Winning a Tournament by Any Means Necessary

2011

In their groundbreaking paper, Bartholdi, Tovey and Trick [1] argued that many well-known voting rules, such as Plurality, Borda, Copeland and Maximin are easy to manipulate. An important assumption made in that paper is that the manipulator's goal is to ensure that his preferred candidate is among the candidates with the maximum score, or, equivalently, that ties are broken in favor of the manipulator's preferred candidate. In this paper, we examine the role of this assumption in the easiness results of [1]. We observe that the algorithm presented in [1] extends to all rules that break ties according to a fixed ordering over the candidates. We then show that all scoring rules are easy to manipulate if the winner is selected from all tied candidates uniformly at random. This result extends to Maximin under an additional assumption on the manipulator's utility function that is inspired by the original model of . In contrast, we show that manipulation becomes hard when arbitrary polynomial-time tie-breaking rules are allowed, both for the rules considered in [1], and for a large class of scoring rules.

Journal of Graph Theory, 1998

We survey results concerning various generalizations of tournaments. The reader will see that tournaments are by no means the only class of directed graphs with a very rich structure. We describe, among numerous other topics mostly related to paths and cycles, results on hamiltonian paths and cycles. The reader will see that although these problems are polynomially solvable for all of the classes described, they can be highly non-trivial, even for these "tournament-like" digraphs.

We study the Maker-Breaker tournament game played on the edge set of a given graph $G$. Two players, Maker and Breaker claim unclaimed edges of $G$ in turns, and Maker wins if by the end of the game she claims all the edges of a pre-defined goal tournament. Given a tournament $T_k$ on $k$ vertices, we determine the threshold bias for the $(1:b)$ $T_k$-tournament game on $K_n$. We also look at the $(1:1)$ $T_k$-tournament game played on the edge set of a random graph ${\mathcal{G}_{n,p}}$ and determine the threshold probability for Maker's win. We compare these games with the clique game and discuss whether a random graph intuition is satisfied.

Combinatorics, Probability and Computing, 2015

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph K n and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament T k on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of T k ; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2 − o(1)) log 2 n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2 − o(1)) log 2 n Breaker can prevent that the underlying graph of Maker's graph contains a k-clique. Moreover the precise value of our lower bound differs from the upper bound only by an additive constant of 12.

2011

Computational complexity of voting manipulation is one of the most actively studied topics in the area of computational social choice, starting with the groundbreaking work of . Most of the existing work in this area, including that of , implicitly assumes that whenever several candidates receive the top score with respect to the given voting rule, the resulting tie is broken according to a lexicographic ordering over the candidates. However, till recently, an equally appealing method of tiebreaking, namely, selecting the winner uniformly at random among all tied candidates, has not been considered in the computational social choice literature. The first paper to analyze the complexity of voting manipulation under randomized tiebreaking is , where the authors provide polynomial-time algorithms for this problem under scoring rules and-under an additional assumption on the manipulator's utilitiesfor Maximin. In this paper, we extend the results of by showing that finding an optimal vote under randomized tie-breaking is computationally hard for Copeland and Maximin (with general utilities), as well as for STV and Ranked Pairs, but easy for the Bucklin rule and Plurality with Runoff.

Journal of Institutional and Theoretical Economics JITE, 2010

We use a real-effort task to investigate the responsiveness of both sabotage and performance in a tournament to: (1) changes in the payoff structure of the tournament, and (2) changes in the identity of competitors over a series of tournaments (rematching versus constant pairings). Constant pairings shows significantly lower performance than rematching because of weak performance by low-ability participants. Constant pairings also depresses the rate at which participants choose sabotage, but causes higher sabotage levels given that the sabotage option is selected. Finally, sabotage is used far less effectively in the constant-pairings than it is in the rematching condition.

American Economic Journal: Microeconomics, 2013

Artificial Intelligence, 2009

We investigate the problem of coalitional manipulation in elections, which is known to be hard in a variety of voting rules. We put forward efficient algorithms for the problem in Scoring rules, Maximin and Plurality with Runoff, and analyze their windows of error. Specifically, given an instance on which an algorithm fails, we bound the additional power the manipulators need in order to succeed. We finally discuss the implications of our results with respect to the popular approach of employing computational hardness to preclude manipulation. * A significantly shorter version of this paper (with most of the proofs omitted) appears in the

2012

Let G be an undirected graph and let T be a tournament on the same vertex set as G. Define the cost of G relative to T to be ∑ u,v∈E(G) (T (u, v) + T (v, u)) + |E|, where T (u, v) denotes the number of two-step paths from u v, in T . In this paper, we determine for several classes of graphs which tournaments minimize the cost. Pelsmajer, et al. [5] conjecture that for each graph there is a transitive tournament that minimizes the graph’s cost. We prove that a transitive tournament minimizes the cost for complete graphs, nearly complete graphs, paths, star graphs, and cycles.

Springer eBooks, 2015

Determining the winner of a Parity Game is a major problem in computational complexity with a number of applications in verification. In a parameterized complexity setting, the problem has often been considered with parameters such as (directed versions of) treewidth or clique-width, by applying dynamic programming techniques. In this paper we adopt a parameterized approach which is more inspired by well-known (non-parameterized) algorithms for this problem. We consider a number of natural parameterizations, such as by Directed Feedback Vertex Set, Distance to Tournament, and Modular Width. We show that, for these parameters, it is possible to obtain recursive parameterized algorithms which are simpler, faster and only require polynomial space. We complement these results with some algorithmic lower bounds which, among others, rule out a possible avenue for improving the bestknown sub-exponential time algorithm for parity games.