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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.
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
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