8
the dollar is proposed, the previous question is moved, and the proposal passes with the votes of
at least a majority of the legislature. The legislative process under an open rule is presented in
Figure 2.
[Insert Figure 2 about here]
Legislators have a common discount factor on payoffs 0
≤ δ ≤ 1, which is realized each
time the legislature goes to a new round of bargaining without having divided the dollar in the
previous round—that is, whenever a legislator makes a new proposal for dividing the dollar.
Specifically, in the closed rule case, the dollar’s value is discounted to
δ times its previous value
if a proposal fails and a new proposal is made. In the open rule case, discounting occurs
similarly upon the defeat of a proposal or when an amendment is made.
A Closed Rule Stationary Equilibrium
Following the Baron and Ferejohn approach, we characterize the stationary equilibrium
for the closed and open rule games. Stationarity is defined by identical continuation values for
each structurally equivalent subgame. Under stationarity, the equilibrium strategy of the initial
proposer is the same as that of future proposers facing the same game structure. Without this
stationarity assumption, an infinite number of strategies could be sustained in equilibrium under
various punishment strategies. With this assumption, it is fairly straightforward to characterize
the unique subgame perfect equilibrium, here given for the closed rule game form.
Closed Rule Equilibrium: For all
δ ∈ [0, 1] a configuration of pure strategies is a stationary
subgame-perfect equilibrium in an infinite round, majority rule, n-member (n odd) legislature
governed by closed rule if and only if it has the following form of collective, mixed, and
particularistic divisions of the dollar:
(1) If )
,
0
[
CM
α
α ∈
, a member recognized offers to keep 0 for himself in particularistic goods, to
give 0 to all other members in particularistic goods, and to contribute the entire dollar
Convention