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A Model of Adversarial Committees
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The second type of equilibrium occurs when there is some voter who votes for status quowith positive probability. So it must be the case that 1 − p ≤ t
m
or t
n
≤ p.
Suppose a matcher upon receiving signal s votes for Q with positive probability, that is,σ
m
(s) < 1 for some s. Also note that since the alternative is selected with strictly positive
probability, it follows that each voter is pivotal with strictly positive probability. Sinceσ
m
(s) < 1, it follows that
t
m
≥ Pr (1|piv, s)
=
Pr (1, piv, s)
Pr (piv, s)
=
Pr (piv, s|1) Pr (1)
Pr (piv, s|0) Pr (0) + Pr (piv, s|1) Pr (1)
since Pr (piv, s, 1) = Pr (piv, s|1) Pr (1) and Pr (piv, s) = Pr (piv, s, 1) + Pr (piv, s, 0),
=
Pr (piv, s|1)
Pr (piv, s|0) + Pr (piv, s|1)
since Pr (1) = Pr (0) =
12
,
=
Pr (piv|1) Pr (s|1) Pr (1)
Pr (piv|0) Pr (s|0) Pr (0) + Pr (piv|1) Pr (s|1) Pr (1)
since a voter is assumed to be pivotal and Pr (piv, s|1) = Pr (piv|1) Pr (s|1) Pr (1).
Since Pr (piv|1) =
Pr(piv,1)
Pr(1)
=
Pr(1|piv) Pr(piv)
Pr(1)
, i.e., Pr (1|piv) = Pr (piv|1)
Pr(1)
Pr(piv)
,
Pr (1|piv, s) =
Pr (piv|1) Pr (s|1) Pr (1)
Pr (piv|0) Pr (s|0) Pr (0) + Pr (piv|1) Pr (s|1) Pr (1)
=
Pr (1|piv)
Pr(piv)
Pr(1)
Pr (s|1) Pr (1)
Pr (0|piv)
Pr(piv)
Pr(0)
Pr (s|0) Pr (0) + Pr (1|piv)
Pr(piv)
Pr(1)
Pr (s|1) Pr (1)
=
Pr (1|piv) Pr (piv) Pr (s|1)
Pr (1|piv) Pr (piv) Pr (s|1) + (1 − Pr (1|piv)) Pr (piv) (1 − Pr (s|1))
=
Pr (1|piv) Pr (s|1)
Pr (1|piv) Pr (s|1) + (1 − Pr (1|piv)) (1 − Pr (s|1))
.
Thus,
Pr (1|piv) ≤
t
m
p
(1 − t
m
) (1 − p) + t
m
p
.
Recall that A denotes the event that the alternative is selected, and let
γ
0
= (1 − η) [pσ
m
(s
0
) + (1 − p) σ
m
(s
1
)] + η [pσ
n
(s
0
) + (1 − p) σ
n
(s
1
)]
19
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The second type of equilibrium occurs when there is some voter who votes for status quo
with positive probability. So it must be the case that 1 − p ≤ t
m
or t
n
≤ p.
Suppose a matcher upon receiving signal s votes for Q with positive probability, that is,
σ
m
(s) < 1 for some s. Also note that since the alternative is selected with strictly positive
probability, it follows that each voter is pivotal with strictly positive probability. Since
σ
m
(s) < 1, it follows that
t
m
≥ Pr (1|piv, s)
=
Pr (1, piv, s)
Pr (piv, s)
=
Pr (piv, s|1) Pr (1)
Pr (piv, s|0) Pr (0) + Pr (piv, s|1) Pr (1)
since Pr (piv, s, 1) = Pr (piv, s|1) Pr (1) and Pr (piv, s) = Pr (piv, s, 1) + Pr (piv, s, 0),
=
Pr (piv, s|1)
Pr (piv, s|0) + Pr (piv, s|1)
since Pr (1) = Pr (0) =
1
2
,
=
Pr (piv|1) Pr (s|1) Pr (1)
Pr (piv|0) Pr (s|0) Pr (0) + Pr (piv|1) Pr (s|1) Pr (1)
since a voter is assumed to be pivotal and Pr (piv, s|1) = Pr (piv|1) Pr (s|1) Pr (1).
Since Pr (piv|1) =
Pr(piv,1)
Pr(1)
=
Pr(1|piv) Pr(piv)
Pr(1)
, i.e., Pr (1|piv) = Pr (piv|1)
Pr(1)
Pr(piv)
,
Pr (1|piv, s) =
Pr (piv|1) Pr (s|1) Pr (1)
Pr (piv|0) Pr (s|0) Pr (0) + Pr (piv|1) Pr (s|1) Pr (1)
=
Pr (1|piv)
Pr(piv)
Pr(1)
Pr (s|1) Pr (1)
Pr (0|piv)
Pr(piv)
Pr(0)
Pr (s|0) Pr (0) + Pr (1|piv)
Pr(piv)
Pr(1)
Pr (s|1) Pr (1)
=
Pr (1|piv) Pr (piv) Pr (s|1)
Pr (1|piv) Pr (piv) Pr (s|1) + (1 − Pr (1|piv)) Pr (piv) (1 − Pr (s|1))
=
Pr (1|piv) Pr (s|1)
Pr (1|piv) Pr (s|1) + (1 − Pr (1|piv)) (1 − Pr (s|1))
.
Thus,
Pr (1|piv) ≤
t
m
p
(1 − t
m
) (1 − p) + t
m
p
.
Recall that A denotes the event that the alternative is selected, and let
γ
0
= (1 − η) [pσ
m
(s
0
) + (1 − p) σ
m
(s
1
)] + η [pσ
n
(s
0
) + (1 − p) σ
n
(s
1
)]
19
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