18
proposal in which he receives utility at least equal to
δα
δ
α
α
δα
+
−
−
−
)
1
)(
1
(
)
1
(
q
q
, and the first
proposal receives unanimous support;
(3a) If
]
1
,
1
3
(
−
∈
δ
and
(
)
]
1
,
[
1
=
∈
m
MP
α
α
, a member recognized offers
2
2
2
4
4
δ
δ
δ
−
+
−
to one other
legislator, chosen at random, keeps
2
2
4
2
δ
δ
δ
−
+
for himself, and allocates no portion of the
dollar into the collective good. Each member votes for any proposal in which he realizes
utility at least equal to
(
)
2
2
2
4
4
δ
δ
δ
α
−
+
−
. If the non-coalition member is selected, he makes the
same proposal to the same coalition member, excluding the original proposer. The game
ends when the coalition member is recognized. That member moves the previous question,
and the proposal receives the support of the proposer and his coalition partner;
(3b) If
]
1
3
,
0
[
−
∈
δ
and
(
)
]
1
,
[
2
=
∈
m
MP
α
α
, a member recognized offers
1
2
+
δ
δ
to each other
member of the legislature, keeps
1
2
1
+
δ
for himself, and allocates no portion of the dollar
into the collective good. Each member votes for any proposal in which he realizes utility at
least equal to
1
2
+
δ
δα
, and the first proposal receives unanimous support;
where
,
1 q
q
CM
+
=
α
(
)
,
4
2
2
)
4
2
(
2
2
1
q
q
q
q
m
MP
δ
δ
δ
δ
δ
α
−
+
+
−
+
=
=
(
)
.
3
1
3
2
q
q
m
MP
+
=
=
α
Derivations for this equilibrium are given in the Appendix.
We are once again interested in conditions under which collective goods are provided, the
amount of those goods chosen by the legislature, and the benefits that accompany proposal
power. With the exception of the comparative statics regarding the legislature size, all three
propositions from the closed rule section hold for the general open rule case examined in Table 1
and for the special open rule three-person legislature.
Specifically, Proposition 1 above stated that collective goods (rather than only
particularistic goods) are provided by legislators who value collective goods highly (
α low), see
a substantial return on their collective investment (q high), and are highly impatient (
δ low).
Based on the comparative statics for the
α cutpoint between the mixed and particularistic parts of
Convention