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).

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 if the state is \lt math\gt \theta_j\;[/math]
• whenever [math]\mu_j(m_j|\theta_j)\gt 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) \gt 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:

1. [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]
2. [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]
3. [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
4. [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.