of policies that uniformly explore each model before exploiting. Utilizing the estimated conditional mean-squared error, we propose a consistent algorithm, adaptive Explore-Then-Commit (AETC), and establish a corresponding trajectory-wise optimality result. These results are then extended to the case of vector-valued responses, where we demonstrate that the algorithm is efficient without the need to worry about estimating high-dimensional parameters. The main advantage of our approach is that we require neither hierarchical model structure nor \textit{a priori} knowledge of statistical information (e.g., correlations) about or between models. Instead, the AETC algorithm requires only knowledge of which model is a trusted high-fidelity model, along with (relative) computational cost estimates of querying each model. Numerical experiments are provided at the end to support our theoretical findings.
The conditionality principle $C$ plays a key role in attempts to characterize the concept of statistical evidence. The standard version of $C$ considers a model and a derived conditional model, formed by conditioning on an ancillary statistic for the model, together with the data, to be equivalent with respect to their statistical evidence content. This equivalence is considered to hold for any ancillary statistic for the model but creates two problems. First, there can be more than one maximal ancillary in a given context and this leads to $C$ not being an equivalence relation and, as such, calls into question whether $C$ is a proper characterization of statistical evidence. Second, a statistic $A$ can change from ancillary to informative (in its marginal distribution) when another ancillary $B$ changes, from having one known distribution $P_{B},$ to having another known distribution $Q_{B}.$ This means that the stability of ancillarity differs across ancillary statistics and raises the issue of when a statistic can be said to be truly ancillary. It is therefore natural, and practically important, to limit conditioning to the set of ancillaries whose distribution is irrelevant to the ancillary status of any other ancillary statistic. This results in a family of ancillaries for which there is a unique maximal member. This also gives a new principle for inference, the stable conditionality principle, that satisfies the criteria required for any principle whose aim is to characterize statistical evidence.
In this paper, we analyze the problem of how to adapt the concept of proportionality to situations where several perfectly divisible resources have to be allocated among certain set of agents that have exactly one claim which is used for all resources. In particular, we introduce the constrained proportional awards rule, which extend the classical proportional rule to these situations. Moreover, we provide an axiomatic characterization of this rule.
Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of free modules $X\xrightarrow{f} Y \xrightarrow{g} Z$ such that $M\cong \ker{g}/\mathrm{im}{f}$. It runs in time $O(|X|^3+|Y|^3+|Z|^3)$ and requires $O(|X|^2+|Y|^2+|Z|^2)$ memory, where $|\cdot |$ denotes the rank. Given the presentation computed by our algorithm, the bigraded Betti numbers of $M$ are readily computed. Our approach is based on a simple matrix reduction algorithm, slight variants of which compute kernels of morphisms between free modules, minimal generating sets, and Gr\"obner bases. Our algorithm for computing minimal presentations has been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our implementation outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.
We study the stochastic multi-player multi-armed bandit problem. In this problem, $m$ players cooperate to maximize their total reward from $K > m$ arms. However the players cannot communicate and are penalized (e.g. receive no reward) if they pull the same arm at the same time. We ask whether it is possible to obtain optimal instance-dependent regret $\tilde{O}(1/\Delta)$ where $\Delta$ is the gap between the $m$-th and $m+1$-st best arms. Such guarantees were recently achieved in a model allowing the players to implicitly communicate through intentional collisions. We show that with no communication at all, such guarantees are, surprisingly, not achievable. In fact, obtaining the optimal $\tilde{O}(1/\Delta)$ regret for some regimes of $\Delta$ necessarily implies strictly sub-optimal regret in other regimes. Our main result is a complete characterization of the Pareto optimal instance-dependent trade-offs that are possible with no communication. Our algorithm generalizes that of Bubeck, Budzinski, and the second author and enjoys the same strong no-collision property, while our lower bound is based on a topological obstruction and holds even under full information.
Power laws and power laws with exponential cut-off are two distinct families of distributions on the positive real half-line. In the present paper, we propose a unified treatment of both families by building a family of distributions that interpolates between them, which we call Interpolating Family (IF) of distributions. Our original construction, which relies on techniques from statistical physics, provides a connection for hitherto unrelated distributions like the Pareto and Weibull distributions, and sheds new light on them. The IF also contains several distributions that are neither of power law nor of power law with exponential cut-off type. We calculate quantile-based properties, moments and modes for the IF. This allows us to review known properties of famous distributions on $\mathbb{R}^+$ and to provide in a single sweep these characteristics for various less known (and new) special cases of our Interpolating Family.
Singleton arc consistency is an important type of local consistency which has been recently shown to solve all constraint satisfaction problems (CSPs) over constraint languages of bounded width. We aim to characterise all classes of CSPs defined by a forbidden pattern that are solved by singleton arc consistency and closed under removing constraints. We identify five new patterns whose absence ensures solvability by singleton arc consistency, four of which are provably maximal and three of which generalise 2-SAT. Combined with simple counter-examples for other patterns, we make significant progress towards a complete classification.
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