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Alonso Dessein Matouschek (2008) - When Does Coordination Require Centralization (view source)
Revision as of 18:59, 23 November 2010
, 18:59, 23 November 2010no edit summary
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>
By taking expectations and subbing back in, we get:
:<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>
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 \;</math> (the need for coordination).
