Introduction complexity has been suggested as a means of precluding strategic behavior. Previous...
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Introduction 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 new concepts; we consider elections distributed with respect to junta distributions , which concentrate on hard instances, and introduce a notion of heuristic polynomial time. We use our techniques to prove that scoring protocols are susceptible to manipulation by coalitions, when the number of candidates is constant. This fundamental impossibility result implies that one cannot design a voting rule which is average-case hard to manipulate on the basis of scoring protocols. E ncouraging voters to truthfully reveal their preferences in an election has long been an important issue. In particular, computational Conitzer & Sandholm Complexity of Manipulating Elections with Few Candidates. AAAI, pp. 314- 319, 2002. Elkind & Lipmaa Small Coalitions Cannot Manipulate Voting. FC, pp. 285-297, 2005. Bartholdi & Orlin Single Transferable Vote Resists Strategic Voting. Social Choice and In multiagent settings, agents often have conflicting preferences – reaching consensus may be difficult. Voting is often applied to this end, as it is well-studied and provides theoretical guarantees. In an election, each of the n voters ranks m candidates; the winner is determined according to a voting protocol . Strategic behavior on the part of voters may be a problem when voting is used – and computational complexity may be the solution. Indeed, Conitzer & Sandholm have demonstrated that many prominent voting protocols are NP-hard to manipulate by coalitions. Nevertheless, it may still be the case that manipulation is usually possible. Preliminaries Definition: A Scoring protocol is defined by a vector = 1 ,..., m . Each candidate receives j points for every voter which ranks it in j’th place. The candidate with the highest score is elected. Definition: In the Coalitional- Weighted-Manipulation (CWM) problem, we are given a set of weighted votes S, a set of votes T, and a preferred candidate p. We are asked whether there is a way to cast the votes in T so that p wins. Our Approach We consider distributions over the CWM problem which concentrate on hard instances. Presumably, if the problem can usually be decided w.r.t. such a distribution, then it can usually • Hardness: the restriction of the problem to instances with non- zero probability is NP-hard. • Balance: there is at least constant probability for “yes” and “no” instances. • Dichotomy: all instances are either probable or impossible. • Symmetry: different candidates have same probability of receiving a certain initial score. • Refinement: Manipulators cannot succeed by voting identically. Definition: A distribution over the instances of the CWM problem is a junta distribution iff it satisfies: Definition: An algorithm A is a heuristic polynomial time algorithm for a decision problem, w.r.t. a distribution , if the probability that the algorithm errs is inverse-polynomial, when the instances of the problem are distributed w.r.t. . Definition: We say that a voting protocol P is susceptible to coalitional manipulation if there exists a heuristic polynomial time algorithm for CWM in P, w.r.t. a junta distribution . Main Result Theorem: Let P be a scoring protocol, and let m=O(1). Then P is susceptible to coalitional manipulation. On the other hand, it is known that CWM is NP-hard in scoring protocols where m-1 < m , even when m=O(1). If so, this is an explicit class of important voting protocols (which includes Borda and Veto) which are NP-hard to manipulate in the worst-case,
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Transcript of Introduction complexity has been suggested as a means of precluding strategic behavior. Previous...
Introduction
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 new concepts; we consider elections distributed with respect to junta distributions, which concentrate on hard instances, and introduce a notion of heuristic polynomial time. We use our techniques to prove that scoring protocols are susceptible to manipulation by coalitions, when the number of candidates is constant. This fundamental impossibility result implies that one cannot design a voting rule which is average-case hard to manipulate on the basis of scoring protocols.
E ncouraging voters to truthfully reveal their preferences in an election has long been an important issue. In particular, computational
Conitzer & Sandholm
Complexity of Manipulating Elections with Few Candidates. AAAI, pp. 314-319, 2002.
Elkind & Lipmaa
Small Coalitions Cannot Manipulate Voting. FC, pp. 285-297, 2005.
Bartholdi & Orlin
Single Transferable Vote Resists Strategic Voting. Social Choice and Welfare 8(4):341-354, 1991.
In multiagent settings, agents often have conflicting preferences – reaching consensus may be difficult. Voting is often applied to this end, as it is well-studied and provides theoretical guarantees. In an election, each of the n voters ranks m candidates; the winner is determined according to a voting protocol.
Strategic behavior on the part of voters may be a problem when voting is used – and computational complexity may be the solution. Indeed, Conitzer & Sandholm have demonstrated that many prominent voting protocols are NP-hard to manipulate by coalitions. Nevertheless, it may still be the case that manipulation is usually possible.
Preliminaries
Definition: A Scoring protocol is defined by a vector =1,...,m. Each candidate receives j points for every voter which ranks it in j’th place. The candidate with the highest score is elected.
Definition: In the Coalitional-Weighted-Manipulation (CWM) problem, we are given a set of weighted votes S, a set of votes T, and a preferred candidate p. We are asked whether there is a way to cast the votes in T so that p wins.
Our Approach
We consider distributions over the CWM problem which concentrate on hard instances. Presumably, if the problem can usually be decided w.r.t. such a distribution, then it can usually be decided w.r.t. any distribution.
• Hardness: the restriction of the problem to instances with non-zero probability is NP-hard.
• Balance: there is at least constant probability for “yes” and “no” instances.
• Dichotomy: all instances are either probable or impossible.
• Symmetry: different candidates have same probability of receiving a certain initial score.
• Refinement: Manipulators cannot succeed by voting identically.
Definition: A distribution over the instances of the CWM problem is a junta distribution iff it satisfies:
Definition: An algorithm A is a heuristic polynomial time algorithm for a decision problem, w.r.t. a distribution , if the probability that the algorithm errs is inverse-polynomial, when the instances of the problem are distributed w.r.t. .
Definition: We say that a voting protocol P is susceptible to coalitional manipulation if there exists a heuristic polynomial time algorithm for CWM in P, w.r.t. a junta distribution .
Main Result
Theorem: Let P be a scoring protocol, and let m=O(1). Then P is susceptible to coalitional manipulation.
On the other hand, it is known that CWM is NP-hard in scoring protocols where m-1<m, even when m=O(1). If so, this is an explicit class of important voting protocols (which includes Borda and Veto) which are NP-hard to manipulate in the worst-case, but easy to manipulate in the average-case.
