Manipulation of Copeland Elections Piotr Faliszewski AGH University of Science and Technology, Poland faliszew@agh.edu.pl Edith Hemaspaandra Rochester Institute of Technology, USA eh@cs.rit.edu Henning Schnoor Christian-Albrechts-Universität Kiel, Germany schnoor@ti.informatik.uni- kiel.de ABSTRACT We resolve an open problem regarding the complexity of unweighted coalitional manipulation, namely, the complex- ity of Copeland α -manipulation for α ∈{0, 1}. Copeland α , 0 ≤ α ≤ 1, is an election system where for each pair of can- didates we check which one is preferred by more voters (i.e., we conduct a head-to-head majority contest) and we give one point to this candidate and zero to the other. However, in case of a tie both candidates receive α points. In the end, candidates with most points win. It is known [13] that Copeland α -manipulation is NP-complete for all rational α’s in (0, 1) −{0.5} (i.e., for all the reasonable cases except the three truly interesting ones). In this paper we show that the problem remains NP-complete for α ∈{0, 1}. In addi- tion, we resolve the complexity of Copeland α -manipulation for each rational α ∈ [0, 1] for the case of irrational voters. Categories and Subject Descriptors I.2.11 [Artificial Intelligence]: Distributed Artificial In- telligence—Multiagent Systems General Terms Algorithms, Theory Keywords preferences, computational complexity, multiagent systems 1. INTRODUCTION The complexity of manipulation in various voting systems is one of the most thoroughly studied topics in computational social choice. Tremendous progress achieved in the last few years have left only very few voting systems for which the complexity of coalitional manipulation is unknown. One such system is Copeland α , for α ∈{0, 0.5, 1}. The idea of Copeland α elections is that for each pair of candidates we compare which one is preferred by more candidates and we give one point to that candidate and zero points to the other one. However, if the result of such a head-to-head contest is a tie, both candidates receive α points. Faliszewski, Hemas- paandra, and Schnoor [13] have shown that coalitional ma- Cite as: Proc. of 9th Int. Conf. on Autonomous Agents and Multiagent Systems (AAMAS 2010) c nipulation is NP-hard for rational α’s in (0, 1) −{0.5}. How- ever, exactly the cases of α ∈{0, 0.5, 1} are the really inter- esting ones. For α ∈{0, 0.5} we get exactly the two systems that are known under the name Copeland, 1 and α = 1 gives a system that is sometimes called Llull. Copeland voting is named after A. H. Copeland who proposed the method over 50 years ago in a lecture [6], but the method is very natural and its variants were often reinvented throughout history (e.g., by Jech and by Zermelo), with the earliest record go- ing 700 years back, to Ramon Llull, a 13th century mystic and philosopher (see, e.g., [18]). In this paper we resolve the complexity of coalitional ma- nipulation in Copeland 1 and in Copeland 0 . In addition, we show why the proof approach of Faliszewski, Hemaspaan- dra, and Schnoor [13] could not have succeeded for these cases. We also derive the exact complexity of manipulation in Copeland α for the case of irrational voters. Manipulation in elections means that some voter (or, a group of voters) decides to vote in a way that does not re- flect his or her (their) true preference, but that does guar- antee an outcome of the election that he or she (they) prefer to the one that their true votes would yield. Unfortunately, the famous Gibbard-Satterthwaite theorem [17, 25] says that essentially every practically useful voting system sometimes creates incentives for voters to attempt manipulation. As a response to this depressing result, Bartholdi, Tovey, Trick, and Orlin [2, 1] suggested that even though manipulation might be possible in principle, for some election systems finding a successful manipulation might be computationally so expensive as to prevent any possibility of finding a ma- nipulative vote, short of luckily guessing one. They have set off to find such voting rules for the case of a single manip- ulator and they indeed showed that so-called second-order Copeland is NP-hard to manipulate by a single voter (see [2]) and so is STV (see [1]). The issue of manipulation in elections is particularly rele- vant for researchers working on multiagent systems. A situ- ation where agents need to make a joint decision often arises in multiagent systems and voting is one of the most natural ways of making such decisions. For example, Ephrati and Rosenschein [10] suggest a way of using voting in multiagent planning problems, Ghosh et al. [16] use voting to develop a recommender system, and Dwork et al. [8] show how voting is useful in designing a metasearch engine for the web. How- ever, voters—in particular when they are software agents— 1 When Copeland is mentioned without the α argument, it typically refers to Copeland 0.5 , but in some papers it means Copeland 0 (see, e.g., [24] and an early version of [11]). 367 367-374