Communication complexity of common voting rules

Predicting winner and estimating margin of victory in elections using sampling

Proportional representation (PR) is often discussed in voting settings as a major desideratum. For the past century or so, it is common both in practice and in the academic literature to look towards the single transferable vote (STV) rule as the solution for achieving PR. Some of the most prominent electoral reform movements around the globe are pushing for the adoption of STV. It has been termed a major open problem to design a voting rule that satisfies the same PR properties as STV and better monotonicity properties. In this paper, we first present a taxonomy of proportional representation axioms for general weak order preferences, some of which generalise and strengthen previously introduced concepts. We then present a rule called the expanding approvals rule (EAR) that satisfies properties stronger than the central PR axiom satisfied by STV, can handle indifferences in a convenient and computationally efficient manner, and also satisfies better candidate monotonicity properties. In view of this, our proposed rule seems to be a compelling solution for achieving proportional representation in voting settings.

Single Transferable Vote (STV) is used in large political elections around the world. It is easy to understand and has desirable normative properties such as clone-proofness. However, voters need to report full rankings, which can make it less practical than plurality voting. We study ways to minimize the amount of communication required to use single-winner STV. In the first part of the paper, voters are assumed to report their top-k alternatives in a single shot. We empirically evaluate the extent to which STV with truncated ballots approximates STV with full information. We also study the computational complexity of the possible winner problem for top-k ballots. For $k=1$, it can be solved in polynomial time, but is NP-complete when $k\geq 2$. In the second part, we consider interactive communication protocols for STV. Building on a protocol proposed by Conitzer and Sandholm (2005), we show how we can reduce the amount of communication required in practice. We then study empirically the average communication complexity of these protocols, based on randomly generated profiles, and on real-world election data. Our conclusion is that STV needs, in practice, much less information than in the worst case.

Various voting rules (or social choice procedures) have been proposed to select a winner from the preferences of an entire population: Plurality, veto, Borda, Minimax, Copeland, etc. Although in theory, these rules may yield drastically different outcomes, for real-world datasets, behavioral social choice analyses have found that the rules are often in perfect agreement with each other! This work attempts to give a mathematical explanation of this phenomenon. We quantify the gap between the outcomes of two voting rules by the pairwise margin between their winners. We show that for many common voting rules, the gap between them can be almost as large as 1 when the votes are unrestricted. As a counter, we study the behavior of voting rules when the vote distribution is a uniform mixture of a small number of multinomial logit distributions. This scenario corresponds to a population consisting of a small number of groups, each voting according to a latent preference ranking. We show that for any such voting profile on g groups, at least 1/2g fraction of the population prefers the winner of a Borda election to any other candidate.

Lu and Boutilier proposed a novel approach based on "minimax regret" to use classical score based voting rules in the setting where preferences can be any partial (instead of complete) orders over the set of alternatives. We show here that such an approach is vulnerable to a new kind of manipulation which was not present in the classical (where preferences are complete orders) world of voting. We call this attack "manipulative elicitation." More specifically, it may be possible to (partially) elicit the preferences of the agents in a way that makes some distinguished alternative win the election who may not be a winner if we elicit every preference completely. More alarmingly, we show that the related computational task is polynomial time solvable for a large class of voting rules which includes all scoring rules, maximin, Copeland α for every α ∈ [0,1], simplified Bucklin voting rules, etc. We then show that introducing a parameter per pair of alternatives which specifies the minimum number of partial preferences where this pair of alternatives must be comparable makes the related computational task of manipulative elicitation NP-complete for all common voting rules including a class of scoring rules which includes the plurality, k-approval, k-veto, veto, and Borda voting rules, maximin, Copeland α for every α ∈ [0,1], and simplified Bucklin voting rules. Hence, in this work, we discover a fundamental vulnerability in using minimax regret based approach in partial preferential setting and propose a novel way to tackle it.

Public vs. private randomness in simultaneous multi-party communication complexity

Private information retrieval (PIR) protocols allow a user to retrieve a data item from a database without revealing any information about the identity of the item being retrieved. Specifically, in information-theoretic k-server PIR, the database is replicated among k non-communicating servers, and each server learns nothing about the item retrieved by the user. The cost of PIR protocols is usually measured in terms of their communication complexity, which is the total number of bits exchanged between the user and the servers. However, another important cost parameter is the storage overhead, which is the ratio between the total number of bits stored on all the servers and the number of bits in the database. Since single-server information-theoretic PIR is impossible, the storage overhead of all existing PIR protocols is at least 2 (or k, in the case of k-server PIR). In this work, we show that information-theoretic PIR can be achieved with storage overhead arbitrarily close to the optimal value of 1, without sacrificing the communication complexity. Specifically, we prove that all known k-server PIR protocols can be efficiently emulated, while preserving both privacy and communication complexity but significantly reducing the storage overhead. To this end, we distribute the n bits of the database among s + r servers, each storing n/s coded bits (rather than replicas). Notably, our coding scheme remains the same, regardless of the specific k-server PIR protocol being emulated. For every fixed k, the resulting storage overhead (s +r)/s approaches 1 as s grows; explicitly we have equation. Moreover, in the special case k = 2, the storage overhead is only 1 + 1/s. In order to achieve these results, we introduce and study a new kind of binary linear codes, called here k-server PIR codes. Finally, we show how such codes can be constructed from multidimensional cubic, from Steiner systems, and from one-step majority-logic decodable codes.

On Data Structures and Asymmetric Communication Complexity

Distributed intrusion detection in the presence of correlated sensor readings: Signal-space and communication-complexity view-point

Let D/sub 0/,..., D/sub n/ be the {0,l}-matrices forming an association scheme. Since /spl Sigma//sub k=0//sup n/D/sub k/ is the all-one matrix, a linear combination /spl Sigma//sub k=0//sup n/c/sub k/D/sub k/ can be regarded as the matrix (f(x,y))/sub x,y/ representing the function f defined by f(x,y)=c/sub k/ if (x,y) is in relation corresponding to matrix D/sub k/. Parameters of functions thus obtained may now be studied exploiting properties of the association scheme. One such parameter is the communication complexity C(f), which is the number of bits that two persons have to exchange in order to evaluate f(x,y), when initially one person only knows x and the other person only knows y.

Fast periodic graph exploration with constant memory

A simulation of an interactive protocol entails the use of interactive communication to produce the output of the protocol to within a fixed statistical distance e. Recent works have proposed that the information complexity of the protocol plays a central role in characterizing the minimum number of bits that the parties must exchange for a successful simulation, namely the distributional communication complexity of simulating the protocol. Several simulation protocols have been proposed with communication complexity depending on the information complexity of the simulated protocol. However, in the absence of any general lower bounds for distributional communication complexity, the conjectured central role of information complexity is far from settled. We fill this gap and show that the distributional communication complexity of e-simulating a protocol is bounded below by the e-tail λe of the information complexity density, a random variable with information complexity as its expected value. For protocols with bounded number of rounds, we give a simulation protocol that yields a matching upper bound. Thus, it is not information complexity but λe that governs the distributional communication complexity.As applications of our bounds, in the amortized regime for product protocols, we identify the exact second order term, together with the precise dependence on e. For general protocols such as a mixture of two product protocols or for the amortized case when the repetitions are not independent, we derive a general formula for the leading asymptotic term. These results sharpen and significantly extend known results in the amortized regime. In the single-shot regime, our lower bound sheds light on the dependence of communication complexity on e. We illustrate this with an example that exhibits an arbitrary separation between distributional communication complexity and information complexity for all sufficiently small $\ep$.

Can quantum communication be more efficient than its classical counterpart? Holevo's theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can be communicated but no more. In apparent contradiction, there are distributed computational tasks for which quantum communication cannot be simulated efficiently by classical means. In some cases, the effect of transmitting quantum bits cannot be achieved classically short of transmitting an exponentially larger number of bits. In a similar vein, can entanglement be used to save on classical communication? It is well known that entanglement on its own is useless for the transmission of information. Yet, there are distributed tasks that cannot be accomplished at all in a classical world when communication is not allowed, but that become possible if the non-communicating parties share prior entanglement. This leads to the question of how expensive it is, in terms of classical communication, to provide an exact simulation of the spooky power of entanglement.

Suppose (X,Y) is a pair of random variables distributed over a support set S. Person P/sub x/ knows X, person P/sub y/ knows Y, and both know S. Using a predetermined protocol, they exchange binary messages in order for P/sub y/ to learn X. P/sub x/ may or may not learn Y. Bounds on communication complexity are obtained and used to obtain efficient protocols for the correlated files problem where X and Y are binary strings (files) within a small edit distance from each other. The average number of bits required for P/sub y/ to learn X when at most m messages are permitted is also determined.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The multiparty communication complexity concerns the least number of bits that must be exchanged among a number of players to collaboratively compute a Boolean function $f ( x_1 , \ldots ,x_k )$, while each player knows at most t inputs for some fixed $t < k$. The relation of the multiparty communication complexity to various hypergraph properties is investigated. Many of these properties are satisfied by random hypergraphs and can be classified by the framework of quasi randomness. Namely, many disparate properties of hypergraphs are shown to be mutually equivalent, and, furthermore, various equivalence classes form a natural hierarchy. In this paper, it is proved that the multiparty communication complexity problems are equivalent to certain hypergraph properties and thereby establish the connections among a large number of combinatorial and computational aspects of hypergraphs or Boolean functions.