28
Appendix
Closed rule case
Collective:
The collective part of the equilibrium takes the form that the proposer gives nothing
to any legislator, putting the whole dollar into the collective good. Where this proposal gives a
higher value to the proposer than would any proposal with particularistic spending, the proposal
is also more attractive to each other member than is delaying and simply generating the same
proposal in the next period. The proposer’s utility is
(
)
q
U
C
α
−
= 1
,
1
.
Mixed:
The mixed part takes the form that the proposer keeps x
1
for himself, and puts the
remaining 1- x
1
in the collective.
coalition weigh their utility from the current proposal (
(
)(
)
q
x
1
1
1
−
−
α
) against the expected
utility from having a new proposer selected following a delay (
(
)(
)
q
x
x
n
1
1
1
1
−
−
+
α
δ
α
δ
).
Thus non-proposers vote for the proposal iff:
(
)(
)
(
)(
)
q
x
x
n
q
x
1
1
1
1
1
1
1
−
−
+
≥
−
−
α
δ
α
δ
α
δα
δ
α
δ
α
+
−
−
−
−
≤
⇔
)
1
)(
1
(
)
1
)(
1
(
1
qn
qn
x
To maximize his utility, the proposer sets
δα
δ
α
δ
α
+
−
−
−
−
=
)
1
)(
1
(
)
1
)(
1
(
1
qn
qn
x
, putting
δα
δ
α
δα
+
−
−
=
)
1
)(
1
(
qn
y
in the collective.
7
This proposal is accepted by all legislators. The
proposer’s utility is
δα
δ
α
δ
δ
α
α
+
−
−
+
−
−
=
)
1
)(
1
(
)
)(
1
(
,
1
qn
n
n
q
U
M
.
Particularistic:
The particularistic part of the equilibrium takes the form that the proposer gives
x
m
to
2
1
−
n
other legislators, and keeps x
1
for himself. As he receives no utility from leaving part
of the dollar undivided, and as no money is put in the collective,
2
1
1
1
−
−
=
n
x
x
m
. At the critical
voting stage, non-proposing members of the coalition weigh their utility from the current
proposal (
α x
m
) against the expected utility from having a new proposer selected following a
delay (
n
δα
). Assuming acceptance when indifferent, the proposer sets
n
x
m
δ
= , leaving
(
)
n
n
x
2
1
1
1
−
−
=
δ
. This proposal is accepted by the proposer and the
2
1
−
n
other coalition
members. The proposer’s utility is
(
)
n
n
U
P
2
1
,
1
−
−
=
δα
α
.
6
It is trivial to show that no proposal giving a nonzero amount to non-proposers, to the proposer, and to the
collective is sustained in equilibrium. Intuitively, where the proposer prefers to give particularistic rather than
collective goods to others, such a case collapses to the particularistic case; and where he prefers to give collective
rather than particularistic goods to others, such a case collapses to the mixed case described here.
7
Note that, for the range of a sustaining this mixed particularistic and collective equilibrium, the proposer prefers
particularistic spending for his own district over spending on the collective good.
Convention