Alonso Dessein Matouschek (2008) - When Does Coordination Require Centralization

Reference(s)

• Alonso, Ricardo, Wouter Dessein and Niko Matouschek (2008), "When Does Coordination Require Centralization?" American Economic Review, Vol. 98(1), pp. 145-179. 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:

1. Decision rights are allocated
2. Managers learn states [math]\theta_1\;[/math] and [math]\theta_2\;[/math] respectively
3. Managers send messages [math]m_1\;[/math] and [math]m_2\;[/math] respectively
4. 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} \lt 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]

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\;[/math] (the need for coordination).

Communication Equilibria

The paper uses a Crawford and Sobel (1982) type model, which is covered in 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:

1. Communication rules are optimal given the decision rules
2. Decision rules are optimal given belief functions
3. Beliefs are derived from the communication rules using Bayes' rule (whenever possible).