A Comment on the Gibbard–Satterthwaite Theorem

Proof of Theorem 1:

Let X be an axiom that allows for an upper weakening of WARP. Let C ₁ = ρ(≻¹) be the choice function produced by 1’s context-dependent ranking ≻¹. Let B C 1 ⊆ A be the set of all alternatives x ∈ B ^C such that if the choice function C′ is such that C′(B) = C(B), B ≠ A, and C′(A) = x, then C ′ ∈ X . Let Φ : Ψ X n → X be a social aggregator such that

Given that C 1 ( A ) ∈ B C 1 (because C 1 ∈ X ), it follows that Φ is well-defined. By construction, C s ∈ X . If C ₁ satisfies WARP, then for any x ∈ A, if C′ is such that C′(B) = C(B), B ≠ A, and C′(A) = x, then C ′ ∈ X u . Thus, if C ₁ satisfies WARP, then B C 1 = A . It follows that if C ₁ = ρ(≻¹) satisfies WARP and C ₂ = ρ(≻²) is such that C ₂(A) ≠ C ₁(A), then C _s (A) = C ₂(A) ≠ C ₁(A). Hence, Φ is not dictatorial. The context-dependent ranking of individuals i ∈ {3, …, n} do not influence the social choice. Thus, i ∈ {3, …, n} cannot manipulate Φ. Individual 2 has no influence on any issue B ≠ A, and, hence, 2 cannot manipulate Φ in any issue B ≠ A. The social choice is the same as 1’s choice in any issue B ≠ A and, hence, 1 cannot manipulate Φ in any issue B ≠ A. Individual 2 cannot manipulate Φ in any issue B ≠ A because 2 has no influence in issue A. Individual 1 cannot manipulate Φ in issue A because, given the choices made in issues other than A, 1 has no influence in the social choice in A. Individual 2 cannot manipulate Φ in any issue A because the social choice is the one that maximizes ≻ A 2 among a set determined by choices made in other issues. Finally, Φ has at least three outcomes because any alternative in A can be the social choice in A.□

Proof of Theorem 2:

Let X be an axiom that allows for lower weakening of WARP. Let B ⊆ B ̄ be the set of all binary issues (i.e., a binary issue B contains two options). If ♯ ( B ̄ ) > n , then let B ̂ ⊂ B ̄ be an arbitrary set such that ♯ ( B ̂ ) = n . If ♯ ( B ̄ ) < n , then let B ̂ = B ̄ . Let ν : B ̂ → { 1 , … , n } be an arbitrary function such that for any pair of issues B ∈ B ̄ and B ′ ∈ B ̄ , ν(B) ≠ ν(B′) when B ≠ B′. Let ≻ _A be an arbitrary preference order on A (without indifference), which is not necessarily related to the any individual ranking on A. Let f : P X n → X be a general social aggregator such that

Given that in all C _s violations of WARP, the subissue is binary, it follows that C s ∈ X l . Thus, C s ∈ X . Hence, f is well-defined. Given that ν(B) ≠ ν(B′) for two different issues B and B′, it follows that as many individuals as the cardinality of B ̂ have influence on f and, hence, f is non-dictatorial.

To see that f is non-dictatorial, consider an arbitrary individual i, and some other individual ı ̃ ≠ i such that ν ( B ı ̃ ) = ı ̃ for some issue B ı ̃ . Consider a profile ( P 1 , … , P n ) ∈ P X n of preferences such that ≻ B ı ̃ ı ̃ and ≻ B ı ̃ i have a different maximum in B ı ̃ . Let C _s  = f(P ₁, …, P _n ) and let C′ be such that C′(B) = C _s (B) when B ≠ B ı ̃ and C ′ ( B ı ̃ ) ≻ B ı ̃ i y , for all y ∈ B ı ̃ , y ≠ C ′ ( B ı ̃ ) . Given that B ı ̃ ∈ B ̄ and in all non-binary issues C′ is equal to C _s , and in all non-binary issues C _s optimizes a single order ≻ _A , it follows that in any C′ violation of WARP, the subissue is binary. Hence, C ′ ∈ X l . Thus, C ′ ∈ X . By definition, C ′ ( B ı ̃ ) ≻ B ı ̃ i C s ( B ı ̃ ) . Moreover, C′ and C _s produce the same choice on any issue other than B ı ̃ . By (3.1), C′ P _i C _s . Thus, f is not dictatorial.

Consider a profile ( P 1 , … , P n ) ∈ P X n and an alternative P i ′ ∈ P X for individual i. Let

Let B ⁱ be the only issue such that ν(B ⁱ ) = i. If B ⁱ does not exist, then, by construction, C s = C s ′ and, therefore, it cannot be that C s ′ P _i C _s because P _i is asymmetric. By construction, C s ( B ) = C s ′ ( B ) for any issue B ≠ B ⁱ . By (3.1), C s ′ P i C s if and only if C s ′ ( B i ) ≻ B i C s ( B i ) . By definition of f, C _s (B ⁱ ) = C _i (B ⁱ ) and C s ′ ( B i ) = C i ′ ( B i ) . By definition of ≻ B i either C i ( B i ) = C i ′ ( B i ) or C i ( B i ) ≻ B i C i ′ ( B i ) . In the former case, C s = C s ′ and, hence, it cannot be that C s ′ P i C s because P _i is asymmetric. In the latter case, C s P i C s ′ and, therefore, it cannot be that C s ′ P i C s because P _i is asymmetric. Thus, f is non-manipulable. Finally, f has more than 3 outcomes because any alternative in A may the aggregate choice in A.□

Proof of Theorem 3:

Let X = X l . Let B ⊆ B ̄ be the set of all binary issues (i.e., a binary issue B contains two options). Let f : P X n → X be a general social aggregator such that

Consider a profile ( P 1 , … , P n ) ∈ P X n and an alternative P 1 ′ ∈ P X for individual 1. Let

Assume that C s ≠ C s ′ . Let B ⊆ ̂ B ̄ be the set of all binary issues B such that C s ( B ) ≠ C s ′ ( B ) . Note that if B is a non-binary issues, then C s ( B ) = C s ′ ( B ) because aggregate choices do not depend upon individual 1’s preferences. Let’s say that B ̂ has k elements and let’s order these issues arbitrarily so that B ̂ = { B 1 , … . , B k } . Let C ¹ be the choice function such that C 1 ( B 1 ) = C s ′ ( B 1 ) ≠ C s ( B 1 ) = C 1 ( B 1 ) and C ¹(B) = C _s (B) for every issue B ≠ B ₁. Let ≻¹ be the context-dependent ranking determined by P ¹. By definition, C s ( B 1 ) = C 1 ( B 1 ) ≻ B 1 1 C 1 ( B 1 ) = C s ′ ( B 1 ) . Given that C ¹ and C _s are equivalent outside B ₁, it follows that C _s P ¹ C ¹. Now let C ² be the choice function such that C 2 ( B 2 ) = C s ′ ( B 2 ) ≠ C 1 ( B 2 ) = C 1 ( B 2 ) and C ²(B) = C ¹(B) for every issue B ≠ B ₂. By definition, C 1 ( B 2 ) = C 1 ( B 2 ) ≻ B 2 1 C s ′ ( B 2 ) = C 2 ( B 2 ) . Given that C ¹ and C ² are equivalent outside B ₂, it follows that C ¹ P ¹ C ². Given C ^e , e = 1, …, k − 1, let C ^e+1 be the choice function such that C e + 1 ( B e ) = C s ′ ( B e ) ≠ C s ( B e ) = C 1 ( B e ) and C ^e+1(B) = C ^e (B) for every issue B ≠ B _e . By the same argument as above, C ^e P ¹ C ^e+1, e = 1, …, k − 1. By transitivity, C ¹ P ¹ C ^k . Thus, by transitivity, C _s P ¹ C ^k . By construction, C k = C s ′ . Thus, C s P 1 C s ′ . It follows that individual 1 cannot manipulate this general social aggregator. The proof that individual 2 cannot manipulate this general social aggregator is entirely analogous. The preferences of all other individuals (i.e. not 1 or 2) do not affect the aggregate choices and, hence they cannot manipulate this general social aggregator as well.

It is immediate that this general social aggregator is non-dictatorial, has more than 3 possible outcomes, two or more individuals have influence on f, and it satisfies unanimity. As mentioned, as long as C 2 ∈ X l , it follows that for any profile ( P 1 , … , P n ) ∈ P X n , C _s  = f(P ₁, …, P _n ) ∈  X l because C _s (B) = C ₂(B) for any non-binary issues. If C 2 ∈ X l , then there are no WARP violation in 2’s choices on two non-binary issues. Therefore, there are no WARP violation in the aggregate choices on two non-binary issues as well.□