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Gilligan Krehbiel (1987) - Collective Decision Making And Standing Committees (view source)
Revision as of 18:14, 29 September 2020
, 18:14, 29 September 2020no edit summary
{{Article
|Has page=Gilligan Krehbiel (1987) - Collective Decision Making And Standing Committees
|Has bibtex key=
|Has article title=Collective Decision Making And Standing Committees
|Has author=Gilligan Krehbiel
|Has year=1987
|In journal=
|In volume=
|In number=
|Has pages=
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}}
*This page is referenced in [[BPP Field Exam Papers]]
==Reference(s)==
==The Model==
<b>Players</b>: There are two bodies:. *c: A committee *f: The legislature, or parent chamber, or 'floor', that uses a majority rule
<b>Choice space</b>: There are two proceeduresprocedures:
*<math>P^R\,</math> is the restrictive proceedure (closed rule) where no amendments are allowed and the policy is voted against the status quo
*<math>P^U\,</math> is the unrestrictive proceedure (open rule) where the parent body may choose any alternative to the policy.
* ... and a policy diminson <math>p\in P=R</math> the
<b>Policy outcomes</b>: The outcome (<math>x\,</math>) is linear in both the policy (<math>p\,</math>) and random variable (<math>\omega \sim U[0,1]\,</math>, such that <math>\mathbb{E}(\omega) = \overline{\omega}\,</math> and <math>\mathbb{V}(\omega) = \sigma_{\omega}^2\,</math>) concerning the state of the world. That is:
:<math>x = p+ \omega\,</math>
<b>Utilities </b> are negative quadratice about ideal points (<math>x_f = 0\,</math> and <math>x_c > 0\,</math>). The committee can incur a cost <math>k\,</math> to learn the state of the world if it chooses to specialize (<math>s \in \{0,1\}\,</math>). The floor knows if the committee has specialized but not what it has learnt.
:<math>u_f = -(x-x_f)^2 = -x^2\,</math>
:<math>u_c = -(x-x_c)^2 - sk\,</math>
<b>Information</b>: S is known (everybody knows whether specialization happened). Specific realization of <math>\omega</math> is unknown. Priors about <math>\omega</math> are common.
The sequence of the game is as follows:
#Nature chooses the state of the world <math>\omega \sim U[0,1]\,</math>
#The floor chooses <math>P \in \{P^U,P^R\}\,</math> (Note not to be confused with the policy space <math>P\,</math>)
#The committee chooses <math>s \in \{0,1\}\,</math> (i.e. symmetric or asymmetric uncertainty-- whether to specialize or not.)#Nature chooses the state of the world <math>\omega \sim U[0,1]\,</math>
#The committee reports a bill <math>b \in P \subset R^1\,</math>
#The floor updates its beliefs <math>g \in [0,1]\,</math>
#A policy is choosen chosen <math>p \in P \subset R^1 \mbox{ if } P^U\,</math> or <math>p \in \{p_0,b\} \mbox{ if } P^R\,</math>. If the floor can offer amendments, it does.
#There are consequences and payoffs: <math>x, u_f, u_c\,</math> all determined
===Open rule, no specialization===
<b>PROPOSITION 1</b>: The equilibrium is:
:<math>b^* \in P, \quad p*(b) = -\overline{\omega}, \quad g^*(b) = \{w|w \in [0,1]\}\,</math>
This is found by taking <math>\mathbb{E}u_f = \mathbb{E}(-(\overline{\omega}+\omega)^2) = -(\mathbb{E}(\omega^2)-\overline{\omega}^2) = -\sigma_{\omega}^2\,</math>. And likewise for the committee. Note that both have informational losses, and the committee has a distributional loss.
<b>Proof of Prop 1</b> (same as above): <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>.
Outcomes are Pareto Optimal iff:
:<math>\omega \in [\overline{\omega}, x_c + \overline{\omega}]\,</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.
===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. (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:
\,</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>.Note that <math>b^{\ast}(\omega)</math> is a correspondance!
See figure 4, p310 of the paper for a graphic intuitiion.
As in cheap talk models, the number of partitions that can be sustained is a function of the bias of the reporting agent. If <math>x_c \ge 3\sigma_{\omega}^2\,</math> only one partition can be supported and the model reverts to the unspecialized case. Likewise, as the bias decreases the number of partitions increases and the floor is able to make more refined inferences about <math>\omega\,</math>, and there is less loss due to uncertainty.
The committee chooses to specialize if its expected utility is higher. Put another way, it specializes if the cost of specialization <math>k\,</math> is less or equal to than some cut off <math>k^U\,</math>, where:
:<math>K^U = \sigma_{\omega}^2 \left( 1- \frac{1}{N^2}\right) - \frac{x_c^2(N^2-1)}{3}\,</math>
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>
b^*(\omega) =
\begin{cases}
x_c-\overline{\omega} & \mbox{if}\; p_0 \le -x_c-\overline{\omega} \mbox{ or } p_0 \ge x_c-\overline{\omega} \\
-p_0-1 & \mbox{if}\; p_0 \in (-x_c-\overline{\omega}, -\overline{\omega}) \\
b' & \mbox{if}\; P_0 \in \left[ -\overline{\omega}, x_c-\overline{\omega}\right)\mbox{ where } \mathbb{E}u_f(b') \le \mathbb{E}u_f(p_0)
\end{cases}
\,</math>
:<math>
p^*(b) =
\begin{cases}
b & \mbox{if}\;\mathbb{E}u_f(b') \ge \mathbb{E}u_f(p_0) \\
p_0 & \mbox{if}\; \mathbb{E}u_f(b') < \mathbb{E}u_f(p_0)
\end{cases}
\,</math>
:<math>
g^*(b) = \{\omega|\omega \in [0,1]\}
\,</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:
:<math>k \le k^R + x_c^2 \mbox{ where } k^R = \sigma_{\omega}^2(1-(4x_c)^3)\,</math>
The paper also implicity defines two bias values:
:<math>x_c'' \mbox{ solves } K^R = K^U + (x_c'')^2\,</math>
:<math>x_c' \mbox{ solves } K^R = (x_c')^2\,</math>
A committee is called:
*Moderate iff <math>x_c \le x_c'' \,</math>
*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 '''moderate''' committees - the '''closed rule''' is preferred irrespective of the cost of specialization (the informational gains outweight the distributive losses)
*For '''extreme''' committees - the choice of rule depends on the cost of specialization
**The cost cut-off for specialization under the open rule is lower than that under the closed rule.
**However, for costs '''below the open rule cut off''' the informational gains from a closed rule are less than the distributional losses - suggesting '''an open rule should be chosen'''.
**For costs '''between the cut-offs''' using a closed rule forces specialization, and the informational gains outweigh the losses - so '''a closed fule should be choosen'''.
**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
