and bounded above by
max
1
2
,
1 − t
n
1 − t
n
(2p − 1)
p (1 − η + pη)
.
PROOF. See Appendix.
To see the intuition behind proposition 1, consider two cases. First consider the case in
which the alternative is always chosen. In other words, both majority-type and minority-
type voters vote for the alternative regardless of their private signals. Then the probability
that a selected alternative is not majority-type-preferred is simply 1/2, that is, the prior
probability that each state occurs.
Second, consider the case in which the alternative is sometimes selected. This case is
divided into two subcases; in one the majority-type voter votes for the status quo when
he receives s
0
with strictly positive probability while the minority-type voter votes for the
alternative independent of the private signal, and in the other subcase the minority-type
voter votes for the status quo with strictly positive probability when she gets s
1
while the
majority-type voter votes for the alternative independent of the private information.
Now consider the first subcase. The majority-type voter will vote for the status quo
if the fact that a vote is pivotal is not overwhelming evidence of the state being 1. In
other words, the probability that the state is 1 conditional on a majority-type voter’s vote
being pivotal and on his private signal s
0
, must be less than or equal to his threshold t
m
.
Analogously in the second subcase the minority-type voter will vote for the status quo
if the probability that the state is 1 conditional on a minority-type voter’s being pivotal
and on her private signal s
1
, is greater than or equal to her threshold t
n
.
The implication from proposition 1 is that the unanimity rule under heterogenous ob-
jectives does not work very well, not only as an information aggregation mechanism, but
also as a preference aggregation device. Moreover, given that the alternative is passed
under the unanimity rule, the probability that the alternative is not majority-preferred is
1/2. Of course, when the status quo is maintained the probability that the status quo is
not majority-type-preferred is also 1/2. Therefore, unanimity rule does not increase social
welfare over a random choice rule when voter have heterogenous objectives.
5
Unanimity Rule: A Status Quo Bias
In this section we compute an equilibrium under unanimity rule and then we show that in
the equilibrium the status quo is almost always maintained when the committee size is very
9
Convention