A tight quantitative version of Arrows impossibility theorem

Consider an election procedure in which a society of n members selects a ranking amongst k alternatives. In the voting process, each member of the society gives a ranking of the alternatives (the ranking is a full linear ordering; that is, indifference between alternatives is not allowed). The set of the rankings given by the individual members is called a profile. Given the profile, the ranking of the society is determined according to some function, called a generalized social welfare function (GSWF). The GSWF is a function F : (S k ) n → {0, 1} ( k 2 ) , where S k is the set of linear orderings on k elements. In other words, given the profile consisting of linear orderings supplied by the voters, the function determines the preference of the society amongst each of the k 2 pairs of alternatives. If the output of F can be represented as a full linear ordering of the k alternatives, then F is called a social welfare function (SWF). Throughout this paper we consider GSWFs satisfying the Independence of Irrelevant Alternatives (IIA) condition: For any pair of alternatives A and B, the preference of the entire society between A and B depends only on the preference of each individual voter between A and B. This natural condition on GSWFs can be traced back to Condorcet [6]. The Condorcet's paradox demonstrates that if the number of alternatives is at least three and the GSWF is based on the majority rule amongst every pair of alternatives, then there exist profiles for which the voting procedure cannot yield a full order relation. That is, there exist alternatives A, B, and C, such that the majority of the society prefers A over B, the majority prefers B over C, and the majority prefers C over A. Such situation is called non-transitive outcome of the election. In his well-known Impossibility theorem [1], Arrow showed that such paradox occurs for any "reasonable" GSWF satisfying the IIA condition: Theorem 1.1 (Arrow). Consider a generalized social welfare function F 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 A over another alternative B, then A is preferred over B in the outcome of F , • F is not a dictatorship (that is, the preference of the society is not determined by a single member), then the probability of a non-transitive outcome is positive (i.e., there necessarily exists a profile for which the outcome is non-transitive). Since the existence of profiles leading to a non-transitive outcome has significant implications on voting procedures, an extensive research has been conducted in order to evaluate the probability of non-transitive outcome for various GSWFs. Most of the results in this area are summarized in [9]. In addition to its significance in Social Choice theory, this area of research leads to interesting questions in probabilistic and extremal combinatorics (see [19]). In 2002, Kalai [14] suggested an analytic approach to this study. He showed that for GSWFs on three alternatives satisfying the IIA condition, the probability of a non-transitive outcome with respect to a uniform distribution of the individual preferences can be computed by a formula related to the Fourier-Walsh expansion of the GSWF. Using this formula he presented a new proof of Arrow's impossibility theorem under additional assumption of neutrality (i.e., invariance of the GSWF under permutation of the alternatives), and established upper bounds on the probability of non-transitive outcome for specific classes of GSWFs. While providing an analytic proof to Arrow's theorem does not seem such an important goal (as there are several simple proofs of it, see [10]), Kalai aimed at establishing a quantitative version of the theorem. Such version would show that for any ǫ > 0, there exists δ = δ(ǫ) such that if a GSWF on three alternatives satisfies the IIA condition and its probability of non-transitive outcome is at most δ, then the GSWF is at most ǫ-far from being a dictatorship or from breaching the Unanimity condition. Kalai indeed proved such statement for neutral GSWFs on three alternatives, with δ(ǫ) = C • ǫ for a universal constant C. Kalai [15] asked whether his techniques can be extended to general GSWFs, and suggested to use the Bonami-Beckner hypercontractive inequality [4,3] in order to get such an extension. However, Keller [17] showed by an example that a direct extension cannot hold -if there exists δ(ǫ) as above, then it cannot depend linearly on ǫ. Keller asked whether for general GSWFs on three alternatives, a quantitative version holds with δ(ǫ) = C • ǫ 2 . A few months ago Mossel [20] succeeded to prove a quantitative version of Arrow's theorem for general GSWFs on three alternatives. Furthermore, he generalized his result to GSWFs on more than three alternatives, and to more general probability distributions on the individual preferences. Unlike Kalai's techniques, Mossel's proof is quite complex. While Kalai's proof uses only simple analytic tools but no combinatorial tools, Mossel's proof extends and exploits a combinatorial proof of Arrow's theorem given by Barbera [2]. Furthermore, it uses "heavier" analytic tools, including a reverse hypercontractive inequality of Borell [5] and a non-linear invariance principle introduced by Mossel et al. [19]. The only drawback in Mossel's result is the dependence of δ on ǫ: δ(ǫ) = exp(-C/ǫ 21 ) for a universal constant C, which seems far from being optimal. Mossel conjectured that the "correct" dependence of δ on ǫ should be polynomial. 1In this paper we present a tight quantitative version of Arrow's theorem for general GSWFs. We show that the dependence of δ on ǫ is indeed polynomial, and compute the exact dependence, up to logarithmic factors. Before we present our results, we should specify the notion of "the distance of a GSWF on k alternatives satisfying the IIA condition from a dictatorship or from breaching the Unanimity condition". We consider two different definitions of this notion. In both definitions, the underlying probability measure is the uniform measure on (S k ) n (the set of all possible profiles). The first definition measures the distance of the GSWF under examination from the family of GSWFs on k alternatives which satisfy the IIA condition and whose output is always transitive. This family was partially characterized by Wilson [22], and fully characterized by Mossel [20]. It essentially consists of combinations of dictatorships with constant functions (see Section 2.3 for the exact characterization). Definition 1.2. Denote by F k (n) the family of GSWFs on k alternatives which satisfy the IIA condition and whose output is always transitive. For a GSWF F on k alternatives that satisfies the IIA condition, let We note that this is the definition that was used in [20]. Our main result with respect to this definition is the following: Theorem 1.3. There exists an absolute constant C such that for any k and for any GSWF F on k alternatives that satisfies the IIA condition, if the probability of non-transitive outcome in For the second definition, we note that a GSWF F on k alternatives that satisfies the IIA condition actually consists of k 2 independent Boolean functions F ij that represent the choice functions amongst the pairs of alternatives (i, j) (for 1 ≤ i < j ≤ k). The second definition is given in terms of these functions. Definition 1.4. Denote by G 2 (n) the set of constant functions and dictatorships on two alternatives. For a GSWF F on k alternatives that satisfies the IIA condition, let where {F ij } 1≤i 0, if D 1 (F ) = ǫ, then P (F ) ≥ min(C, 1 50000 • ǫ 3 ), for a universal constant C. We shall prove this for where C ′ is the constant in Mossel's Theorem 2.13. Let F be a GSWF on three alternatives satisfying the IIA conditions, and denote the choice functions of F by f, g, and h, as in the proof of Lemma 3.2. If D 1 (F ) ≥ 2 -500003 , then by Theorem 2.13, P (F ) ≥ C. Thus, we may assume that , and similarly for g. This implies that , and thus we can apply Lemma 3.2 to F and get as asserted. Thus, we may assume that G is a dictatorship. The following part of the proof is similar to the proof of Theorem 7.1 in [20]. W.l.o.g., we assume that the output of G is determined by the first voter. We "split" the choice functions according to the first voter. Let and similarly for g and h. Furthermore, for any profile (σ 1 , σ 2 , . . . , σ n ) ∈ S n 3 , denote and similarly for G. The Boolean choice functions of F σ are f a 1 , g a 2 , and h a 3 , where (a 1 , a 2 , a 3 ) ∈ {0, 1} 3 represents the preference σ of the first voter (note that only six of the eight possible combinations of (a 1 , a 2 , a 3 ) represent elements of S 3 ). Denote by f a 1 , ḡa 2 , ha 3 the choice functions of G σ . Since G is a dictatorship of the first voter, the functions f a 1 , ḡa 2 , and ha 3 are constant. Clearly, we have and thus, for all σ ∈ S 3 , and since G σ is constant, this implies that and similarly for g a 2 and h a 3 . The rest of the proof is divided into two cases: • Case B: There exists We first show that Case A leads to a contradiction by constructing a GSWF Then we show that in Case B, the assertion of the theorem follows by applying Lemma 3.2 to the function F σ 0 . Case A: Consider a GSWF G ′ whose choice functions f ′′ , g ′′ , and h ′′ are defined as follows: For and similarly for g ′ and h ′ . We claim that the output of G ′ is always transitive, and thus G ′ ∈ F 3 (n). Indeed, by assumption, for any σ ∈ S 3 , there exists Ḡσ ∈ F 3 (n -1) such that Pr[F σ = Ḡσ ] ≤ D 1 (F )/4. The GSWF Ḡσ cannot be a dictatorship since by Equation (38), the choice functions f a 1 , g a 2 , and h a 3 of F σ satisfy and thus, for any dictatorship H, Therefore, Ḡσ always ranks one alternative at the top/bottom. Denote the choice functions of Ḡσ by f , g, and h, and assume w.l.o.g. that Ḡσ always ranks alternative 1 at the top, and thus f = 1 and h = 0. Since Hence, by the definition of G ′ , its choice functions satisfy f ′′ = 1 and h ′′ = 0, which means that G ′ always ranks alternative 1 at the top, and is thus always transitive. Therefore, G ′ ∈ F 3 (n), and on the other hand, we have contradicting the definition of D 1 (F ). Case B: Let σ 0 ∈ S 3 be such that D 1 (F σ 0 ) > D 1 (F )/4. By Equation (38), the choice functions and thus (in the notation of Lemma 3.2), D ′ 2 (F σ 0 ) ≤ 6D 1 (F ) < 2 -500000 . Hence, we can apply Lemma 3.2 to the GSWF G σ 0 , and get Finally, This completes the proof of the theorem. Theorem 1.3 follows immediately from Theorem 4.1 using Theorem 2.14 (the generic reduction lemma of Mossel). Theorem 4.2. There exists an absolute constant C such that for any GSWF F on three alternatives that satisfies the IIA condition, if the probability of non-transitive outcome in F is at most then D 2 (F ) ≤ ǫ. Proof: By Equation ( 8), it is sufficient to prove that for any ǫ > 0, if D 2 (F ) = ǫ, then for a universal constant C. We shall prove this for where C ′ is the constant in Mossel's Theorem 2.13. Let F be a GSWF on three alternatives satisfying the IIA conditions, and denote the choice functions of F by f, g, and h, as in the proof of Lemma 3.2. First we consider the case D 2 (F ) ≥ 2 -500003 . We show that in general, D 1 (F ) ≥ D 2 (F ), and thus in this case we have D 1 (F ) ≥ D 2 (F ) ≥ 2 -500003 , which by Theorem 2.13 implies that P (F ) ≥ C. Let G ∈ F 3 (n) satisfy Pr[F = G] = D 1 (F ), and denote the Boolean choice functions of G by f ′ , g ′ , and h ′ . Clearly, By Theorem 2.15, G either always ranks one alternative at the top/bottom or is a dictatorship. In the first case, at least two of the functions f ′ , g ′ , and h ′ are constant, and thus Equation (41) implies that at least two of the functions f, g, and h are at most D 1 (F )-far from a constant function. In the latter case, the functions f ′ , g ′ , and h ′ are dictatorships, and thus Equation (41) implies that f, g, and h are at most D 1 (F )-far from a dictatorship. Hence, in both cases, , and thus we can apply Lemma 3.2 to F and get • RHC(1/2, D 2 (F )), as asserted. Thus, we may assume that f is a dictatorship. Assume w.l.o.g. that f is a dictatorship of the first voter. Define the functions F σ , f 0 , f 1 , g 0 , g 1 , h 0 , and h 1 as in the proof of Theorem 4.1, and let Clearly, we have and thus, for Since f 0 and f 1 are constant functions, this implies that The rest of the proof is divided into two cases: • Case B: There exists Case A: In this case, for any σ ∈ S 3 , at least one of the choice functions of F σ is at most D 2 (F )/4-far from a constant function. Note that if f 0 is at most D 2 (F )/4-far from a constant function, then f 1 must be at least 7D 2 (f )/4-far from a constant function, since otherwise, f is less than D 2 (F )-far either from a constant function or from a dictatorship, contradicting the definition of D 2 (F ). The same holds also for the pairs (g 0 , g 1 ) and (h 0 , h 1 ). Thus, the only two possibilities are that either the functions f 1 , g 1 , h 1 or the functions f 0 , g 0 , h 0 are simultaneously at most D 2 (F )/4-far from a constant function. (For example, if f 1 , g 1 , and h 0 are at most D 2 (F )/4-far from a constant function, then f 0 , g 0 , and h 1 are at least 7D 2 (F )/4-far from a constant function, and thus, for the preference σ = (0, 0, 1), we have D ′ 2 (F σ ) ≥ 7D 2 (F )/4, a contradiction. The other possibilities are discarded in a similar way). Assume w.l.o.g. that f 1 , g 1 , and h 1 are at most D 2 (F )/4-far from a constant function. Furthermore, since amongst Consider the GSWF F σ 0 for the preference σ 0 = (1, 1, 0). Since h 0 is at least 7D 2 (F )/4-far from the constant zero function, it follows that and thus, as asserted. Case B: Let σ 0 ∈ S 3 be such that D ′ 2 (F σ 0 ) > D 2 (F )/4. By Equation (43), the choice function and thus, D ′ 2 (F σ 0 ) ≤ 2D 2 (F ) < 2 -500000 . Hence, we can apply Lemma 3.2 to the GSWF G σ 0 , and get Finally, This completes the proof of the theorem. The generalization to k alternatives for all k ≥ 3 follows immediately by applying Theorem 4.2 to any subset of three alternatives. In this section we show that for GSWFs on three alternatives, the assertions of Theorems 1.3 and 1.5 are tight up to logarithmic factors. In all our examples below, the Boolean choice functions f, g, h of the GSWF F are monotone threshold functions, that is, functions of the form: for different values of l. We note that in ([19], Theorem 2.9), Mossel et al. showed that amongst neutral GSWFs on three alternatives, a GSWF based on the majority rule is the "most rational" in the asymptotic sense (i.e., has the least probability of non-transitive outcome as the number of voters tends to infinity). To some extent, our examples generalize this result to general GSWFs on three alternatives. The examples show that GSWFs based on monotone threshold Boolean choice functions are "close to be the most rational" amongst GSWFs whose choice functions have the same expectations, in the sense that their probability of non-transitive outcome is logarithmic close to the lower bound. In fact, we conjecture that such GSWFs are indeed the most rational amongst GSWFs whose choice functions have the same expectations. However, such exact result is not known even for neutral GSWFs. We use the following proposition of Mossel et al. [18], showing that Borell's reverse Bonami-Beckner inequality is essentially tight for diametrically opposed Hamming balls. Since we use the proposition only for noise of rate ǫ = 1/3, we state it in this particular case. Theorem 5.1 ( [18], Proposition 3.9). Fix s, t > 0, and let f n , g n : {0, 1} n → {0, 1} be defined by In order to show the tightness of Theorem 1.3, we fix a constant ǫ > 0 and define the choice functions according to Equation (44), choosing the values of l such that It is clear that D 1 (F ) = ǫ. By Equation (25), By our construction, the pair of functions (1 -g, h) is of the form considered in Theorem 5.1, with s = t ≈ 2 log(1/ǫ), and thus by the theorem, for n sufficiently large, The lower bound asserted by Theorem 1.3 is P (F ) ≥ C ′ • ǫ 3 , and thus the example shows the tightness of the assertion up to logarithmic factors. The tightness of Theorem 1.5 is shown similarly, with choice functions chosen such that It is clear that D 2 (F ) = ǫ, and by Equation (25), The pairs (f ′ , h) and (1 -g, h) are both of the form considered in Theorem 5.1, and application of the theorem to both of them yields tightness up to a logarithmic factor, like in the previous case. Finally, we note that while the examples above deal with GSWFs whose choice functions have constant expectation, it also makes sense to consider choice functions whose expectation tends to zero, as n (the number of voters) tends to infinity. In particular, one may ask what is the least possible probability of non-transitive outcome, as function of n, for GSWFs with D 1 (F ) > 0 or D 2 (F ) > 0. It appears that the question is of interest mainly for D 2 (F ), as for D 1 (F ), one can easily check that the minimal possible probability of 6 -n is obtained by a GSWF whose choice functions are chosen according to Equation (44), such that For D 2 (F ), it was shown in [17] that for a GSWF whose choice functions are chosen according to Equation (44), such that we have P (F ) ≤ 0.471 n . Furthermore, it was conjectured that this is the most rational GSWF on three alternatives that satisfies the assumptions of Arrow's theorem (and in particular, the minimal possible probability 6 -n is not obtained). Our results show that this function is at least "close" to be the most rational, as by Theorem 1.5, for any GSWF F such that D 2 (F ) > 0, we have 6 Questions for Further Research We conclude this paper with several open problems related to our results. • Our main lemma (Lemma 3.2) gives an essentially tight lower bound on the probability of non-transitive outcome for GSWFs in which at least one of the Boolean choice functions is "close" to a constant function. In the case where the distance from constant functions is greater than a fixed constant, our technique fails, and we use Mossel's theorem [20] instead. As a result, the constant multiplicative factor in the assertions of Theorems 1.3 and 1.5 is extremely small, and clearly non-optimal. It will be interesting to find a "direct" proof also for GSWFs whose Boolean choice functions are "far" from constant functions, thus removing the reliance of the proof on the non-linear invariance principle (used in Mossel's argument) that seems "unnatural" in our context, and improving the constant factor. • While the results of Kalai [14] and Mossel [20] hold also for more general distributions of the individual preferences called "even product distributions" or "symmetric distributions" (see [17,20]), our proof does not extend directly to such distributions. The reason is that for highly biased distributions of the preferences, the lower bound obtained by Borell's reverse Bonami-Beckner inequality is weaker, and cannot "beat" the upper bound obtained by the Bonami-Beckner inequality. Thus, obtaining a tight quantitative version of Arrow's theorem for general even product distributions of the preferences is an interesting open problem. • We believe that GSWFs whose Boolean choice functions are monotone threshold functions are the most rational amongst GSWFs whose choice functions have the same expectations, not only in the asymptotic sense, but also for any particular (large enough) n. However, this conjecture seems quite challenging, as it includes the Majority is Stablest conjecture (whose proof by Mossel et al. [19] holds only in the limit as n → ∞). • Another direction of research is using our techniques to obtain quantitative versions of other theorems in social choice theory. In [8], Friedgut et al. presented a quantitative version of the Gibbard-Satterthwaite theorem [11,21] for neutral GSWFs on three alternatives. Recently, Isaksson et al. [12] generalized the result of [8] to neutral GSWFs on k alternatives, for all k ≥ 4. One of the main ingredients in the proof of [8] is Kalai's quantitative Arrow theorem for neutral GSWFs. It seems interesting to find out whether our quantitative version of Arrow's theorem can lead to a quantitative Gibbard-Satterthwaite theorem for general GSWFs (without the neutrality assumption). • Finally, our results (as well as the previous results of Kalai [14] and Mossel [20]) apply only to GSWFs that satisfy the IIA condition, since such GSWFs can be represented by their Boolean choice functions, which allows to use the tools of discrete harmonic analysis. It will be very interesting to find a quantitative version of Arrow's theorem that will not assume the IIA condition, but rather will relate the probability of non-transitive outcome to the distance of the GSWF from satisfying IIA. We note that Mossel also obtained another variant of his theorem, in which the dependence of δ on ǫ is δ(ǫ) = Cǫ 3 n -3 , where n is the number of voters, and C is a "decent" constant. As follows from our results presented below, this variant is essentially tight for very small values of ǫ (dependent on n). Moreover, for certain choices of parameters (specifically, "relatively small" n and ǫ very small as a function of n), this result gives a stronger bound than our result, due to the better value of the constant.