Changes

In general there is a policy maker who choses a policy to implement <math>p\,</math>, based on some facts about the state of the world <math>\theta\,</math>. There is also a Special Interest Group (SIG) who observes the facts about the state of the world but has a bias <math>\delta > 0\,</math>. The utility functions of both types of players are inverse quadratic(i.e. quadratic loss functions).

:<math>\threfore therefore \delta \le \frac{\theta_H - \theta_L}{2}\,</math>

Again there is no incentive for the SIG to misreport <math>\theta_H\,</math> when it is the true state. When <math>\theta_M\,</math> is the true state we can perform the bias restriction calculation as before to get: :<math>\delta \le \frac{\theta_H-\theta_M}{2}\,</math> If we were just distinguishing between all three states this would hold for <math>\theta_L\,</math> too:  :<math>\delta \le \frac{\theta_M-\theta_L}{2}\,</math> However, the SIG now has the option of reporting <math>\theta_L\,</math> or not <math>\theta_L\,</math>, in the latter case expecting the policy maker to implement <math>\frac{(\theta_H-\theta_M)}{2}\,</math>. In this case of partial information transmission we have to recalculate the bias restriction for when the real state of the world is <math>\theta_L\,</math> (For <math>\theta_M\,</math> the bias restriction is less binding). Here the SIG's ideal point is <math>\theta_L+\delta\,</math> so the SIG will not report falsely iff:  :<math>(\theta_L+\delta) - \theta_L \ge \frac{(\theta_H + \theta_M)}{2} - (\theta_L + \delta)\,</math> :<math>\therefore \delta \le \frac{\theta_H - \theta_M}{4} + {\theta_M - \theta_L}{2}\,</math> This allows for an equilibrium with partial transmission of information - this can be possible when the bias conditions for full transmission are violated.  ===One Lobby - Continuous States===  As the number of states grows it becomes harder to get full relevation - the bias must be less than half of the distance between two states and this tends to zero as the number of states tends towards infinity. However, we can partition a continuous range of information and report credible with range the true state of the world falls in.   Suppose that: :<math>\tilde{\theta} \sim U[\theta_{min},\theta_{max}]\,</math>  The SIG then reports <math>\theta_1\,</math> if <math>\theta_{min} \le \theta \le \theta_1\,</math> and so forth. The policy maker getting such a report implements <math>\frac{(\theta_{min} + \theta_1)}{2}\,</math>. The temptation to lie is greatest when <math>\theta\,</math> is on the boundary of <math>\theta_1\,</math>, and <math>\theta_2\,</math> is the greatest temptation (all other higher values greater exagerate the outcome and will overshoot the bliss point). Credibility therefore requires:  :<math>\frac{(\theta_1 + \theta_2)}{2} - (\theta_1 + \delta) \ge (\theta_1 + \delta) - \frac{(\theta_{min} + \theta_1)}{2}\,</math> This solves to:  :<math>\theta_2 \ge 2\theta_1 +4\delta - \theta_{min}\,</math> If the truth is in <math>\theta_2\,</math>, then the SIG would prefer to report this rather than <math>\theta_1\,</math> iif:  :<math>(\theta_2 + \delta) - \frac{(\theta_1 + \theta_2)}{2} \le (\theta_1 + \delta) - \frac{(\theta_{min} + \theta_1)}{2}\,</math> This solves to:  :<math>\theta_2 \le 2\theta_1 +4\delta - \theta_{min}\,</math> Putting these two together we have that:  :<math>\theta_2 = 2\theta_1 +4\delta - \theta_{min}\,</math> More generally, as these equations have to hold for each and every interval:  :<math>\theta_j = 2\theta_{j-1} +4\delta - \theta_{j-2}\,</math> There are two boundaries where <math>\theta_n = \theta_{max}\,</math> and <math>\theta_0 = \theta_{min}\,</math>, which can be used together to solve the general solutions for <math>\theta_j\,</math>:  :<math>\theta_j = \frac{j}{n}\theta_{max} +\frac{n-j}{n}\theta_{min} -j(n-j)\delta\,</math> Considering that <math>\theta_1 > \theta_{min}\,</math> it must be the case that:  :<math>2n(n-1)\delta < \theta_{max} - \theta_{min}\,</math>  There are some things to note:#Partitions get bigger to the right#There always exists an equilibrium with <math>n=1\,</math> - it is the babbling equilibrium#As <math>\delta\,</math> gets smaller, more partitions can be sustained. #If <math>n\,</math> partitions can be sustained then so can <math>k<n\,</math>.  ===Welfare considerations=== More partitions are more informative and yield greater welfare. The equilibrium with the greatest number of partitions is the one preferred by both the SIG and the policymaker.  ===Two Lobbies - Like bias=== Suppose that <math>0 < \delta_1 \le \delta_2\,</math> so that SIG 1 is more moderate. There are three cases to consider: secret meetings, private meetings and public meetings. With secret meetings the SIGs do not know if the other SIG has met with the policy maker, and with private meetings they know the meeting has taken place but do not know the contents.  ====Secret Meetings==== The SIGs behave as if they alone are providing information and the policy maker combines informations. The combined information is more informative, but there is no actual equilibrium (because of the implicit assumption of no strategic response).  ====Private Meetings==== Using each SIG to discipline the other, the policy maker can gain full relevation. However, this is a fragile equilibrium.  ====Public Meetings==== When information is reported sequentially there can not be full relevation. There can be a partition equilibrium from combining the two reports. The number of partitions can not be greater than the number that would arise from a single lobby with the smaller (more moderate) bias, and so both would agree to allow only this SIG to report.  ===Two Lobbies - Opposite Bias=== With two lobbies where <math>\delta_1 < 0\,</math> and <math>\delta_2 > 0 \,</math> the policy maker can use the competition to become more informed, though not fully informed. Crucially, define <math>\theta_2^* \equiv \theta_{max} - \delta_2\,</math>  It is not possible to reveal the state of the world when <math>\theta > \theta_2^*\,</math>. See the paper for further information. ===Multi-dimensional Information=== When the groups differ in their relative bias on a single dimension there is full revelation. By choosing new dimensions through the multidimensional space it is possible (given the necessary dimensionality of the choices and alignment conditions) to establish full relevation through-out the space.