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Alonso Dessein Matouschek (2008) - When Does Coordination Require Centralization (view source)
Revision as of 19:14, 29 September 2020
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|Has article title=When Does Coordination Require Centralization
|Has author=Alonso Dessein Matouschek
|Has year=2008
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*This page is referenced in [[PHDBA602 (Theory of the Firm)]]
==Reference(s)==
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>
====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).
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).
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).
Communication rules are optimal given the decision rules if (for <math>l \in \{C,D\}\;</math>):
*<math>\mu_j(m_j|\theta_j)\;</math> is the probability of sending message <math>m_j\;</math> if the state is <math>\theta_j\;</math>
*whenever <math>\mu_j(m_j|\theta_j)>0\;</math>, <math> m_j \in arg \max_{m \in M} \mathbb{E} (\lambda \pi_j^l + (q-\lambda)\pi_k^l | theta_j )\;</math>
*where <math>\pi_j\;</math> takes <math>d_j\;</math> and <math>d_k\;</math> as given.
Decision rules are optimal given belief functions if:
*Under centralization, <math>d_1^C\;</math> and <math>d_2^C\;</math> solve <math>\max_{(d_1,d_2)} \mathbb{E} (\pi_1 + \pi_2 |m)\;</math>
*Under decentralization <math>d_j^D\;</math> solves <math>\max_{d_j} \mathbb{E} ( \lambda \pi_j + (1-\lambda) \pi_k)|m,\theta_k)\;</math>
Beliefs satisfy:
*<math>g(\theta_j|m) = \frac{\mu_j(m_j|\theta_j)}{\int_P \mu_j(m_j|\theta_j) d\theta_j}\;</math>
*where <math>P = \{\theta_j: \mu_j(m_j|\theta_j) >0 \}\;</math>
The equilibria are finite interval equilibria with end points symmetric around zero. Adopting the notation <math>a_{j,i}\;</math> to indicate an interval end point for manager <math>j\;</math>, number <math>i\;</math>, we have:
*<math>a_{j,-N} = -s_j\;</math>
*<math>a_{j,0} = 0\;</math>
*<math>a_{j,N} = s_j\;</math>
*and <math>a_{j,i}=a_{j,-i}\;</math> by symmetry
Proposition 1 in the paper gives the communication equilibria: If <math>\delta \in (0, \infty)\;</math>, then for every positive interger <math>N_j\;</math>, there exists at least one equilibrium <math>(\mu_1(\cdot),\mu_2(\cdot),d_1(\cdot),d_2(\cdot),g_1(\cdot),g_2(\cdot))\;</math> where:
#<math>\mu_j(m_j|\theta_j) \sim U[a_{j,i-1},a_{j,i}]\;</math> if <math>\theta_j\;</math> in <math>(a_{j,i-1},a_{j,i})\;</math>
#<math>g_j(\theta_j|m_j) \sim U[a_{j,i-1},a_{j,i}]\;</math> if <math>m_j\;</math> in <math>(a_{j,i-1},a_{j,i})\;</math>
#<math>a_{j,i+1}-a_{j,i} = a_{j,i} -a_{j,i-1} + 4b a_{j,i}\;</math> for <math>i=1,\ldots,N_j-1\;</math>, where <math>b = b_C\;</math> or <math>b_D\;</math> appropriately
#<math>d_j(m,\theta_j) = d_j^l\;</math>
Because there is truth telling at 0, and as intervals get smaller towards zero, there can exist an infinite number of intervals. As in Crawford and Sobel (1982), the highest sustainable number of intervals is the most efficient, and so an infinite number of intervals should be used for the best communication, as this will yield the highest expected profits (see proposition 2).
===The Remainder of the Paper===
The remainder of the paper is concerned with:
*A comparison of communication - the quality of communication (in terms of the residual variance <math>\mathbb{E}((\theta_j - \mathbb{E}(\theta_j|m_j))^2)\;</math> ) is weakly higher under decentralization, and increases in <math>\lambda\;</math> faster.
*Organizational Performance - First best profits could be achieved with a perfectly informed HQ, but since this is not possible, it depends on the bias of managers and the need for coordination as to which is the better form
*A comparison of Centralization vs Decentralization - There is an adaptational advantage (to changes in local conditions) to the decentralized form, and a coordination advantage to the centralized one. For various parameter values (of <math>\lambda\;</math> and <math>\delta\;</math>) these are compared.
*A brief discussion of empirical implications, with some support for the model from two cases
*A discussion of changes to the model to handle asymmetries, with regards to either a first mover advantage or parameterization. Rantakari (2006) provides a model to address these in detail apparently.
