In computational social choice, one appealing escape from the Gibbard-Satterthwaite theorem [Gibbard1973, Satterthwaite1975] was proposed in [Bartholdi, Tovey, and Trick1989]. Whilst manipulation may always be possible, perhaps it is computationally too difficult to find? Many results have subsequently been proven showing that various voting rules are NP-hard to manipulate [Bartholdi and Orlin1991, Conitzer and Sandholm2003, Elkind and Lipmaa2005, Conitzer, Sandholm, and Lang2007, Faliszewski, Hemaspaandra, and Schnoor2008, Xia et al.2009, Faliszewski, Hemaspaandra, and Schnoor2010] in various senses. However, recent results suggest that computing a manipulation is easy on average or in many cases. Therefore, computational complexity seems to be a weak barrier against manipulation. See [Faliszewski, Hemaspaandra, and Hemaspaandra2010, Faliszewski and Procaccia2010] for some surveys of this recent research.
It is normally assumed that the manipulator has full information about the votes of the non-manipulators. The argument often given is that if it is NP-hard with full information, then it only can be at least as computationally difficult with partial information. However, when there is only one manipulator, computing a manipulation is polynomial for most common voting rules, including all positional scoring rules, Copeland, maximin, and voting trees. The only known exceptions are STV [Bartholdi and Orlin1991] and ranked pairs [Xia et al.2009]. Therefore, it is not clear whether a single manipulator has incentive to lie when the manipulator only has partial information.
In this paper, we study the problem of how one manipulator computes a manipulation based on partial information about the other votes. For example, the manipulator may know that some voters prefer one alternative to another, but might not be able to know all pairwise comparisons for all voters. We suppose the knowledge of the manipulator is described by an information set . This is some subset of possible profiles of the non-manipulators which is known to contain the true profile. Given an information set and a pair of votes and , if for every profile in , the manipulator is not worse off voting than voting , and there exists a profile in such that the manipulator is strictly better off voting , then we say that dominates . If there exists a vote that dominates the true preferences of the manipulator then the manipulator has an incentive to vote untruthfully. We call this a dominating manipulation. If there is no such vote, then a risk-averse manipulator might have little incentive to vote strategically.
We are interested in whether a voting rule is immune to dominating manipulations, meaning that a voter’s true preferences are never dominated by another vote. If is not immune to dominating manipulations, we are interested in whether is resistant, meaning that computing whether a voter’s true preferences are dominated by another vote is NP-hard, or vulnerable, meaning that this problem is in P. These properties depend on both the voting rule and the form of the partial information. Interestingly, it is not hard to see that most voting rules are immune to manipulation when the partial information is just the current winner. For instance, with any majority consistent rule (for example, plurality), a risk averse manipulator will still want to vote for her most preferred alternative. This means that the chairman does not need to keep the current winner secret to prevent such manipulations. On the other hand, if the chairman lets slip more information, many rules stop being immune. With most scoring rules, if the manipulator knows the current scores, then the rule is no longer immune to such manipulation. For instance, when her most preferred alternative is too far behind to win, the manipulator might vote instead for a less preferred candidate who can win.
In this paper, we focus on the case where the partial information is represented by a profile of partial orders, and the information set consists of all linear orders that extend . The dominating manipulation problem is related to the possible/necessary winner problems [Konczak and Lang2005, Walsh2007, Betzler, Hemmann, and Niedermeier2009, Betzler and Dorn2010, Xia and Conitzer2011]. In possible/necessary winner problems, we are given an alternative and a profile of partial orders that represents the partial information of the voters’ preferences. We are asked whether is the winner for some extension of (that is, is a possible winner), or whether is the winner for every extension of (that is, is a necessary winner). We note that in the possible/necessary winner problems, there is no manipulator and represents the chair’s partial information about the votes. In dominating manipulation problems, represents the partial information of the manipulator about the non-manipulators.
We start with the special case where the manipulator has complete information. In this setting the dominating manipulation problem reduces to the standard manipulation problem, and many common voting rules are vulnerable to dominating manipulation (from known results). When the manipulator has no information, we show that a wide range of common voting rules are immune to dominating manipulation. When the manipulator’s partial information is represented by partial orders, our results are summarized in Table 1.
|dominating manipulation111All hardness results hold even when the number of undetermined pairs in each partial order is no more than a constant.|
|STV||Resistant (Proposition 2)|
|Ranked pairs||Resistant (Proposition 2)|
|Borda||Resistant (Theorem 4)|
|Copeland||Resistant (Corollary 2)|
|Voting trees||Resistant (Corollary 2)|
|Maximin||Resistant (Theorem 7)|
|Plurality||Vulnerable (Algorithm 2)|
Our results are encouraging. For most voting rules we study in this paper (except plurality and veto), hiding even a little information makes resistant to dominating manipulation. If we hide all information, then is immune to dominating manipulation. Therefore, limiting the information available to the manipulator appears to be a promising way to prevent strategic voting.
Let be the set of alternatives (or candidates). A linear order on is a transitive, antisymmetric, and total relation on . The set of all linear orders on is denoted by . An -voter profile on consists of linear orders on . That is, , where for every , . The set of all -profiles is denoted by . We let denote the number of alternatives. For any linear order and any , is the alternative that is ranked in the th position in . A voting rule is a function that maps any profile on to a unique winning alternative, that is, . The following are some common voting rules. In this paper, if not mentioned specifically, ties are broken in the fixed order .
(Positional) scoring rules: Given a scoring vector of integers, for any vote and any , let , where is the rank of in . For any profile , let . The rule will select so that is maximized. We assume scores are integers and decreasing. Some examples of positional scoring rules are Borda
, for which the scoring vector is, plurality, for which the scoring vector is , and veto, for which the scoring vector is .
Copeland: For any two alternatives and , we conduct a pairwise election in which we count how many votes rank ahead of , and how many rank ahead of . wins if and only if the majority of voters rank ahead of . An alternative receives one point for each such win in a pairwise election. Typically, an alternative also receives half a point for each pairwise tie, but this will not matter for our results. The winner is the alternative with the highest score.
Maximin: Let be the number of votes that rank ahead of minus the number of votes that rank ahead of in the profile . The winner is the alternative that maximizes .
Ranked pairs: This rule first creates an entire ranking of all the alternatives. In each step, we will consider a pair of alternatives that we have not previously considered; specifically, we choose the remaining pair with the highest . We then fix the order , unless this contradicts previous orders that we fixed (that is, it violates transitivity). We continue until we have considered all pairs of alternatives (hence we have a full ranking). The alternative at the top of the ranking wins.
Voting trees: A voting tree is a binary tree with leaves, where each leaf is associated with an alternative. In each round, there is a pairwise election between an alternative and its sibling : if the majority of voters prefer to , then is eliminated, and is associated with the parent of these two nodes. The alternative that is associated with the root of the tree (i.e. wins all its rounds) is the winner.
Single transferable vote (STV): The election has rounds. In each round, the alternative that gets the lowest plurality score (the number of times that the alternative is ranked in the top position) drops out, and is removed from all of the votes (so that votes for this alternative transfer to another alternative in the next round). The last-remaining alternative is the winner.
For any profile , we let denote the weighted majority graph of , defined as follows. is a directed graph whose vertices are the alternatives. For , if , then there is an edge with weight .
We say that a voting rule is based on the weighted majority graph (WMG), if for any pair of profiles such that , we have . A voting rule is Condorcet consistent if it always selects the Condorcet winner (that is, the alternative that wins each of its pairwise elections) whenever one exists.
3 Manipulation with Partial Information
We now introduce the framework of this paper. Suppose there are non-manipulators and one manipulator. The information the manipulator has about the votes of the non-manipulators is represented by an information set . The manipulator knows for sure that the profile of the non-manipulators is in . However, the manipulator does not know exactly which profile in it is. Usually is represented in a compact way. Let denote the set of all possible information sets in which the manipulator may find herself.
Suppose the voting rule is .
If the manipulator has no information, then the only information set is . Therefore .
If the manipulator has complete information, then .
If the manipulator knows the current winner (before the manipulator votes), then the set of all information sets the manipulator might know is , where for any , .
Let denote the true preferences of the manipulator. Given a voting rule and an information set , we say that a vote dominates another vote , if for every profile , we have , and there exists such that . In other words, when the manipulator only knows the voting rule and the fact that the profile of the non-manipulators is in (and no other information), voting is a strategy that dominates voting . We define the following two decision problems.
Given a voting rule , an information set , the true
preferences of the manipulator, and two votes and , we
are asked the following two questions.
Does dominate ? This is the domination problem.
Does there exist a vote that dominates ? This is the dominating manipulation problem.
We stress that usually is represented in a compact way, otherwise the input size would already be exponentially large, which would trivialize the computational problems. Given a set of information sets, we say a voting rule is immune to dominating manipulation, if for every and every that represents the manipulator’s preferences, is not dominated; is resistant to dominating manipulation, if dominating manipulation is NP-hard (which means that is not immune to dominating manipulation, assuming PNP); and is vulnerable to dominating manipulation, if is not immune to dominating manipulation, and dominating manipulation is in P.
4 Manipulation with Complete/No Information
In this section we focus on the following two special cases: (1) the manipulator has complete information, and (2) the manipulator has no information. It is not hard to see that when the manipulator has complete information, dominating manipulation coincides with the standard manipulation problem. Therefore, our framework of dominating manipulation is an extension of the traditional manipulation problem, and we immediately obtain the following proposition from the Gibbard-Satterthwaite theorem [Gibbard1973, Satterthwaite1975].
When and the manipulator has full information, a voting rule satisfies non-imposition and is immune to dominating manipulation if and only if it is a dictatorship.
The following proposition directly follows from the computational complexity of the manipulation problems for some common voting rules [Bartholdi, Tovey, and Trick1989, Bartholdi and Orlin1991, Conitzer, Sandholm, and Lang2007, Zuckerman, Procaccia, and Rosenschein2009, Xia et al.2009].
When the manipulator has complete information, STV and ranked pairs are resistant to dominating manipulation; all positional scoring rules, Copeland, voting trees, and maximin are vulnerable to dominating manipulation.
Next, we investigate the case where the manipulator has no information. We obtain the following positive results. Due to the space constraint, most proofs are omitted.
When the manipulator has no information, any Condorcet consistent voting rule is immune to dominating manipulation.
When the manipulator has no information, Borda is immune to dominating manipulation.
When the manipulator has no information and , any positional scoring rule is immune to dominating manipulation.
These results demonstrate that the information that the manipulator has about the votes of the non-manipulators plays an important role in determining strategic behavior. When the manipulator has complete information, many common voting rules are vulnerable to dominating manipulation, but if the manipulator has no information, then many common voting rules become immune to dominating manipulation.
5 Manipulation with Partial Orders
In this section, we study the case where the manipulator has partial information about the votes of the non-manipulators. We suppose the information is represented by a profile composed of partial orders. That is, the information set is . We note that the two cases discussed in the previous section (complete information and no information) are special cases of manipulation with partial orders. Consequently, by Proposition 1, when the manipulator’s information is represented by partial orders and , no voting rule that satisfies non-imposition and non-dictatorship is immune to dominating manipulation. It also follows from Theorem 2 that STV and ranked pairs are resistant to dominating manipulation. The next theorem states that even when the manipulator only misses a tiny portion of the information, Borda becomes resistant to dominating manipulation.
domination and dominating manipulation with partial orders are NP-hard for Borda, even when the number of unknown pairs in each vote is no more than .
Proof. We only prove that domination is NP-hard, via a reduction from Exact Cover by 3-Sets (x3c). The proof for dominating manipulation is omitted due to space constraint. The reduction is similar to the proof of the NP-hardness of the possible winner problems under positional scoring rules in [Xia and Conitzer2011].
In an x3c instance, we are given two sets
, , where for any
, and . We are asked whether
there exists a subset of such that each element in
is in exactly one of the 3-sets in . We construct a domination instance as follows.
Alternatives: , where is an auxiliary alternative. Therefore, . Ties are broken in the following order: .
Manipulator’s preferences and possible manipulation: . We are asked whether is dominated by .
The profile of partial orders: Let , defined as follows.
First part () of the profile: For each ,
We define a partial order as follows.
That is, is a partial order that agrees with , except that the pairwise relations
between and are not determined (and these are the
only unknown relations).
Second part () of the profile: We first give the properties that we need to satisfy, then show how to construct in polynomial time. All votes in are linear orders that are used to adjust the score differences between alternatives. Let . That is, () is an extension of (in fact, is the set of linear orders that we started with to obtain , before removing some of the pairwise relations). Let . is a set of linear orders such that the following holds for :
(1) For any , , .
(2) For any , the scores of and are higher than the score of in any extension of and in any extension of .
(3) The size of is polynomial in .
We now show how to construct in polynomial time. For any
alternative , we define the following two votes:
, where is the reversed order of
the alternatives in . We note that for any
alternative , and . Let
. is composed
of the following parts:
(1) copies of .
(2) copies of .
(2) For each , there are copies of .
We next prove that is dominated by if and only if is the winner in at least one extension of . We note that for any , the score of in is the same as the score of in . The score of in is lower than the score of in . Therefore, for any extension of , if , then (because cannot win). Hence, for any extension of , voting can result in a different outcome than voting only if . If there exists an extension of such that , then we claim that the manipulator is strictly better off voting than voting . Let denote the extension of in . Then, because the total score of is no more than the total score of , is ranked lower than at least times in . Meanwhile, for each , is not ranked higher than more than one time in , because otherwise the total score of will be strictly higher than the total score of . That is, the votes in where make up a solution to the x3c instance. Therefore, the only possibility for to win is for the scores of , and all alternatives in to be the same (so that wins according to the tie-breaking mechanism). Now, we have . Because , the manipulator is better off voting . It follows that is dominated by if and only if there exists an extension of where is the winner.
The above reasoning also shows that is dominated by if and only if the x3c instance has a solution. Therefore, domination is NP-hard.
Theorem 4 can be generalized to a class of scoring rules similar to the class of rules in Theorem 1 in [Xia and Conitzer2011], which does not include plurality or veto. In fact, as we will show later, plurality and veto are vulnerable to dominating manipulation.
We now investigate the relationship to the possible winner problem in more depth. In a possible winner problem , we are given a voting rule , a profile composed of partial orders, and an alternative . We are asked whether there exists an extension of such that . Intuitively, both domination and dominating manipulation seem to be harder than the possible winner problem under the same rule. Next, we present two theorems, which show that for any WMG-based rule, domination and dominating manipulation are harder than two special possible winner problems, respectively.
We first define a notion that will be used in defining the two special possible winner problems. For any instance of the possible winner problem , we define its WMG partition as follows. For any , let . That is, is composed of all WMGs of the extensions of , where the winner is . It is possible that for some , is empty. For any subset , we let denote the weighted majority graph where for each , there is an edge with weight , and these are the only edges in . We are ready to define the two special possible winner problems for WMG-based voting rules.
Let be an alternative and let be a nonempty subset of . For any WMG-based voting rule , we let PW denote the set of possible winner problems satisfying the following conditions:
For any , .
For any and any , .
For any , .
We recall that and are elements in the WMG partition of the possible winner problem.
Let be an alternative and let be a nonempty subset of . For any WMG-based voting rule , we let PW denote the problem instances of PW, where for any , .
Let be a WMG-based voting rule. There is a polynomial time reduction from PW to domination with partial orders, both under .
Proof. Let be a PW instance. We construct the following domination instance. Let the profile of partial orders be , , and . Let be an extension of . It follows that , and . Therefore, the manipulator can change the winner if and only if , which is equivalent to being a possible winner. We recall that by the definition of PW, for any , ; for any and any , ; and . It follows that (=) is dominated by if and only if the PW instance has a solution.
Theorem 5 can be used to prove that domination is NP-hard for Copeland, maximin, and voting trees, even when the number of undetermined pairs in each partial order is bounded above by a constant. It suffices to show that for each of these rules, there exist and such that PW is NP-hard. To prove this, we can modify the NP-completeness proofs of the possible winner problems for Copeland, maximin, and voting trees by Xia and Conitzer [Xia and Conitzer2011]. These proofs are omitted due to space constraint.
domination with partial orders is NP-hard for Copeland, maximin, and voting trees, even when the number of unknown pairs in each vote is bounded above by a constant.
Let be a WMG-based voting rule. There is a polynomial-time reduction from PW to dominating manipulation with partial orders, both under .
Proof. The proof is similar to the proof for Theorem 5. We note that is the manipulator’s top-ranked alternative. Therefore, if is not a possible winner, then () is not dominated by any other vote; if is a possible winner, then is dominated by .
Similarly, we have the following corollary.
dominating manipulation with partial orders is NP-hard for Copeland and voting trees, even when the number of unknown pairs in each vote is bounded above by a constant.
It is an open question if PW with partial orders is NP-hard for maximin. However, we can directly prove that dominating manipulation is NP-hard for maximin by a reduction from x3c.
dominating manipulation with partial orders is NP-hard for maximin, even when the number of unknown pairs in each vote is no more than .
For plurality and veto, there exist polynomial-time algorithms for both domination and dominating manipulation. Given an instance of domination, denoted by , we say that is a possible improvement of , if there exists an extension of such that . It follows that dominates if and only if is a possible improvement of , and is not a possible improvement of . We first introduce an algorithm (Algorithm 1) that checks whether is a possible improvement of for plurality.
Let (resp., ) denote the top-ranked alternative in (resp., ). We will check whether there exists , with , and an extension of , such that if the manipulator votes for , then the winner is , whose plurality score in is , and if the manipulator votes for , then the winner is . We note that if such exist, then either or (or both hold). To this end, we solve multiple maximum-flow problems defined as follows.
Let denote a set of alternatives. Let be an arbitrary vector composed
of natural numbers such that . We define
a maximum-flow problem as follows.
For any , there is an edge from to with capacity .
For any and , there is an edge with capacity if and only if can be ranked in the top position in at least one extension of .
For any , there is an edge with capacity .
For any , there is an edge with capacity .
There is an edge with capacity .
For example, is illustrated in Figure 1.
It is not hard to see that has a solution whose value is if and only if there exists an extension of , such that (1) for each , the plurality of is exactly , and (2) for each , the plurality of is no more than . Now, for any pair of alternatives such that and either or , we define the set of admissible maximum-flow problems to be the set of maximum flow problems where , and if has a solution, then the manipulator can improve the winner by voting for . Details are omitted due to space constraint. Algorithm 1 solves all maximum-flow problems in to check whether is a possible improvement of .
The algorithm for dominating manipulation for plurality simply runs Algorithm 2 times. In the input we always have that , and for each alternative in , we solve an instance where that alternative is ranked first in . If in any step is dominated by , then there is a dominating manipulation; otherwise is not dominated by any other vote. The algorithms for domination and dominating manipulation for veto are similar. We omit the details due to space constraint.
6 Future Work
Analysis of manipulation with partial information provides insight into what needs to be kept confidential in an election. For instance, in a plurality or veto election, revealing (perhaps unintentionally) part of the preferences of non-manipulators may open the door to strategic voting. An interesting open question is whether there are any more general relationships between the possible winner problem and the dominating manipulation problem with partial orders. It would be interesting to identify cases where voting rules are resistant or even immune to manipulation based on other types of partial information, for example, the set of possible winners. We may also consider other types of strategic behavior with partial information in our framework, for example, coalitional manipulation, bribery, and control. We are currently working on proving completeness results for higher levels of the polynomial hierarchy for problems similar to those studied in this paper.
Vincent Conitzer and Lirong Xia acknowledge NSF CAREER 0953756 and IIS-0812113, and an Alfred P. Sloan fellowship for support. Toby Walsh is supported by the Australian Department of Broadband, Communications and the Digital Economy, the ARC, and the Asian Office of Aerospace Research and Development (AOARD-104123). Lirong Xia is supported by a James B. Duke Fellowship. We thank all AAAI-11 reviewers for their helpful comments and suggestions.
- [Bartholdi and Orlin1991] Bartholdi, III, J., and Orlin, J. 1991. Single transferable vote resists strategic voting. SCW 8(4):341–354.
- [Bartholdi, Tovey, and Trick1989] Bartholdi, III, J.; Tovey, C.; and Trick, M. 1989. The computational difficulty of manipulating an election. SCW 6(3):227–241.
- [Betzler and Dorn2010] Betzler, N., and Dorn, B. 2010. Towards a dichotomy for the possible winner problem in elections based on scoring rules. JCSS 76(8):812–836.
- [Betzler, Hemmann, and Niedermeier2009] Betzler, N.; Hemmann, S.; and Niedermeier, R. 2009. A multivariate complexity analysis of determining possible winners given incomplete votes. In Proc. IJCAI, 53–58.
- [Conitzer and Sandholm2003] Conitzer, V., and Sandholm, T. 2003. Universal voting protocol tweaks to make manipulation hard. In Proc. IJCAI, 781–788.
- [Conitzer, Sandholm, and Lang2007] Conitzer, V.; Sandholm, T.; and Lang, J. 2007. When are elections with few candidates hard to manipulate? JACM 54(3):1–33.
- [Elkind and Lipmaa2005] Elkind, E., and Lipmaa, H. 2005. Hybrid voting protocols and hardness of manipulation. In Proc. ISAAC, 24–26.
- [Faliszewski and Procaccia2010] Faliszewski, P., and Procaccia, A. D. 2010. AI’s war on manipulation: Are we winning? AI Magazine 31(4):53–64.
- [Faliszewski, Hemaspaandra, and Hemaspaandra2010] Faliszewski, P.; Hemaspaandra, E.; and Hemaspaandra, L. A. 2010. Using complexity to protect elections. Commun. ACM 53:74–82.
- [Faliszewski, Hemaspaandra, and Schnoor2008] Faliszewski, P.; Hemaspaandra, E.; and Schnoor, H. 2008. Copeland voting: Ties matter. In Proc. AAMAS, 983–990.
- [Faliszewski, Hemaspaandra, and Schnoor2010] Faliszewski, P.; Hemaspaandra, E.; and Schnoor, H. 2010. Manipulation of copeland elections. In Proc. AAMAS, 367–374.
- [Gibbard1973] Gibbard, A. 1973. Manipulation of voting schemes: A general result. Econometrica 41:587–602.
- [Konczak and Lang2005] Konczak, K., and Lang, J. 2005. Voting procedures with incomplete preferences. In Multidisciplinary Workshop on Advances in Preference Handling.
- [Satterthwaite1975] Satterthwaite, M. 1975. Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions. JET 10:187–217.
- [Walsh2007] Walsh, T. 2007. Uncertainty in preference elicitation and aggregation. In Proc. AAAI, 3–8.
- [Xia and Conitzer2011] Xia, L., and Conitzer, V. 2011. Determining possible and necessary winners under common voting rules given partial orders. To appear in JAIR.
- [Xia et al.2009] Xia, L.; Zuckerman, M.; Procaccia, A. D.; Conitzer, V.; and Rosenschein, J. 2009. Complexity of unweighted coalitional manipulation under some common voting rules. In Proc. IJCAI, 348–353.
- [Zuckerman, Procaccia, and Rosenschein2009] Zuckerman, M.; Procaccia, A. D.; and Rosenschein, J. S. 2009. Algorithms for the coalitional manipulation problem. Artificial Intelligence 173(2):392–412.