A Social Choice Function (SCF), or an election rule, aggregates the preferences of all members of a society towards a common social choice. The study of SCFs dates back to the works of Condorcet in the 18th century, and has expanded greatly in the last decades.
One of the obviously desired properties of an SCF is strategy-proofness: a voter should not gain from voting strategically, that is, from reporting false preferences instead of his true preferences (such voting is called in the sequel manipulation). However, it turns out that this property cannot be obtained by any reasonable SCF. This was shown in a landmark theorem of Gibbard [Gib73] and Satterthwaite [Sat75]:
Theorem 1.1 (Gibbard, Satterthwaite).
Any SCF which is not a dictatorship (i.e., the choice is not made according to the preferences of a single voter), and has at least three alternatives in its range, can be strategically manipulated.
The Gibbard-Satterthwaite theorem implies that we cannot hope for full truthfulness in the context of voting, since any reasonable election rule can be manipulated. However, it still may be that such a manipulation is possible only very rarely, and thus can be neglected in practice.
In this paper we prove a quantitative version of the Gibbard-Satterthwaite theorem in the case of three alternatives, showing that if the SCF is not very close to a dictatorship or to having only two alternatives in its range, then even a random manipulation by a randomly chosen voter will succeed with a non-negligible probability. Thus, one cannot hope that manipulations will be negligible for any reasonable election rule.111We note that functions that are very close to being a dictatorship may have a very small number of manipulable profiles (see e.g. [MPS04]). However, all of the prominent SCFs are far from being a dictatorship.
In order to present our results we need a few standard definitions. First we formally define an SCF and a profitable manipulation.
An SCF on voters and alternatives is a function , where is the set of linear orders on alternatives. A set of preferences given by the voters, i.e., , is called a profile. When we want to single out the th voter, we write the profile as , where denotes the preferences of the other voters.
A profitable manipulation by voter at the profile is a preference , such that is preferred by voter (according to his “true” preference order ) over . A profile is called manipulable if there exists a profitable manipulation for some voter at that profile.
Now we define the quantitative settings we consider. Throughout the paper, we make the impartial culture assumption [Bla58], meaning that the profiles are distributed uniformly.222Note that we cannot hope for an impossibility result for every distribution, e.g. since for every SCF one may consider a distribution on its non-manipulable profiles. Under the uniform probability measure, the distance of from a dictatorship is simply the fraction of values that has to be changed in order to turn into a dictatorship. Similarly, in the case of three alternatives, the distance of from having only two alternatives in its range is the minimal probability that an alternative is elected.
We quantify the probability of manipulation in the following way:
The manipulation power of voter on an SCF , denoted by , is the probability that is a profitable manipulation of by voter at profile , where and are chosen uniformly at random from .
We note that our notion of manipulation power
resembles notions of power and influence which play important roles in voting theory and in theoretical computer science. Specifically, our reliance on the uniform probability distribution makes our notion analogous to theBanzhaf Power Index from voting theory [Fis73], and to Ben-Or and Linial’s notion of influence [BL85]. While the latter two notions coincide for monotone Boolean functions, the notion of Ben-Or and Linial deals also with the influence of coalitions (i.e., larger sets of voters) and with much more general protocols for aggregation. Similarly, the notion of manipulation power of one and more voters can be of interest in much greater generality.
Finally, we define a notion related to Generalized Social Welfare Functions which plays a central role in our proof.
A Generalized Social Welfare Function (GSWF) on voters and alternatives is a function . That is, given the preference orders of the voters, outputs the preferences of the society amongst each pair of alternatives.
A GSWF satisfies the Independence of Irrelevant Alternatives (IIA) condition if the preference of the society amongst any pair of alternatives depends only on the individual preferences between and , and not on other alternatives.
Now we are ready to state our main theorem.
There exist universal constants such that for every and any the following holds:
If is an SCF on voters and three alternatives, such that the distance of from a dictatorship and from having only two alternatives in its range is at least , then
If, in addition, is neutral (that is, invariant under permutation of the alternatives), then:
We note that the value of the constant obtained in our proof of Theorem 1.5 is extremely low (see Remark 5.2), and thus the first assertion applies only in the asymptotic setting. Unlike the value of , the obtained value of is reasonable.
The proof of the theorem consists of three steps:
Reduction from low manipulation power to low dependence on irrelevant alternatives: We show that if is small, then in some sense, the question whether the output of is alternative or alternative depends only a little on alternatives other than and . Specifically, the probability of changing the outcome of from to by altering the individual preferences between all other alternatives (and leaving the preferences between and unchanged) is low. This reduction is obtained by a directed isoperimetric inequality, which we prove using the FKG correlation inequality [FKG71] (or, more precisely, using Harris’ inequality [Har60]).
Reduction from an SCF with low dependence on irrelevant alternatives to a GSWF with a low paradox probability: We show that given an SCF on three alternatives with low dependence on irrelevant alternatives, one can construct a GSWF on three alternatives which satisfies the IIA condition and has a low probability of paradox (a paradox occurs if for some profile, the society prefers over , over and over ). Furthermore, the distance of from dictatorship and from always ranking one alternative at the top/bottom is roughly the same as the distance of from dictatorship and from having only two alternatives in its range, respectively.
Applying a quantitative version of Arrow’s impossibility theorem: We use the quantitative versions of Arrow’s theorem333Arrow’s theorem and its quantitative versions are described in Section 4. obtained by Kalai [Kal02] (in the neutral case), Mossel [Mos09], and Keller [Kel10] to show that since has low paradox probability, it has to be close either to a dictatorship or to always ranking one alternative at the top/bottom. Translation of the result to yields the assertion of the theorem. We note that the proofs of the quantitative Arrow theorem are quite complex and use discrete Fourier analysis on the Boolean hypercube and hypercontractive inequalities.
For a fixed value of , Theorem 1.5 implies lower bounds of and on and , respectively. It is easy to see that the lower bound on cannot be improved (up to the value of and the dependence on ), and that the lower bound on cannot be improved to . The latter follows since for the plurality SCF on voters, only an fraction of the profiles can be manipulated at all by any single voter, and thus for all . However, it is still possible that one can obtain a better lower bound than on , and we leave this as our first open problem.
Our second open problem concerns the case of more than three alternatives, . While some parts of our proof extend to this case (see Section 6), we were not able to extend all required parts of the proof. After the preliminary version of this paper was written, several papers tried to resolve this case, and the most notable result is by Isaksson, Kindler, and Mossel [IKM09], who obtained a quantitative Gibbard-Satterthwaite theorem for alternatives, under the only additional assumption of neutrality (see Theorem 2.1 below). However, the case of general SCFs on more than three alternatives is still open, and we leave it as our second open problem. We do conjecture that the theorem generalizes to , perhaps with the exact form of the bound decreasing polynomially in (like the bound obtained by Isaksson et al. in the neutral case).
Our result can be viewed as part of the study of computational complexity as a barrier against manipulation in elections. A brief overview of the work in this direction, several follow-up results, and a short discussion of their implications is presented in Section 2. In Sections 3, 4, and 5 we present the three steps of our proof. Finally, we discuss the case of more than three alternatives in Section 6.
2 Related Work
Since the Gibbard-Satterthwaite theorem was presented, numerous works studied ways to overcome the strategic voting obstacle. The two best-known ways are allowing payments (see, e.g., [Gro73]) and restricting the voters’ preferences (see [Mou80]).
Another way, suggested in 1989 by Bartholdi, Tovey, and Trick [BTT89], is to use a computational barrier. That is, to show that there exist reasonable SCFs for which, while a manipulation does exist, it cannot be found efficiently, and thus in practice, the SCF can be considered strategy-proof. Bartholdi et al. [BTT89] constructed a concrete SCF for which they proved that finding a profitable manipulation is -hard as an algorithmic problem. This approach was further explored by Bartholdi and Orlin [BO91] who proved that manipulation is -hard also for the well-known Single Transferable Vote (STV) election rule. In a related line of research, several papers showed that for various SCFs, the problem of coalitional manipulation (i.e., when a coalition of voters tries to coordinate their ballots in order to get their favorite alternative elected), is -hard for some SCFs, even for a constant number of alternatives (see [CS03, CSL07, EL05, FHH09, HHR07]).
However, while the results in this direction are encouraging, the computational barrier they suggest against manipulation may be practically insufficient. This is because all the results study the worst case complexity of manipulation, and show that manipulation is computationally hard for specific instances. In order to practically prevent manipulation, one should show that it is computationally hard for most instances, or at least in the average case.
In the last few years, several papers considered the hardness of manipulation in the average case [CS06, PR07, PR07b, XC08b, ZPR09].444It should be noted that the success probability of a random manipulation for SCFs with a small number of voters and alternatives was studied a long time ago in a paper of Kelly [Kel93]. Their results suggest that unlike worst-case complexity, it appears that various SCFs can be manipulated relatively easily in the average case – that is, for an instance chosen at random according to some typical distribution. However, all these results consider specific families of SCFs, and manipulation by coalitions rather than by individual voters.
Our results also study manipulation in the “average case” by examining the success probability of a random manipulation by a randomly chosen voter, and yield a general impossibility result in the case of three alternatives, showing that for any reasonable SCF, such manipulation succeeds with a non-negligible probability. However, our result does not have direct implications on the study of computational hardness of manipulation, since in the case of a constant number of alternatives, the number of possible manipulations by a single voter is constant, and thus manipulation by a single voter cannot be computationally hard in this setting.
Since the preliminary version of this paper [FKN08] appeared in FOCS’08, three follow-up works generalized its results to more than three alternatives, under various additional constraints.
The first follow-up work is by Xia and Conitzer [XC08], who use similar techniques to show that a random manipulation will succeed with probability of for any SCF on a constant number of alternatives satisfying the following five conditions:
Homogeneity – The outcome of the election does not change if each vote is replaced by copies of it.
Anonymity – The SCF treats all the voters equally.
Non-Imposition – Any alternative can be elected.
Cancelling out – The outcome is not changed by adding the set of all possible linear orders of the alternatives as additional votes.
A complex stability condition (see [XC08] for the exact formulation).
While the conditions look a bit restrictive, Xia and Conitzer show that they hold for several well-known SCFs, including all positional scoring rules, STV, Copeland, Maximin, and Ranked Pairs.
The second follow-up work is by Dobzinski and Procaccia [DP09]. They consider the case of two voters and an arbitrary number of alternatives, and show that if an SCF is -far from a dictatorship and satisfies Pareto optimality (i.e., if both voters prefer alternative over , then is not elected), then a random manipulation will succeed with probability at least . The techniques used in the proof of [DP09] are relatively simple, and the authors suggest that possibly their result can be generalized to any number of voters, by modifying an inductive argument of Svensson [Sve99] that extends the proof of the classical Gibbard-Satterthwaite theorem from two voters to voters, for any .
The most recent, and most notable, work is by Isaksson, Kindler and Mossel [IKM09] who prove a quantitative version of the Gibbard-Satterthwaite theorem for a general number of alternatives, under the only additional assumption of neutrality.
Theorem 2.1 (Isaksson, Kindler, and Mossel).
Let be a neutral SCF on alternatives which is at least -far from a dictatorship. Consider a random manipulation generated by choosing a profile and a manipulating voter at random, and replacing four adjacent alternatives in the preference order of that voter by a random permutation of them. Then
The techniques used by Isaksson et al. are combinatorial and geometric, and contain a generalization of the canonical path method which allows to prove isoperimetric inequalities for the interface of three bodies.
The result of Isaksson et al. shows that for any neutral SCF which is far from a dictatorship, a random manipulation by a single randomly chosen voter will succeed with a non-negligible probability. Thus, a single voter with black-box access to the SCF can find a manipulation efficiently.
However, this result still does not imply that the agenda of using computational hardness as a barrier against manipulation is completely hopeless, for three reasons:
The result relies on the assumption that the votes are distributed uniformly. It is possible to argue that in real-life situations, the distribution of the votes is far from uniform, and thus the result does not apply.
The result applies only to neutral SCFs.
While the result implies that with a non-negligible probability, a random manipulation by a randomly chosen voter succeeds, it is still possible that for most of the voters, manipulation cannot be found efficiently (or even at all), while only for a polynomially small portion of the voters a manipulation can be found efficiently. Thus, it is possible that only a few voters can manipulate efficiently, while most voters cannot.
For an extensive overview of the study of computational complexity as a barrier against manipulation, and a further discussion on the implication of our results and the results of the follow-up works, we refer the reader to the survey [FP10] by Faliszewski and Procaccia.
3 Reduction from Low Manipulation Power to Low Dependence on Irrelevant Alternatives
In this section we show that if is an SCF on three alternatives such that the manipulation power555See Definition 1.3. of the voters on is small, then in some sense, the dependence of on irrelevant alternatives is low. We quantify this notion as follows:
Let be an SCF on three alternatives and let be two alternatives. For a profile , denote by
the vector which represents the preferences of the voters betweenand , where if the th voter prefers over , and otherwise. The dependence of the choice between and on the (irrelevant) alternative is:
where are chosen at random, subject to the restriction .
By the definition, measures how often a change of the individual preferences between the alternative and the alternatives leads to changing the output of from to or vice versa. Thus, the notion measures how much the (irrelevant) alternative affects the question of whether or is elected.
We note that can be also viewed as measuring kind of a manipulation, where all the voters together attempt to change the output of to be rather than by re-choosing at random all their preferences – except for those between and . However, this definition does not require that anyone in particular gains from changing the output.
The result we prove is the following:
Let be an SCF on three alternatives. Then for every pair of alternatives ,
In order to prove Lemma 3.2, we define a certain combinatorial structure and relate it both to and to .
We begin with a convenient way to represent a profile , given the individual preferences between and (denoted by ). Note that for any specific value of , there are exactly possible values of that agree with it. Indeed, the agreement of with fixes the preferences of all voters between and in , and each voter may choose one of three locations for : above both and , below both of them, or between them. Thus, for every fixed we can view the set as isomorphic to . We use to denote an element in this set. Thus, once is fixed, encodes both and . For example, and encodes the preference .
Next, we define two sets which are closely related to the definition of .
For every value of the preferences between and , let
Both and are viewed as residing in the space .
In terms of these definitions, we clearly have:
In order to relate to the sets and , we endow the set with a structure of a directed graph, whose edges correspond to (some of) the profitable manipulations by voter . For each fixed value of , for each , and for each , the graph has three directed edges in direction between the possible values of : , , and . The following definition counts the directed edges going “upward” from a subset of .
Let . The upper edge border of in the th direction, denoted by , is the set of directed edges in the th direction defined above whose tail is in and whose head is not in . That is,
The upper edge border of is .
We now relate to the upper edge borders in the th direction of and .
For any , we have:
We compute a lower bound on by choosing and at random, differing only (possibly) in the preferences of the th voter, and providing a lower bound on the probability that the th coordinate of is a profitable manipulation of . We perform the random choice as follows: First we choose at random , , and . With probability , we have , and the rest of the analysis is conditioned on this event indeed occurring (a conditioning that does not affect the distribution chosen). We next choose , and finally we choose and .
We claim that if , then either is a manipulation of or is a manipulation of .
Indeed, note that by the definition of , the condition implies that when moving from to , voter lowered his relative preference of without changing his ranking of the pair , and this changed the output of from to some other result . We have two possible cases:
If, according to , voter prefers to , then is a manipulation of .
If ranks above , then this is definitely true for too, since when moving from to , ’s rank relative to did not change, whereas it improved relative to . Thus, is a manipulation of .
Thus, in both cases either is a manipulation of or is a manipulation of , as claimed.
The claim implies that every edge in corresponds to a different pair for which the th coordinate of is a profitable manipulation of . Since each such edge is chosen with probability , the total contribution of such pairs to the lower bound on is A similar contribution comes from the case . ∎
Summing the two sides of Equation (2) over , we get:
Now we are ready to establish the relation between and . Recall that Equation (1) above states:
By combination of these two equations, the application of the following proposition to the sets and yields the assertion of Lemma 3.2.
For every pair of disjoint sets , we have:
We start by “shifting” both and upward, using a standard monotonization technique (see, e.g., [Fra87]). The shifting is performed by a process of steps. We denote , and for each , at step we replace by a set of the same size that is monotone in the ’th coordinate (which means that if and then ). This is done by moving every with to have if the obtained element is not already in , and then moving every that remained with to have if the obtained element is not already in . Clearly such steps do not change the size of the set, and thus for all . As usual in such operations, it is not hard to check that the step operation does not increase for any , and in particular, does not destroy the monotonicity in previous indices (see, e.g., [Fra87] for similar arguments). Hence, the sequence is monotone decreasing in for all .
Let and be the sets we obtain after all steps. We claim that , the set of “new” elements added in the monotonization process, satisfies
Indeed, it is clear that every new element added in the th step of the monotonization corresponds to either one or two edges in and these edges are disjoint. Thus, denoting by the number of new elements added in the th step, we get by the monotonicity of the sequence , that:
Similarly, we have
Since both and are monotone in the partial order of the lattice , they are “positively correlated”, by Harris’ theorem [Har60], or by its better known generalization, the FKG inequality [FKG71]. This means that
This completes the proof of the proposition, and thus also of Lemma 3.2. ∎
4 Reduction from an SCF with Low Dependence on Irrelevant Alternatives to an Almost Transitive GSWF
In this section we present a reduction which allows to pass from an SCF with low dependence on irrelevant alternatives to a GSWF to which one can apply a quantitative version of Arrow’s impossibility theorem. In order to present the results, we need a few more definitions related to GSWFs and to the quantitative Arrow theorem.
Recall that a GSWF on alternatives is a function which is given the preference orders of the voters, and outputs the preference of the society amongst each pair of alternatives. The output preference of between and for a given profile is denoted by , where if is preferred over , and is is preferred over . satisfies the IIA condition if for any pair , the function depends only on the vector (which represents the preferences of the voters between and ), and not on other alternatives.
As was shown by Condorcet in 1785, a GSWF based on the majority rule amongst pairs of alternatives can result in a non-transitive outcome, that is, a situation in which there exist alternatives , such that is preferred by the society over , is preferred over , and is preferred over . The seminal impossibility theorem of Arrow [Arr50, Arr63] asserts that such non-transitivity occurs in any “non-trivial” GSWF on at least three alternatives satisfying the IIA condition:
Theorem 4.1 (Arrow).
Consider a GSWF with at least three alternatives. If the following conditions are satisfied:
The IIA condition,
Unanimity — if all the members of the society prefer some alternative over another alternative , then is preferred over in the outcome of ,
is not a dictatorship (that is, the preference of the society is not determined by a single member),
then there exists a profile for which the outcome is non-transitive.
Since we would like to use quantitative versions of Arrow’s theorem on three alternatives, we use the following notation:
For a GSWF on three alternatives, let
The family of all GSWFs on three alternatives satisfying the IIA condition whose output is always transitive (i.e., those trivial GSWFs for which the conclusion of Arrow’s theorem does not apply) was partially characterized by Wilson [Wil72], and fully characterized by Mossel [Mos09]. It consists exactly of all the dictatorships and the anti-dictatorships (i.e., GSWFs whose output is either the preference order of a single voter or its inverse), and the GSWFs which rank a fixed alternative always at the top (or always at the bottom). (See Theorem 6.1 for the exact formulation.) Clearly, all such GSWFs are undesirable from the point of view of Social choice theory, and one may assume that a reasonable GSWF is “far” from being contained in this set. To quantify this notion, we denote
and for a GSWF on three alternatives, denote by
the minimal fraction of output values that should be changed in order to make always transitive, while maintaining the IIA condition. The quantitative versions of Arrow’s theorem which we use in the next section assert that if is small (i.e., is almost transitive), then must be small as well (and thus, is close to the family of “bad” GSWFs).
Another definition that will be used in the proof is the following:
For a GSWF on alternatives, and a profile , we say that an alternative is a Generalized Condorcet Winner (GCW) at profile if for any alternative , we have . A Generalized Condorcet Loser (GCL) is defined similarly.
Now we are ready to present our result.
Let , and let be an SCF on three alternatives, such that:
for all pairs .
is at least -far from a dictatorship and from an anti-dictatorship (i.e., an SCF which always outputs the bottom choice of a fixed voter).
is at least -far from breaching non-imposition. That is, for each alternative , .
Then one can construct a GSWF on three alternatives, such that:
satisfies the IIA condition.
Given , we define the GSWF as follows:
For each pair of alternatives , and a profile , we set if
and if the reverse inequality holds. In the case of equality we break the tie according to the preference of some fixed voter between and .
Intuitively, considers all profiles which agree with on the preferences of the voters between and , and checks whether occurs more often then or the opposite, while ignoring all cases where equals some other alternative. It is clear from the definition that satisfies the IIA condition, and that if is neutral (i.e., invariant under permutation of the alternatives), then is neutral as well.
In order to analyze , we introduce an auxiliary definition:
A profile is called a minority preference on the pair of alternatives if while , or if while . is called a minority preference if it is a minority preference for at least some pair . For a fixed pair of alternatives , denote
It is easy to relate to , using the Cauchy-Schwarz inequality:
For every SCF and every pair of alternatives we have
Given , and a vector representing the preferences of the voters between and , define
In these terms,
Thus, by the Cauchy-Schwarz inequality,
as asserted. ∎
We are now ready to prove that satisfies the desired properties.
Consider a profile that is not a minority preference and denote . Note that by the definition of a minority preference, for all we must have that and thus, is a Generalized Condorcet Winner of at .
This immediately implies that satisfies Assertion 3 of the lemma. Indeed,
In order to prove Assertion 2, let , and let be such that can be transformed to by changing only fraction of the values. We consider four cases:
Case 1: always ranks alternative at the top. In this case, . Note that by the argument above, if is not a minority preference and is a of at then . Hence,
However, by the assumption,
and thus, , as asserted.
Case 2: always ranks alternative at the bottom. In this case, . As in the previous case, if is not a minority preference and is a of at then . Thus,
However, by assumption we have , and thus , as asserted.
Case 3: is a dictatorship of voter . For a profile , denote the top alternative in the preference order of voter by . We have
As in the previous cases, this implies that
However, since by assumption, is at least -far from a dictatorship of voter , we have , and the assertion follows.
Case 4: is an anti-dictatorship of voter . By the same argument as in the previous case, if is the bottom alternative in the preference order of voter , then
However, since is also at least -far from anti-dictatorship of voter , the assertion follows.
This completes the proof of Condition 2 and of Lemma 4.4. ∎
We note that a certain converse of Lemma 4.4 is true as well. If we have a GSWF satisfying the IIA condition such that , then we can define an SCF to be equal to the GCW of if such GCW exists, and to the top choice of a fixed voter if the GCW does not exist. Since satisfies the IIA condition, the event and with can occur only if either or does not have a GCW, and thus, .
5 Application of a Quantitative Arrow Theorem
The only ingredient required for concluding the proof of Theorem 1.5 is a quantitative version of Arrow’s impossibility theorem. In order to get the optimal result for various assumptions on the SCF , we use two such versions, due to Kalai [Kal02], and to Keller [Kel10].
Now we are ready to present the proof of Theorem 1.5.
Let be an SCF on three alternatives, and assume on the contrary that:
The distance of from a dictatorship is at least ,666We note that there is no need to add the condition that is far from an anti-dictatorship, since an SCF which is close to an anti-dictatorship can be clearly manipulated by the “anti-dictator”.
For each alternative , , but
(where is a constant that will be specified below).
. (The second inequality holds for sufficiently small.)
However, for small enough (concretely, where is the constant in Theorem 5.1), this contradicts the first version of Theorem 5.1 above. This proves the first assertion of Theorem 1.5. The second assertion follows similarly using the second version of Theorem 5.1 instead of the first one (note that by the construction of , if is neutral then is neutral as well and thus Kalai’s version of the quantitative Arrow theorem can be applied). This completes the proof of Theorem 1.5. ∎
Since the value of the constant in the first version of Theorem 5.1 is extremely low (i.e., of order ), for certain values of and , a better result can be obtained by using another version of the quantitative Arrow theorem. In that version, obtained by Mossel [Mos09], the lower bound is replaced by . Applying Mossel’s theorem instead of the version we used above, we get the lower bound
where . While this bound depends also on , for certain values of the parameters it is still stronger, due to the bigger value of the constant.
6 SCFs with More than Three Alternatives
In this section we consider SCFs with more than three alternatives. We show that the second step of our proof (reduction from an SCF with low dependence on irrelevant alternatives to an almost transitive GSWF) can be generalized to SCFs on alternatives, and that the third step (application of a quantitative Arrow theorem) can be generalized under an additional assumption of neutrality. However, we weren’t able to generalize the first step (reduction from low manipulation power to low dependence on irrelevant alternatives), and thus we do not obtain any variant of the main theorem for more than three alternatives.
We would like to mention again two related follow-up works. Xia and Conitzer [XC08] showed that the first step of our proof can be generalized to any constant number of alternatives under several additional assumptions (see Section 2). Furthermore, Isaksson et al. [IKM09] obtained a quantitative Gibbard-Satterthwaite theorem for any number of alternatives under a single additional assumption of neutrality, using a different technique.
Despite these two works, we decided to present the partial generalization of our proof to more than three alternatives, hoping that the technique can be extended to obtain a quantitative Gibbard-Satterthwaite theorem without the neutrality assumption.
6.1 Reduction from an SCF with Low Dependence on Irrelevant Alternatives to a GSWF which Almost Always has a Condorcet Winner
In order to present the results of this section, we have to generalize the notions of and defined in Section 4 to GSWFs on alternatives.
The class of all GSWFs on alternatives which satisfy the IIA condition and whose output is always transitive, was partially characterized by Wilson [Wil72], and fully characterized by Mossel [Mos09] in the following theorem:
Theorem 6.1 (Mossel).
The class consists exactly of all GSWFs on alternatives satisfying the IIA condition, for which there exists a partition of the set of alternatives into disjoint sets such that:
For any profile, ranks all the alternatives in above all the alternatives in , for all .
For all such that , the restriction of to the alternatives in is a dictatorship or an anti-dictatorship.
For all such that , the restriction of to the alternatives in is an arbitrary non-constant function of the individual preferences between the two alternatives in .
While the notion makes sense also for GSWFs on alternatives, we use here a different notion which coincides with in the case of three alternatives:
Let be a GSWF on alternatives. The probability that does not have a Generalized Condorcet Winner (GCW) is denoted by
Similarly, denotes the probability that has a GCW.
Under these definitions, Lemma 4.4 generalizes directly to the case of alternatives. We get:
Let , and let be an SCF on alternatives, such that:
for all pairs .
is at least -far from a dictatorship and from an anti-dictatorship.
There exist alternatives , such that
Then one can construct a GSWF on alternatives, such that:
satisfies the IIA condition.
Furthermore, if is neutral, then is neutral as well.
Note that the third condition imposed on , which means that is -far from having only two alternatives in its range, is weaker than being -far from breaching non-imposition (which means that any alternative is elected with probability at least ).
Sketch of Proof. The definition of and the proofs of Assertions 1 and 3 are the same as in the proof of Lemma 4.4. In order to prove Assertion 2, let , and let be such that can be transformed to by changing only fraction of the values. By Theorem 6.1 applied to , the set of alternatives can be partitioned into disjoint sets such that for any profile, ranks all the alternatives in above all the alternatives in , for all . Note that for any alternative , can occur only if either or is a minority preference. Hence,
Therefore, either (as claimed in Assertion 2 of the lemma), or . In the latter case, and thus, by Theorem 6.1, the restriction of to the alternatives in is a dictatorship or an anti-dictatorship. In both cases, the assertion of the lemma is proved in the same way as cases 3 and 4 in the proof of Lemma 4.4.
6.2 Generalization of the Quantitative Arrow Theorem
The quantitative versions of Arrow’s theorem presented by Mossel [Mos09] and Keller [Kel10] apply also to GSWFs on alternatives, and assert that if is not too small, then is also not too small. However, the reduction given by Lemma 6.3 yields a bound on rather than on , and thus we need a lower bound on (the probability of not having a Generalized Condorcet Winner), which may be much lower than the probability of being non-transitive.
In this subsection we prove a generalization of the quantitative Arrow theorem which allows to obtain a lower bound on . However, we require the additional assumption that is neutral (i.e., invariant under permutation of the alternatives), and our proof relies heavily on this assumption.
Before we present the generalization, we recall a few properties of neutral GSWFs. Let be a GSWF on voters and alternatives denoted by . If satisfies the IIA condition, then its output is determined by Boolean functions , which are given the individual preferences between alternatives and and output the preference of the society between them. If, in addition, is neutral, then all the functions are equal, and thus we denote them by a single function , and write . Note that the neutrality assumption implies also that
is an odd function, that is,, and in particular, . We denote the distance of from a dictatorship or an anti-dictatorship by .
We are now ready to present our result. We start with an equivalent formulation of Kalai’s version of the quantitative Arrow theorem [Kal02].
Theorem 6.4 (Kalai).
There exists a constant such that the following holds. Let be a neutral GSWF on voters and alternatives satisfying the IIA condition. Then
We prove the following generalization:
For any , and for every , there exists a constant such that the following holds. Let be a neutral GSWF on voters and alternatives which satisfies the IIA condition. If , then .
Moreover, for , we can take , where is a universal constant.
Before we present the proof of the theorem, we note that if a neutral GSWF on alternatives is at least -far from a dictatorship and from an anti-dictatorship, then . Thus, Theorem 6.5 implies immediately the following corollary.
For any , and for every , there exists a constant such that the following holds. Let be a neutral GSWF on alternatives satisfying the IIA condition. If is at least -far from a dictatorship and from an anti-dictatorship, then .
Moreover, for , we can take , where is a universal constant.
Proof of Theorem 6.5. The case is exactly Kalai’s theorem above. We first give a direct proof of the cases and , and then show a general inductive argument that allows to leverage the result to any .
GSWFs on four alternatives
We begin by considering the case . For , let be the random variable that indicates the event , where the profile is chosen at random. Note that by the neutrality assumption, the probability is precisely four times the probability that alternative is a GCW of . Hence,
Before expanding this equation, we make three observations. First, from the neutrality of it follows that is balanced (i.e., ), and thus, for all