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Alonso Dessein Matouschek (2008) - When Does Coordination Require Centralization (view source)
Revision as of 19:50, 23 November 2010
, 19:50, 23 November 2010no edit summary
==Reference(s)==
*Alonso, Ricardo, Wouter Dessein and Niko Matouschek (2008), "When Does Coordination Require Centralization?" American Economic Review, Vol. 98(1), pp. 145-179. [http://www.edegan.com/pdfs/Alonso%20Dessein%20Matouschek%20(2008)%20-%20When%20Does%20Coordination%20Require%20Centralization.pdf pdf]
==Abstract==
This paper compares centralized and decentralized coordination when managers are privately informed and communicate strategically. We consider a multidivisional organization in which decisions must be adapted to local conditions but also coordinated with each other. Information about local conditions is dispersed and held by self-interested division managers who communicate via cheap talk. The only available formal mechanism is the allocation of decision rights. We show that a higher need for coordination improves horizontal communication but worsens vertical communication. As a result, decentralization can dominate centralization even when coordination is extremely important relative to adaptation.
==The Model==
===Basic Setup===
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>.
The profits of the divisions are given by:
:<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>
Where:
*<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>
The headquarters (HQ) manager has preferences:
:<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.
There are two organisational forms:
*Under '''centralization''' division managers simultaneously send messages to HQ who makes decisions
*Under '''decentralization''' the division managers simultaneously exchange messages and make decisions
The game proceeds are follows:
#Decision rights are allocated
#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
===Decision Making===
====Under Centralization:====
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>
where:
:<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>d_1^C = \mathbb{E}[\theta_1|m]/;</math>
*When<math> \delta = 1: <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>
====Under Decentralization:====
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>.
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>
====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>
===Strategic Communication===
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).
====With HQ (under centralization)====
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>.
====With each other (under decentralization)====
In the same way we can calculate:
:<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).
===Communication Equilibria===
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>.
The game uses a perfect Bayesian equilibria solution concept which requires:
#Communication rules are optimal given the decision rules
#Decision rules are optimal given belief functions
#Beliefs are derived from the communication rules using Bayes' rule (whenever possible).
