Evaluation of Election Outcomes under Uncertainty Noam Hazon1, Yonatan Aumann1, Sarit Kraus1, Michael Wooldridge2 1Department of Computer Science 2Department of Computer Science Bar-Ilan University University of Liverpool Israel United Kingdom {hazonn,aumann,sarit}@cs.biu.ac.il [email protected] ABSTRACT a computational perspective, many interesting theoretical We investigate the extent to which it is possible to evaluate questions arise. Perhaps the most natural question from a the probability of a particular candidate winning an elec- computer science perspective is: are the voting protocols to tion, given imperfect information about the preferences of select a winning outcome e±ciently computable, given all the electorate. We assume that for each voter, we have a the agents preferences? Fortunately, it seems that relatively probability distribution over a set of preference orderings. few voting protocols are hard to compute in this sense [4]. Thus, for each voter, we have a number of possible prefer- Perhaps more intriguing are questions related to the com- ence orderings { we do not know which of these orderings plexity of manipulating a voting procedure. It can often be actually represents the voters' preferences, but we know for computationally infeasible for an agent to compute the most each one the probability that it does. We give a polynomial bene¯cial manipulation [3], implying that while manipula- algorithm to solve the problem of computing the probability tion is possible in theory, it is infeasible in practice. The that a given candidate will win when the number of candi- complexity of manipulation was studied in [3, 2, 6] under dates is a constant. However, when the number of candi- the assumption that the number of outcomes is unbounded, dates is not bounded, we prove that the problem becomes while [5, 7] analyzed the complexity of manipulation with #P-Hard for the Plurality, Borda, and Copeland voting pro- a constant number of outcomes. However, most of these tocols. We further show that even evaluating if a candidate results assume perfect information about voter preferences, has any chance to win is NP-Complete for the Plurality vot- which is surely a very unrealistic assumption in real world ing protocol, in the weighted voters case. We give a poly- settings. nomial algorithm for this problem when the voters' weights In this work, we investigate voting systems under an im- are equal. perfect information model. We assume that what is known about an electorate is the following. For each voter, we have a probability distribution over a set of preference orderings. Categories and Subject Descriptors The idea is that although we do not know a voter's prefer- I.2.11 [Arti¯cial Intelligence]: Distributed Arti¯cial In- ence ordering exactly, we know that it is one of a set of possi- telligence|Coherence and coordination, Intelligent agents; ble orderings (typically a subset of the overall set of possible F.2.1 [Analysis of Algorithms and Problem Complex- preference orders), and we have a probability distribution ity]: Numerical Algorithms and Problems; J.4 [Social and over these. This information may be estimated using histor- Behaviorial Sciences]: [Economics] ical data. In this setting, the following fundamental question arises: given such an incomplete information model of voter General Terms preferences and a particular voting system, how hard is it to compute the probability that a particular candidate will Algorithms, Economics, Theory win? To the best of our knowledge, this question is not addressed in the existing literature1. Keywords The motivation for investigating this question is not merely Computational social choice, Voting protocols theoretical interest (which is, of course, by itself a legitimate thing). In many situations, it might be bene¯cial to try to foresee the probability of an outcome being chosen using 1. INTRODUCTION only partial knowledge about the other agents preferences, In many multi-agent systems, it is desirable to have a mech- which is modeled by a probability distribution as we have anism which enables the agents within the system to make described. One area is the avoidance of strategic voting by a collective decision on a given issue. The mechanism by a coalition of manipulators. Suppose that agent A wants to which such a collective decision is made is typically a vot- vote for an outcome which is its most preferred one. An- ing procedure. When considering voting procedures from other manipulator agent, B, could try to convince A that Cite as: Evaluation of Election Outcomes under Uncertainty, Noam Ha- its outcome does not have any chance to be the winner so zon, Yonatan Aumann, Sarit Kraus and Michael Wooldridge, Proc. of 7th he should directly vote for his second preferred outcome; Int. Conf. on Autonomous Agents and Multiagent Systems (AAMAS 2008), Padgham, Parkes, Müller and Parsons (eds.), May, 12- 1 16.,2008,Estoril,Portugal,pp.959-966. The exception is the work of [5] but their result holds only Copyright °c 2008, International Foundation for Autonomous Agents and for weighted voters with weights that are not bounded by Multiagent Systems (www.ifaamas.org). All rights reserved. P oly(n) as we will show later. 959 otherwise this outcome will also lose to A's least preferred inators); ­ = f!1;:::;!mg is a non-empty set of outcomes, candidate. Due to lack of exact knowledge how the other or candidates { the things the voters are trying to decide agents will vote, A may be convinced by B. Alternatively, over; and Â= fÂ1;:::; Âng is a non-empty set of preference A can estimate the other agent's probabilities to vote for the relations, Âi⊆ ­ £ ­ is a (strict) preference relation over ­, outcomes, by asking people who know them, or by using the for each i 2 V , which is usually private to i. history of their former votes on the same issue. The ability The preference aggregation problem is that of combining to calculate the probability of an outcome winning should the preference relations Âi to obtain a social preference or- then help A to decide whether B has a valid point. der, and the general problem of social choice theory is to This ability to calculate the probability of an outcome ¯nd some way of aggregating preference relations in such a winning might also be useful in other domains. For example, way that certain principles (such as the Pareto condition) (and somewhat more speculatively), consider large multi- are satis¯ed [1]. Generating the social preference order is agent environments, in which there is a need to keep com- commonly done by a voting system which speci¯es the form munication to a minimum. The voting process inevitably of the ballot, the set of allowable votes, and the voting pro- requires communication between the election o±cer and the tocol (an algorithm for determining the outcome). We are voters in order to elicit their preferences. However, one way concerned with settings in which we simply want to select to reduce the communication load is to calculate the prob- one outcome from ­, and the voting protocol runs in poly- abilities on the agents preferences from their voting history nomial time. and then calculate the probability of each outcome to win: We now review some common voting systems in the case the winning outcome is then the one which gets the high- of un-weighted votes (i.e., the case where wi = 1 for all i). est probability to be the winner. In this way, we simulate The evaluation of a voting protocol for weighted votes is a voting process by choosing the successful outcome with- done by simply replacing the vote of each voter i with wi out the need to use communication at all. (This method identical un-weighted votes. In general, voting systems can might be extended to a more sophisticated protocol which be classi¯ed based on their ballot type. In binary voting uses limited communication by asking only a subset of the systems a voter either votes or does not vote for a given voters about their current preferences, although we do not candidate. In ranked voting systems, each voter ranks the investigate this possibility here.) candidates in order of preference. We represent this ballot We therefore analyze the ability to calculate the probabil- as a vector where the ¯rst candidate is the most preferred ity of an outcome to win in various di®erent settings. We candidate, the second one is the second preferred candidate ¯rst give some background and review some common voting and so on. Condorcet systems (or pairwise systems) are systems in Section 2. We formally de¯ne the above men- a class of ranked voting systems that meet the Condorcet tioned \evaluation" question in De¯nition 1. In Section 3, we criterion. That is, the candidate who, when compared in give a polynomial algorithm to answer the evaluation prob- turn with each of the other candidates, is preferred over the lem if the number of outcomes is a constant number, and we other candidate is always declared to be the winner, if such show that the result of [5] holds only for weighted voting sys- a candidate exists. tems with weights that are not bounded by P oly(n). If the ² Binary voting systems: number of candidates is not bounded, the evaluation prob- lem becomes much harder: we show in Section 4 that even Plurality (aka. ¯rst-past-the-post, relative majority, or for the Plurality, Borda, and Copeland voting protocols the winner-take-all). Each voter votes for one candidate, problem is #P-Hard. We then analyze a simpler question, and the candidate that receives the most votes wins (the CHANCE-EVALUATION problem { De¯nition 8): can (even if it receives less than a majority of votes).