1
Appendix B: Pareto dominance of unanimity when
proposer has private information (Proposition 10)
In this appendix we establish that there are circumstances under which both the proposer
and voters prefer unanimity, even when the proposer has private information. Formally,
we establish Proposition 10, stated in the main text.
Existence of a pooling equilibrium under unanimity
The proof requires us to work with a fixed equilibrium under unanimity. The particular equilibrium we use is of little importance, but the equilibrium utilities need to be
continuous as we change preferences. The easiest equilibrium to work with is a pooling
equilibrium, in which the proposer makes the same offer for all realizations of σ 0 . Our
first two results establish circumstances under which such an equilibrium exists.
First, we show that whenever the information quality of voters is low, the proposer’s
limit expected payoff is convex in the offer x.
Notationally, let vn (x, σ 0 , b, λ) denote the proposer’s expected payoff from an offer x
when he has observed σ 0 , the voters attach belief b to offer x, and have preferences λ,
i.e.,
vn (x, σ 0 , b, λ) = Eω V̄ ω (σ 0 ) + Pnω (x, b, λ, α) V ω (x, σ 0 ) − V̄ ω (σ 0 ) |σ 0 .
(We used the same definition in the proof of Proposition 3 — see (A-4).) Define P ω and
v as the pointwise limits of Pnω and v respectively.
Lemma B-1 Fix preferences λ and suppose unanimity rule is in effect (α = 1). Then
there exists ℓ such that whenever ℓ(σ) > ℓ, the proposer’s limit payoff v is convex in x over
[0, min {1, xU (b, λ)}] for all σ 0 , b. As a consequence, v is maximized at min {1, xU (b, λ)};
and if xU (b, λ) < 1 the maximizer is unique.
Proof of Lemma B-1: For use throughout the proof, write x̌ = min {1, xU (b, λ)}. The
limit acceptance probability is identically equal to zero over [0, xH (λ)], and is increasing
2
thereafter. So to prove the result, it suffices to show that for ω = H, L there exists ℓ
such that whenever ℓ(σ) > ℓ, the second derivative of P ω (x, b, λ) V ω (x, σ 0 ) − V̄ ω (σ 0 )
is positive for x ∈ (xH (λ) , x̌], for all σ 0 , b.
H
(x,σ,λ) b
Let ℓ(σ) denote the likelihood ratio at σ. Write R(x, σ, b, λ) = − ∆
, γH =
∆L (x,σ,λ) 1−b
γω
ℓ(σ) and γ L = 1, so that P ω (x, b, λ) = [R(x, σ, b, λ)ℓ(σ)] 1−ℓ(σ) for x ∈ (xH (λ) , xU (b, λ)).
The second derivative of P ω (x, b, λ) V ω (x, σ 0 ) − V̄ ω (σ 0 ) with respect to x is given by
∂2 ω
V (x, σ 0 )
∂x2
γω
γω
∂
∂
+2 V ω (x, σ 0 )
[R(x, σ, b, λ)ℓ(σ)] 1−ℓ(σ) −1 R(x, σ, b, λ)
∂x
1 − ℓ(σ)
∂x
γω
∂2
γω
[R(x, σ, b, λ)ℓ(σ)] 1−ℓ(σ) −1 2 R(x, σ, b, λ)
+ V ω (x, σ 0 ) − V̄ ω (σ 0 )
1 − ℓ(σ)
∂x
2
γω
γω
γω
∂
−2
ω
ω
+ V (x, σ 0 ) − V̄ (σ 0 )
(
− 1)[R(x, σ, b, λ)ℓ(σ)] 1−ℓ(σ)
R(x, σ, b, λ)
1 − ℓ(σ) 1 − ℓ(σ)
∂x
γω
= P ω (x, b, λ)
K
1 − ℓ(σ)
P ω (x, b, λ)
where
K =
1 − ℓ(σ) ∂ 2 ω
∂
∂
V (x, σ 0 ) + 2 V ω (x, σ 0 ) ℓ(σ)−1
ln R(x, σ, b, λ)
2
γ ω ∂x
∂x
∂x
∂2
+ V ω (x, σ 0 ) − V̄ ω (σ 0 ) R(x, σ, b, λ)−1 ℓ(σ)−1 2 R(x, σ, b, λ)
∂x
∂
γ − 1 + ℓ(σ)
+ V ω (x, σ 0 ) − V̄ ω (σ 0 ) ω
R(x, σ, b, λ)−1 ℓ(σ)−1
ln R(x, σ, b, λ).
1 − ℓ(σ)
∂x
Observe that γ L − 1 + ℓ(σ) = ℓ(σ) and γ H − 1 + ℓ(σ) = 2ℓ(σ) − 1. The term V ω (x, σ 0 ) −
V̄ ω (σ 0 ) is bounded away from zero over [0, x̌]. The term R(x, σ, b, λ)−1 is also bounded
away from zero over (xH (λ) , x̌]. From Assumption 5, the term R(x, σ, b, λ) is strictly
increasing over [0, x̌], and so
∂
∂x
ln R(x, σ, b, λ) is positive and bounded away from zero
over [0, x̌]. From these observations, it is immediate that there exists ℓ such that K > 0
whenever ℓ(σ) > ℓ and x ∈ (xH (λ) , x̌], and for all σ 0 , b. This completes the proof.
Lemma B-1 implies that in the limit, for any given voter beliefs the proposer prefers
the offer xU (b, λ) to all lower offers, unless lower offers lead voters to hold more positive
3
beliefs (i.e., higher b). We use this observation to establish that there exists a pooling
equilibrium in which the proposer offers xU b = pH , λ regardless of his information σ 0 .
Lemma B-2 Fix voter preferences λ and suppose that xU (b = pH , λ) < 1. If the unanimity rule is in effect (α = 1) and voter information is sufficiently poor so that Lemma
B-1 holds, then there exists N such that for all n ≥ N there is a pooling equilibrium in
which the proposer’s offer is independent of σ 0 and lies within 1/n of xU (b = pH , λ).
Proof of Lemma B-2: If the proposer’s signal is completely uninformative (i.e.,
ℓ0 (·|H) ≡ 1), the result is immediate from the uniform convergence of acceptance probabilities (Lemma 4) and the proposer payoff convexity (Lemma B-1).
The remainder
of the proof deals with the case in which the proposer’s signal is informative, and so
b < pH < b̄.
We claim that for all n large enough, there is a pooling equilibrium in
which the proposer offers xU (b = pH , λ) independent of his signal σ 0 ; the voters accept
with probability one; and the voters assign belief b to any downwards deviation by the
proposer.
If the proposer offers xU b = pH , λ and the voters hold beliefs b = pH , the acceptance
probability is one for all n.
So the proposer’s payoff for any finite n in the claimed
equilibrium exactly equals the limit payoff v xU b = pH , λ , σ 0 , b = pH , λ , which for
the remainder of the proof we write as v ∗ .
Fix δ > 0. From the payoff convexity result Lemma B-1, there exists ε > 0 such that
v x, σ 0 , b = pH , λ < v ∗ − ε whenever x < xU b = pH , λ − δ. A fortiori, v (x, σ 0 , b, λ) <
v ∗ − ε also. By uniform convergence (Lemma 4) it follows that there exists N1 such that
vn (x, σ 0 , b, λ) < v ∗ − ε whenever n ≥ N1 .
It remains only to rule out downwards deviations between xU b = pH , λ − δ and
xU b = pH , λ . From Lemma 3, there exists some ϕ > 0 such that the limit acceptance
probabilities satisfy
P ω (x, b, λ) < P ω x, b = pH , λ − ϕ
4
for offers x in this interval. So there exists ϕ̂ > 0 such that the limit proposer payoffs
similarly satisfy
v (x, σ 0 , b, λ) < v x, σ 0 , b = pH , λ − ϕ̂
for offers x in this same interval. By payoff convexity (Lemma B-1), v x, σ 0 , b = pH , λ <
v ∗ . So by uniform convergence, it follows that there exists N2 such that vn (x, σ 0 , b, λ) <
v ∗ whenever n ≥ N2 .
So for all n ≥ max {N1 , N2 } the proposer has no profitable
deviation, completing the proof.
Pareto dominance of unanimity
Given the existence of a pooling equilibrium under unanimity, we are now ready to
establish our main result of this section, Proposition 10. The proof strategy is similar to
that for the special case stated in Proposition 9: by changing preferences, and specifically
the proposer’s status quo payoff V̄ L , by continuity we can find a set of preferences such
that the voters are indifferent between majority and unanimity rules, while total surplus
is higher.
In more detail, the argument is as follows. Suppose preferences are such that (15) does
not hold, i.e., agreement creates surplus in state L, and moreover W (·; λ, α) > 0. For
these preferences, voters strictly prefer unanimity. Next, consider a change in preferences
as V̄ L is decreased. If V̄ L is decreased by enough, W (·; λ, α) < 0, and the voters strictly
prefer majority.
Moreover, provided that the proposer’s signal is non-degenerate, the
voters’ expected payoff under majority varies in a continuous way as V̄ L changes.
The effect of changing preferences on the voters’ payoff under unanimity is harder
to characterize, since in general there may be multiple equilibria. However, if we focus
on the specific pooling equilibrium whose existence we established in Lemma B-2, the
expected payoff of voters also varies in a continuous way as the proposer’s preferences
are changed.
It follows that there exist preferences at which the voters receive the same payoff from
majority voting and unanimity voting (in the pooling equilibrium). At this indifference
5
point, when facing majority the proposer offers xH after some realizations of his own signal σ 0 , and xL after other realizations — for if instead he always offered one or the other,
the voters would have a strict preference between majority and unanimity. But because
the proposer offers xH to voters using majority rule with strictly positive probability,
the offer is rejected by the voters with strictly positive probability also. In contrast, in
the unanimity pooling equilibrium the proposer always offers xU , and the voters always
except.
Given these observations, one can see that at the “indifference” preferences total
surplus is higher under unanimity than majority. Since by construction the voters are
indifferent between the two decision rules, the proposer strictly prefers unanimity. That
is, the unanimity rule Pareto dominates majority voting. Formally:
Proof of Proposition 10: The proof revolves around a family of possible proposer preferences, which we select as follows. First, choose a specification of proposer
preferences such that V ω and V̄ ω are independent of σ 0 , and such that
pH (σ)V H (xH (λ = 0)) + pL (σ)V̄ L > E [V ω (xL (λ = 0)) |σ] .
Under these preferences, W (σ 0 ) > 0 for all σ 0 and all α, for λ small enough. Write V̄0L
for the proposer’s payoff from the status quo in state L under these preferences. Next,
choose V̄1L sufficiently small such that
pH (σ̄)V H (xH (λ = 0; α)) + pL (σ̄)V̄1L < E [V ω (xL (λ = 0)) |σ̄] .
So under these alternate preferences, W (σ 0 ) > 0 for all σ 0 and all α, for λ small enough.
For any η ∈ (0, 1) define V̄ηL = (1 − η) V̄0L + η V̄1L . The family of proposer preferences
we consider is
V=
V H , V L , V̄ H , V̄ L : V̄ L = V̄ηL , some η ∈ [0, 1] .
Let C be the constant whose existence is implied by Assumption 7.
Since at λ = 0,
U ω (xω (λ; α) , σ i , λ) = Ū ω (σ i , λ) for all σ i , and xL (λ = 0) = xL (λ = 0; α) for all α, it
6
follows from xL (λ = 0, α) < 1 and Assumptions 4 and 5 that
U ω (x, σ i , λ = 0) + CV ω (x) > Ū ω (σ i , λ = 0) + C V̄ ω
for all x ∈ [0, 1], σ i ∈ [σ, σ̄], and proposer preferences in the family V. So there exists
some ε > 0 and some λ̌ > 0 such that for λ ≤ λ̌,
U ω (x, σ i , λ) + CV ω (x) > Ū ω (σ i , λ) + C V̄ ω + ε
for all x ∈ [0, 1], σ i ∈ [σ, σ̄], and proposer preferences in the family V. That is, under
agreement at any offer in [0, 1] creates social surplus of at least ε relative to the status
quo.
For the remainder of the proof, we assume that voter information, as measured by
ℓ(σ), is sufficiently poor such that Lemma B-1 holds for all preferences in the set V and
λ ≤ λ̌.
(Note that changing the voter information quality does not affect any of the
expressions used so far in the proof.)
For the remainder of the proof, fix voter preferences at λ ≤ λ̌ and a majority voting
rule αM < 1. From Proposition 4, for any proposer preferences such that W (σ) > 0,
the majority limit payoff of voters is Eσi ,ω Ū ω (σ i ) + pH Eσi ∆H (xH (λ; αM ) , σ i , λ) |H .
Likewise, for any η such that W (σ̄) < 0, the majority limit payoff of voters is Eσi ,ω Ū ω (σ i ) +
Eσi ,ω [∆ω (xL (λ; αM ) , σ i , λ)]. So as the proposer’s preferences range over V as η increases
from 0 to 1, the majority limit payoff of voters varies continuously from the former expression to the latter. In contrast, the unanimity limit payoff of voters in the pooling
equilibrium (Lemma B-2) is simply Eσi ,ω Ū ω (σ i ) + Eσi ,ω ∆ω xU pH , λ , σ i , λ for all
proposer preferences in V.
By Proposition 5, in the limit, the voters are strictly better off under the unanimity
pooling equilibrium for η = 0, while since xU pH , λ < xL (λ; αM ) they strictly prefer
majority equilibrium at η = 1. So by continuity, there exists η̂ ∈ (0, 1) at which they
are indifferent between the majority equilibrium and the unanimity pooling equilibrium.
Moreover, we know that W (σ) < 0 < W (σ̄) at η̂, so the limit rejection probability under
7
majority strictly exceeds 0.
By continuity, select η̌ < η̂ such that the limit rejection
probability under majority in state L equals q > 0, and the unanimity limit payoff of
voters exceeds the majority limit payoff by an amount pL qε/2.
At proposer preferences η̌, the sum of the voter and proposer limit payoffs is lower
under majority than under unanimity by an amount of at least pL qε (that is, the efficiency loss ε multiplied by the rejection probability).
Since the limit voter payoff is
pL qε/2 higher under unanimity, the limit proposer payoff is at least pL qε/2 higher under
unanimity also. So there exists some N such that under the preferences defined, both the
proposer and voters strictly prefer unanimity to the majority rule αM whenever n ≥ N.