Prior work on the complexity of bribery assumes that the bribery happens simultaneously, and that the briber has full knowledge of all voters' votes. But neither of those assumptions always holds. In many real-world settings, votes come in sequentially, and the briber may have a use-it-or-lose-it moment to decide whether to bribe alter a given vote, and at the time of making that decision, the briber may not know what votes remaining voters are planning on casting. In this paper, we introduce a model for, and initiate the study of, bribery in such an online, sequential setting. We show that even for election systems whose winner-determination problem is polynomial-time computable, an online, sequential setting may vastly increase the complexity of bribery, in fact jumping the problem up to completeness for high levels of the polynomial hierarchy or even PSPACE. On the other hand, we show that for some natural, important election systems, such a dramatic complexity increase does not occur, and we pinpoint the complexity of their bribery problems in the online, sequential setting.
Our approach to the briber's goal, which is assuming worst-case revelations of information, is inspired by the approach used in the area known as online algorithms @cite_1 .
@cite_1
2122422466 1493942848 2139297408 2963441643
We consider bandit problems, motivated by applications in online advertising and news story selection, in which the learner must repeatedly select a slate, that is, a subset of size s from K possible actions, and then receives rewards for just the selected actions. The goal is to minimize the regret with respect to total reward of the best slate computed in hindsight. We consider unordered and ordered versions of the problem, and give efficient algorithms which have regret O(√T), where the constant depends on the specific nature of the problem. We also consider versions of the problem where we have access to a number of policies which make recommendations for slates in every round, and give algorithms with O(√T) regret for competing with the best such policy as well. We make use of the technique of relative entropy projections combined with the usual multiplicative weight update algorithm to obtain our algorithms. Encouraging voters to truthfully reveal their preferences in an election has long been an important issue. Recently, computational complexity has been suggested as a means of precluding strategic behavior. Previous studies have shown that some voting protocols are hard to manipulate, but used NP-hardness as the complexity measure. Such a worst-case analysis may be an insufficient guarantee of resistance to manipulation. Indeed, we demonstrate that NP-hard manipulations may be tractable in the average-case. For this purpose, we augment the existing theory of average-case complexity with some new concepts. In particular, we consider elections distributed with respect to junta distributions, which concentrate on hard instances. We use our techniques to prove that scoring protocols are susceptible to manipulation by coalitions, when the number of candidates is constant. In this work, we study the notion of competing campaigns in a social network and address the problem of influence limitation where a "bad" campaign starts propagating from a certain node in the network and use the notion of limiting campaigns to counteract the effect of misinformation. The problem can be summarized as identifying a subset of individuals that need to be convinced to adopt the competing (or "good") campaign so as to minimize the number of people that adopt the "bad" campaign at the end of both propagation processes. We show that this optimization problem is NP-hard and provide approximation guarantees for a greedy solution for various definitions of this problem by proving that they are submodular. We experimentally compare the performance of the greedy method to various heuristics. The experiments reveal that in most cases inexpensive heuristics such as degree centrality compare well with the greedy approach. We also study the influence limitation problem in the presence of missing data where the current states of nodes in the network are only known with a certain probability and show that prediction in this setting is a supermodular problem. We propose a prediction algorithm that is based on generating random spanning trees and evaluate the performance of this approach. The experiments reveal that using the prediction algorithm, we are able to tolerate about 90 missing data before the performance of the algorithm starts degrading and even with large amounts of missing data the performance degrades only to 75 of the performance that would be achieved with complete data. In the Prophet Secretary problem, samples from a known set of probability distributions arrive one by one in a uniformly random order, and an algorithm must irrevocably pick one of the samples as soon as it arrives. The goal is to maximize the expected value of the sample picked relative to the expected maximum of the distributions. This is one of the most simple and fundamental problems in online decision making that models the process selling one item to a sequence of costumers. For a closely related problem called the Prophet Inequality where the order of the random variables is adversarial, it is known that one can achieve in expectation 1 2 of the expected maximum, and no better ratio is possible. For the Prophet Secretary problem, that is, when the variables arrive in a random order, (2015) showed that one can actually get 1-1 e of the maximum. The 1-1 e bound was recently extended to more general settings by (2018). Given these results, one might be tempted to believe that 1-1 e is the correct bound. We show that this is not the case by providing an algorithm for the Prophet Secretary problem that beats the 1-1 e bound and achieves 1-1 e+1 400 times the expected maximum. We also prove a hardness result on the performance of algorithms under a natural restriction which we call deterministic distribution-insensitivity.