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===Other Notes===
<b>Proposition 1</b>: For <math><P^{u}|s=0></math>, the legislative equilibrium is: (i) <math>b^{\ast}\in P</math>, (ii) <math>p^{\ast}(b)=-\bar{\omega}=1/2</math>, (iii) <math>g^{\ast}=\{\omega|\omega\in[0,1]\}</math>, and (iv) EU_{f}(P^{U}|s=0)=-\sigma^{2}_{w}</math>, (v) <math>EU_{c}(P^{U}|s=0)=-\sigma_{\omega}^{2}-x_{c}^{2}</math>.
<b>Proof</b>: <math>g^{\ast}(b)=\{w\in [0,1]\}</math> since b is not a function of w since c does not observe w. FLoor chooses p such that <math>\max_{p}\int_{0}^{1}-(p+w)^{2}f(w)dw, f(w)=1, w\in[0,1]</math>.
<b>Comments</b>: 1) No information transmission, 2) Leads to "info losses" for both the floor and committee, 3) Risk aversion, 4) Incomplete (by which I think he means <i>asymmetric</i>) info is not relevant, 5) Committee plays no role.
Gilligan Krehbiel (1987) - Collective Decision Making And Standing Committees (view source)
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*This page is referenced in [[BPP Field Exam Papers]]
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. (Specifically, see Crawford and Sobel (1982), covered in [[Grossman Helpman (2001) - Special Interest Politics Chapters 4 And 5 |Grossman and Helpman (2001)]]).
====Proposition 2====
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:
Note that it would be efficient for the committee to specialize if <math>k \ge 2K^U\,</math> as well, as this would lead to gains to the floor, but the committee doesn't do so and is therefore 'underspecialized'.
===="Key points" from Rui====
* <math>x_{1},x_{2}</math> have to be equidistant to <math>c</math> at cut point <math>a_{i}</math>.
* Floor beliefs must lie at the midpoint of the interval in <math>v</math>.
* Note that the equilibrium is partially separating.
<b>Proof</b>:
* <b><math>p^{\ast}(b)</math></b>: Start by fixing the committee's strategy. Suppose <math>b\in[x_{c}-a_{i+1},x_{c}-a_{i}]</math>. This implies that <math>\omega\in (a_{i},a_{i+1})</math>. The floor's problem is now <math>\max_{p}-\int_{a_{i}}^{a_{i+1}}(p+\omega)^{2}f(\omega)d(\omega) \implies p=\frac{-(a_{i+1}+a_{i})}{2}</math>. SOC: <math>-2<0 \implies p^{\ast}(b)=\frac{-(a_{i+1}+a_{i})}{2}</math>.
* <b><math>b^{\ast}(\omega)</math></b>: Note that at the cutpoint, the committee is indifferent between which choice the committee makes. <math>[-\frac{a_{i}+a_{1}}{2}]^{2}=[-\frac{a_{i-1}+a_{i}}{2}+a_{i}-x_{c}]^{2}</math>. With <math>a_{i+1}=2a_{i}-a_{i-1}-4x_{c} \implies</math> definitions of <math>a_{i}</math> above.
* Beliefs: Straight forward since when <math>\omega\in[a_{i},a_{i-1}]\implies b^{\ast}(\omega)\in[x_{c}-a_{i+1},x_{c}-a_{i}]\implies g^{\ast}(b)=\{\omega|\omega\in[a_{i},a_{i+1}]\}</math>. Which is consistant because ...
* See paper for proofs of expected utilities.
====Proposition 3====
Compare the expected utilities for the committee in proposition 1 (no specialization) and proposition 2 (specialization) -- both with open amendment rules. With open rules, the committee will specialize iff <math>k<\tilde{k}^{u}</math> where <math>\tilde{k}^{u}=\frac{N^{2}-1}{N^{2}}\sigma^{2}_{\omega}-\frac{N^{2}-1}{3}x_{c}^{2}</math>.
===Closed rule, no specialization===
In this game the committee is not specialized and knows nothing, but it can get an outcome closer to it's ideal point by putting a more attractive bill (than the status quo) on the agenda. <math>p_{0}</math> (the status quo) is exogenous.
The legislative equilibrium is then:
\,</math>
Note that if <math>p_0 = -\overline{\omega}\,</math> then the expected utilities and the Pareto optimality condition are the same as in the open rule with no specialization. ====Rui's Comments ====* All deviated by <math>-\bar{\omega}</math>, so c prefers <math>x_{c}-\omega</math> f prefers <math>-\omega</math>. * "This is exactly Romer and Rosenthal, but with uncertainty."
===Closed rule, specialization===
The case of the closed rule with specialization is much like the open rule with specialization except that for extreme values of <math>\omega\,</math> (specifically <math>\omega \le -3x_c - p_0\,</math> and <math>\omega \ge x_c - p_0)\,</math>, the floor can exactly infer the state of the world and is willing to implement the committee's ideal point, as it prefers this to the status quo. For non-extreme values, noisy signalling again occurs, but now the committee can constrain the floor to choosing between its bill and the status quo. The full details of an equilibrium are in the paper on page 318.
Equilibrium is separating if <math>0<\omega<-3x_{c}-p_{0}</math> or <math>1>\omega>x_{c}-p_{0}</math>. Otherwise pooling. Winner of vote is b iff <math>0<\omega<-x_{c}-p_{0}</math> or <math>x_{c}-p_{0}</math>. Otherwise <math>p_{0}</math> wins.
Again, we can derive a condition for when the committee would wish to specialize. Crucially, under a closed rule, a committee may choose to 'overspecialize', because this results in distributional gains. A committee will specialize iff:
*Extreme iff <math>x_c \in (x_c'',x_c')\,</math>
*Very Extreme iff <math>x_c \ge x_c' \,</math>
We can then derive a set of conditions under which the floor would choose an open or closed rule, and in essence which game to play:
**For costs above the closed rule cut off the floor's expected utility is unaffected.
*For very '''extreme''' committees - the '''open rule''' is preferred irrespective of the cost of specialization
