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Gilligan Krehbiel (1987) - Collective Decision Making And Standing Committees (view source)
Revision as of 17:53, 20 May 2010
, 17:53, 20 May 2010no edit summary
:<math>\omega \in [\overline{\omega}, x_c + \overline{\omega}]\,</math>
===Open rule, specialization===
Exact inference by the floor is not possible - it is not in the committee's interest to allow this. But inference in a partition of the range of the distribution is possible, much like a cheap talk model.
Let <math>a_i\,</math> denote the partition boundaries, with <math>a_0 = 0\,</math> and <math>a_N = 1\,</math>.
A legislative equilibrium is then:
:<math>b^*(\omega) \in [x_c-a_{i+1}, x_c - a_i] \quad \mbox{if}\; \omega \in [a_i,a_{i+1}]\,</math> \\
:<math>
p^*(b) =
\begin{cases}
-\frac{(a_{N-1} + a_N)}{2} & \mbox{if}\; b < x_c -1 \\
-\frac{(a_{i} + a_{i+1})}{2} & \mbox{if}\; b \in [x_c-a_{i+1}, x_c - a_i] \\
-\frac{(a_{0} + a_1)}{2} & \mbox{if}\; b > x_c
\end{cases}
\,</math>
:<math>
g^*(b) =
\begin{cases}
\{\omega|\omega \in [a_{N-1} + a_N]\} & \mbox{if}\; b < x_c -1 \\
\{\omega|\omega \in [a_{i} + a_{i+1}]\} & \mbox{if}\; b \in [x_c-a_{i+1}, x_c - a_i] \\
\{\omega|\omega \in [a_{0} + a_1]\} & \mbox{if}\; b > x_c
\end{cases}
\,</math>
Where <math>a_i = a_1 i + 2i(1-i)x_c\,</math> and <math>N\,</math> is the largest interger such that <math>|2N(1-N)x_c| < 1\,</math>.
The expected utilities are:
:<math>\mathbb{E}u_f = -\frac{\sigma_{\omega}^2}{N^2} - \frac{x_c^2(N^2-1)}{3}\,</math>
:<math>\mathbb{E}u_c = -\frac{\sigma_{\omega}^2}{N^2} - \frac{x_c^2(N^2-1)}{3}- x_c^2 - k\,</math>
Outcomes are Pareto Optimal iff:
:<math>\omega \in [\frac{(a_i+a_{i+1})}{2}, x_c + \frac{(a_i+a_{i+1})}{2}]\quad i = 0,\ldots,N-1\,</math>
