Changes

There are two divisions, <math>j \in \{1,2\}/\;</math>.

Each division makes a decision <math>d/\;</math>, based on local conditions <math>\theta_j in \mathbb{R}/\;</math>.

:<math>\pi = K_1 - (d_1 - \theta_1)^2 - \delta (d_1 - d_2)^2/\;</math>

:<math>\pi = K_2 - (d_2 - \theta_2)^2 - \delta (d_1 - d_2)^2/\;</math>

*<math>K_j \in \mathbb{R}/\;</math>, WLOG <math>K_j = 0/\;</math>*<math>\delta \in [0,\infty]/\;</math> measures the importance of coordination*<math>\theta_j \sim U[-s_j,s_j]/\;</math>, where the distribution is common knowledge but the draw is private

The division managers have preferences (<math>\lambda \in [\frac{1}{2},1]/\;</math> represents bias):

:<math>u_1 = \lambda \pi_1 + (1-\lambda \pi_2)/\;</math>

:<math>u_2 = \lambda \pi_2 + (1-\lambda \pi_1)/\;</math>

:<math>u_h = \pi_1 + \pi_2/\;</math>

The managers can send messages <math>m_1 \in M_1/\;</math> and <math>m_2 \in M_2/\;</math> respectively.

#Managers learn states <math>\theta_1/\;</math> and <math>\theta_2/\;</math> respectively#Managers send messages <math>m_1/\;</math> and <math>m_2/\;</math> respectively#Decisions <math>d_1/\;</math> and <math>d_2/\;</math> are made

HQ determines <math>d_1^C/\;</math> and <math>d_2^C/\;</math> by maximizing <math>u_h/\;</math> with respect to these variables. The solutions are:

:<math>d_1^C - \gamma_C \mathbb{E}[\theta_1|m} ] + (1-\gamma_C) \mathbb{E}[\theta_2|m}/]\;</math>

:<math>d_1^C - \gamma_C \mathbb{E}[\theta_2|m} ] + (1-\gamma_C) \mathbb{E}[\theta_1|m}/]\;</math>

:<math>\gamma_C = \frac{1+2\delta}{1+4\delta}/\;</math>

====Centralization Comparative Statics:====

*<math>\frac{d \gamma_C}{d\delta} < 0, \gamma_C \in [\frac{1}{2},1]/\;</math>*When <math>\delta = 0\;</math>: <math>d_1^C = \mathbb{E}[\theta_1|m]/\;</math>*When<math> \delta = 1\;</math>: <math>d_1^C/\;</math> puts more weight on <math>\mathbb{E}[\theta_2|m]/\;</math>*As <math>\delta \to \infty/\;</math>: equal weight is put on both, <math>d_1^C = \mathbb{E}[\frac{\theta_1 + \theta_2}{2}|m]/\;</math>

Each manager determines their own decision by maximizing <math>u_j/\;</math> with respect to <math>d_j/\;</math>, taking the message from the other party into account. This gives:

:<math>d_1^D = \frac{\lambda}{\lambda + \delta} \theta_1 + \frac{\delta}{\lambda + \delta} \mathbb{E}[d_2|theta_1,m]/\;</math>

:<math>d_1^D = \frac{\lambda}{\lambda + \delta} \theta_2 + \frac{\delta}{\lambda + \delta} \mathbb{E}[d_1|theta_2,m]/\;</math>

Note that the weight each decision puts on local information is increasing the bias <math>\lambda/\;</math>, and decreasing in the need for coordination <math>\delta/\;</math>.

:<math>d_1^D = \frac{\lambda}{\lambda + \delta} \theta_1 + \frac{\delta}{\lambda + \delta} \left(\frac{\delta}{\lambda + 2 \delta} \mathbb{E}[\theta_1|\theta_2,m] + \frac{\lambda+ \delta}{\lambda + 2\delta} \mathbb{E}[\theta_2|theta_1,m] \right )/\;</math>

:<math>d_2^D = \frac{\lambda}{\lambda + \delta} \theta_2 + \frac{\delta}{\lambda + \delta} \left(\frac{\delta}{\lambda + 2 \delta} \mathbb{E}[\theta_2|\theta_1,m] + \frac{\lambda+ \delta}{\lambda + 2\delta} \mathbb{E}[\theta_1|theta_2,m] \right )/\;</math>

====Decentralization Comparative Statics:====*As <math>\delta/\;</math> increases: each manager puts less weight on his own information, and more on a weighted average*As <math>\delta \to \infty/\;</math>: again equal weight is put on both, <math>d_1^C = \mathbb{E}[\frac{\theta_1 + \theta_2}{2}|m]/\;</math>

When <math>\theta=0/\;</math> there is no reason to misrepresent. However, otherwise both under centralization and decentralization their is an incentive to exagerate.

Under centralization, the need for coordination (a high <math>\delta/\;</math>) exacerbates this problem (because the HQ manager is already a little insensitive to local conditions, and now becomes entire insensitive).

Under decentraliztaion, the need for coordination (a high <math>\delta/\;</math>) mitigates this problem (as the managers become more responsive to each other's needs).

Let <math>\nu_1^* = \mathbb{E}[\theta_1|m]/\;</math> be the expection of the local state that 1 would like HQ to have, so that:

:<math>\nu_1^* =arg \max_{\nu_1} \mathbb{E} [ - \lambda(d_1 - \theta_1)^2 -(1-\lambda) (d_2 - \theta_2)^2- \delta (d_1 - d_2)^2 ]/\;</math>

In equilibrium the beliefs of the HQ manager will be correct, so <math>\mathbb{E}_{m_2}( \mathbb{E}[\theta_1|m] ) = \mathbb{E}[\theta_1] = 0/\;</math>, and likewise for <math>\theta_2/\;</math>, so:

:<math>\nu_1^* - \theta_1 = \frac{(2 \lambda - 1) \delta}{\lambda+\delta}\theta_1 = b_C \cdot \theta_1/\;</math>

Where we will call <math>b_C/\;</math> the bias in messages to the HQ. This bias is zero when <math>\theta_1 = 0/\;</math>, and positive otherwise. It is also increasing in <math>| \theta_1 | , \lambda, \delta/\;</math>.

:<math>\nu_1^* - \theta_1 = \frac{(2\lambda -1)(\lambda+\delta)}{\lambda(1-\lambda)+\delta}\theta_1 = b_D \theta_1/\;</math>

Where we will call <math>b_D/\;</math> the bias in messages to the other division manager. This bias is zero when <math>\theta_1 = 0/\;</math>, and positive otherwise. It is also increasing in <math>| \theta_1 |/\;</math> and <math>\lambda/\;</math> (home bias), but decreasing in <math>\delta (the need for coordination).

The paper uses a Crawford and Sobel (1982) type model, which is covered in [[Grossman Helpman (2001) - Special Interest Politics Chapters 4 And 5 | Grossman and Helpman (2001)]], in which the state spaces <math>[-s_1,s_1]/\;</math> and <math>[-s_2,s_2]/\;</math> are partitioned into intervals. The size of the intervals (which determine how informative messages are) depends directly on the biases <math>b_D/\;</math> and <math>b_C/\;</math>.