Local Distance Restricted Bribery in Voting

Studying complexity of various bribery problems has been one of the main research focus in computational social choice. In all the models of bribery studied so far, the briber has to pay every voter some amount of money depending on what the briber wants the voter to report and the briber has some budget at her disposal. Although these models successfully capture many real world applications, in many other scenarios, the voters may be unwilling to deviate too much from their true preferences. In this paper, we study the computational complexity of the problem of finding a preference profile which is as close to the true preference profile as possible and still achieves the briber's goal subject to budget constraints. We call this problem Optimal Bribery. We consider three important measures of distances, namely, swap distance, footrule distance, and maximum displacement distance, and resolve the complexity of the optimal bribery problem for many common voting rules. We show that the problem is polynomial time solvable for the plurality and veto voting rules for all the three measures of distance. On the other hand, we prove that the problem is NP-complete for a class of scoring rules which includes the Borda voting rule, maximin, Copeland$^\alpha$ for any $\alpha\in[0,1]$, and Bucklin voting rules for all the three measures of distance even when the distance allowed per voter is $1$ for the swap and maximum displacement distances and $2$ for the footrule distance even without the budget constraints (which corresponds to having an infinite budget). For the $k$-approval voting rule for any constant $k>1$ and the simplified Bucklin voting rule, we show that the problem is NP-complete for the swap distance even when the distance allowed is $2$ and for the footrule distance even when the distance allowed is $4$ even without the budget constraints.Comment: Accepted as an extended abstract in AAMAS 201

Reduction formulas for symmetric products of spin matrices

We show that, for SU(2) generators of arbitrary dimension $D$, there exist identities that express the completely symmetric product of $D$ matrices in terms of completely symmetric products of fewer number of matrices. We also indicate why such identities are important in characterizing electromagnetic interactions of particles.Comment: 8 pages, no figur

Nothing but Relativity

We deduce the most general space-time transformation laws consistent with the principle of relativity. Thus, our result contains the results of both Galilean and Einsteinian relativity. The velocity addition law comes as a bi-product of this analysis. We also argue why Galilean and Einsteinian versions are the only possible embodiments of the principle of relativity.Comment: 6 pages, Late

At the root of things

Modern theories of fundamental interactions describe strong, electromagnetic and weak interactions as quantum field theories with certain kinds of embedded internal symmetries called `gauge symmetries'. This article introduces quantum field theories and gauge symmetries to the uninitiated.Comment: LaTeX, 22 pages, requires axodraw.sty. A small error has been corrected in this version. Journal reference was also wrong in the previous versio

The road not considered ... the question of photon mass

We consider possibilities of modification of Maxwell's equations of electrodynamics that could have automatically led to a massive photon. Why weren't such questions considered at the time when quantum theory was introduced at the beginning of the 20th century? We try to answer this question.Comment: 8 pages, Late

A reincarnation of R-parity

In supersymmetric theories, R-parity is defined in a way such that it does not commute with the space-time symmetries. We show that, in general sypersymmetric models, one can define a discrete symmetry which commutes with all space-time and gauge symmetries, and whose phenomenological implications are equivalent to those of R-parity.Comment: 3 pages, Late

Sample Complexity for Winner Prediction in Elections

Predicting the winner of an election is a favorite problem both for news media pundits and computational social choice theorists. Since it is often infeasible to elicit the preferences of all the voters in a typical prediction scenario, a common algorithm used for winner prediction is to run the election on a small sample of randomly chosen votes and output the winner as the prediction. We analyze the performance of this algorithm for many common voting rules. More formally, we introduce the $(\epsilon, \delta)$-winner determination problem, where given an election on $n$ voters and $m$ candidates in which the margin of victory is at least $\epsilon n$ votes, the goal is to determine the winner with probability at least $1-\delta$. The margin of victory of an election is the smallest number of votes that need to be modified in order to change the election winner. We show interesting lower and upper bounds on the number of samples needed to solve the $(\epsilon, \delta)$-winner determination problem for many common voting rules, including scoring rules, approval, maximin, Copeland, Bucklin, plurality with runoff, and single transferable vote. Moreover, the lower and upper bounds match for many common voting rules in a wide range of practically appealing scenarios.Comment: Accepted in AAMAS 201

Fishing out Winners from Vote Streams

We investigate the problem of winner determination from computational social choice theory in the data stream model. Specifically, we consider the task of summarizing an arbitrarily ordered stream of $n$ votes on $m$ candidates into a small space data structure so as to be able to obtain the winner determined by popular voting rules. As we show, finding the exact winner requires storing essentially all the votes. So, we focus on the problem of finding an {\em \eps-winner}, a candidate who could win by a change of at most \eps fraction of the votes. We show non-trivial upper and lower bounds on the space complexity of \eps-winner determination for several voting rules, including $k$-approval, $k$-veto, scoring rules, approval, maximin, Bucklin, Copeland, and plurality with run off.Comment: Adding Acknowledgemen

Estimating the Margin of Victory of an Election using Sampling

The margin of victory of an election is a useful measure to capture the robustness of an election outcome. It also plays a crucial role in determining the sample size of various algorithms in post election audit, polling etc. In this work, we present efficient sampling based algorithms for estimating the margin of victory of elections. More formally, we introduce the \textsc{$(c, \epsilon, \delta)$--Margin of Victory} problem, where given an election $\mathcal{E}$ on $n$ voters, the goal is to estimate the margin of victory $M(\mathcal{E})$ of $\mathcal{E}$ within an additive factor of $c MoV(\mathcal{E})+\epsilon n$. We study the \textsc{$(c, \epsilon, \delta)$--Margin of Victory} problem for many commonly used voting rules including scoring rules, approval, Bucklin, maximin, and Copeland$^{\alpha}.$ We observe that even for the voting rules for which computing the margin of victory is NP-Hard, there may exist efficient sampling based algorithms, as observed in the cases of maximin and Copeland$^{\alpha}$ voting rules.Comment: To appear in IJCAI 201

Asymptotic Collusion-Proofness of Voting Rules: The Case of Large Number of Candidates

Classical results in voting theory show that strategic manipulation by voters is inevitable if a voting rule simultaneously satisfy certain desirable properties. Motivated by this, we study the relevant question of how often a voting rule is manipulable. It is well known that elections with a large number of voters are rarely manipulable under impartial culture (IC) assumption. However, the manipulability of voting rules when the number of candidates is large has hardly been addressed in the literature and our paper focuses on this problem. First, we propose two properties (1) asymptotic strategy-proofness and (2) asymptotic collusion-proofness, with respect to new voters, which makes the two notions more relevant from the perspective of computational problem of manipulation. In addition to IC, we explore a new culture of society where all score vectors of the candidates are equally likely. This new notion has its motivation in computational social choice and we call it impartial scores culture (ISC) assumption. We study asymptotic strategy-proofness and asymptotic collusion-proofness for plurality, veto, $k$-approval, and Borda voting rules under IC as well as ISC assumptions. Specifically, we prove bounds for the fraction of manipulable profiles when the number of candidates is large. Our results show that the size of the coalition and the tie-breaking rule play a crucial role in determining whether or not a voting rule satisfies the above two properties.Comment: Accepted as an extended abstract in AAMAS 201