Specialization, implicit coordination and organizational performance: trading off common and idiosyncratic knowledge

Abstract

This paper reconsiders the trade-off between specialization and coordination by focusing on as yet neglected forms of implicit coordination. While specialization draws on idiosyncratic knowledge pertaining to the worker’s specific tasks and environment, implicit coordination is based on knowledge common to all workers. Accordingly, the trade-off between specialization and coordination translates into one between idiosyncratic and common knowledge. Investigating this trade-off I determine the organization’s optimal knowledge structure depending on a number of organizational and environmental parameters. Further extensions of the model allow for explicit coordination through communication, differentiate between common knowledge at the corporate level and departmental level, and apply important results to the issue of corporate culture, interpreted as a specific form of common knowledge that likewise facilitates implicit coordination.

Appendices

Appendix 1

In this appendix, I generalize the model as laid out in the main body by showing that the stated propositions also hold for an arbitrary h(p) subject to $\partial h(p)/\partial p > 0$, $\mathop {\lim}\nolimits_{p \to 0} h(p) = 0$ and $\mathop {\lim}\nolimits_{p \to 1} h(p) = \infty$ as well as for an arbitrary φ(p) subject to $\partial \varphi (p)/\partial p < 0$, φ(0) = 1 and φ(1) = 0. In this case, I derive p* by drawing on the first order condition of the generalized payoff function

$$ E(P(p)) = - n \cdot {\frac{{\alpha + (n - 1) \cdot \beta \cdot (1 - \varphi (p)) \cdot \sigma_{\theta}^{2} \cdot h(p)}}{{{\frac{1}{{\sigma_{\theta}^{2}}}} + h(p)}}}. $$

(22)

p* is then implicitly determined by

$$ 0 = \underbrace {{(n - 1) \cdot \beta \cdot \left({\left({1 - \varphi (p)} \right) \cdot h^{\prime}(p) - \varphi^{\prime}(p) \cdot h(p) - \sigma_{\theta}^{2} \cdot \varphi^{\prime}(p) \cdot h(p)^{2}} \right)}}_{= :A(p)} - \underbrace {{\alpha \cdot h^{\prime}(p)}}_{= :B(p)} $$

(23)

in case of an interior solution, i.e., $p^{*} \in]0,1[$, or takes the value $p^{*} = 1$ in case of a corner solution. Note that as $B(0) > A(0) = 0$, the total absence of specialization (i.e., p = 0) is never optimal. Should there exist multiple local maxima on [0,1], p* clearly takes the value of the global maximum. It can, however, be shown that subject to the additional assumption of the convexity of h(p) as well as of h′(p), multiple maxima can be ruled out.

Generalized Proof of Propositions 1–3

Assume $p_{0}^{*} \in]0,1[$ as the propositions pertain to the case of an interior solution. A decrease of α—an increase of β, $\sigma_{\theta}^{2}$ , n, respectively—then yields $A(p_{0}^{*}) > B(p_{0}^{*})$. We know, however, that B(0) > A(0). From this follows by the intermediate value theorem that $\exists p_{1}^{*} \in]0,p_{0}^{*} [:A(p_{1}^{*}) = B(p_{1}^{*})$. $\square$

Generalized Proof of Proposition 4

Obtaining the second derivative of (22) with respect to n yields

$$ {\frac{{\partial^{2} E(P)}}{{\partial n^{2}}}} = - {\frac{{2 \cdot \beta \cdot (1 - \varphi (p)) \cdot h(p) \cdot \sigma_{\theta}^{2}}}{{{\frac{1}{{\sigma_{\theta}^{2}}}} + h(p)}}} < 0. $$

(24)

$\square$

Generalized Proof of Proposition 5

To incorporate communication I extend (22) by analogy to (12) to

$$ E\left({P(p)} \right) = - n \cdot {\frac{{\alpha + (n - 1) \cdot \beta \cdot (1 - \varphi (p)) \cdot \sigma_{\theta}^{2} \cdot h(p) \cdot \left[{1 - \varphi (p) - (1 - p) \cdot \tau + \varphi (p) \cdot (1 - p) \cdot \tau} \right]}}{{{\frac{1}{{\sigma_{\theta}^{2}}}} + h(p)}}}. $$

(25)

Differentiating this extended payoff function with respect to τ yields

$$ {\frac{\partial E(P)}{\partial \tau}} = {\frac{{n \cdot (n - 1) \cdot \beta \cdot (1 - \varphi (p))^{2} \cdot h(p) \cdot \sigma_{\theta}^{2} \cdot (1 - p)}}{{{\frac{1}{{\sigma_{\theta}^{2}}}} + h(p)}}} \ge 0. $$

(26)

$\square$

Generalized Proof of Proposition 6 (constructive sketch)

Assume again the existence of an interior solution, i.e., $p^{*} \in]0,1[$. Proposition 6 then implies that $\exists \bar{\alpha}\forall \tau \in [0,1]:A_{\tau} (p^{*}) = B(p^{*})$. This $\bar{\alpha}$ can be obtained as follows: Determine the common intersection point p ₀ by choosing two arbitrary $\tau_{1},\tau_{2} \in [0,1]$, $\tau_{1} \ne \tau_{2}$, and solving $A_{{\tau_{1}}} (p) = A_{{\tau_{2}}} (p)$ for p. $\bar{\alpha}$ is then implicitly determined by $A_{\tau} (p_{0}) = B(p_{0})$.$\square$

Generalized Proof of Proposition 7

Identical to the one given in the main body of the paper.$\square$

Appendix 2

In the basic model I assume that workers seek to maximize the outcome of the actions under their control while they neglect other workers’ results. This induces a kind of goal incongruence that transcends orthodox team theory. In this appendix, I discuss the opposite case of perfect goal alignment (and, concluding, varying degrees of alignment). I will show that in this setting, coordination by common knowledge is (partially) crowded out by another form of coordination, namely by adhering to the status quo and reduced adaptation, similar to that ascertained by Dessein and Santos (2006). Nevertheless, the main propositions hold true.

In case of perfect goal alignment, workers jointly seek to maximize (1). Their decision problem therefore constitutes a game with identical payoffs. In equilibrium, complementary actions are set to $\forall j \in \{1, \ldots,n\} \backslash \{i\} :b_{ij} = E(a_{i})$ and primary actions are accordingly chosen to maximize

$$ \tilde{P}\left({a\left| {s_{i},p} \right.} \right) = - \sum\limits_{i = 1}^{n} {\left[{\alpha \cdot \left({E\left({\theta_{i} \left| {s_{i},p} \right.} \right) - a_{i}} \right)^{2} + \left({n - 1} \right) \cdot \beta \cdot \left({1 - \varphi \left(p \right)} \right) \cdot \left({a_{i} - E\left({a_{i}} \right)} \right)^{2}} \right]}, $$

(27)

where $E(\theta_{i} \left| {s_{i},p} \right.)$ is the ex-post expectation value of θ _i after observing signal s _i with the precision h(p) as stated by (3). Plugging (3) into (27), I derive the optimal primary action as

$$ \tilde{a}_{i}^{*} = \left({1 - A} \right) \cdot \theta^{0} + A \cdot s_{i},\quad A: = {\frac{\alpha \cdot h(p)}{{\left({\alpha + \left({n - 1} \right) \cdot \beta \cdot \left({1 - \varphi \left(p \right)} \right)} \right) \cdot \left({{\frac{1}{{\sigma_{\theta}^{2}}}} + h(p)} \right)}}} $$

(28)

and the optimal complementary actions accordingly as

$$ \tilde{b}_{ij}^{*} = E(a_{i}) = \theta^{0}. $$

(29)

Whereas the optimal complementary actions remain the same, the worker choosing the primary action underweights her private information (i.e., the signal) as compared to the basic model discussed in the main body. In other words, she consciously forgoes adaptation and therefore local efficiency to foster smooth integration and global coordination of activities. To keep things tractable, I assume again in the following h(p) = p/(1 − p) and φ(p) = 1 − p (a generalization to an arbitrary h(p) and φ(p) would proceed along the lines of Appendix 1). The adjusted payoff function is then given by

$$ \tilde{E}\left({P(p)} \right) = - \frac{1}{p} \cdot \left[{\left({1 - A} \right)^{2} \cdot \alpha \cdot \sigma_{\theta}^{2} \cdot p + A^{2} \cdot \left({\alpha \cdot \left({1 - p} \right) + \left({n - 1} \right) \cdot \beta \cdot p \cdot \left({p \cdot \left({\sigma_{\theta}^{2} - 1} \right) + 1} \right)} \right)} \right], $$

(30)

and the optimal fraction of idiosyncratic knowledge is

$$ \tilde{p}^{*} = \mathop {\arg \max}\limits_{{p \in \left[{0,1} \right]}} \tilde{E}\left({P\left(p \right)} \right) = {\frac{\sqrt \alpha}{{\sqrt {\beta \cdot \left({n - 1} \right) \cdot \left({\sigma_{\theta}^{2} - 1} \right)}}}}. $$

(31)

It is immediate that Propositions 1–3 hold true. Likewise, the validity of Propositions 5 and 7 can easily be shown in this setting. This is not true, however, for Propositions 4 and 6. Additionally, I yield the general result that in case of goal alignment, the optimal fraction of idiosyncratic knowledge is always greater than in the event of goal incongruence, as

$$ \tilde{p}^{*} - p^{*} = {\frac{\sqrt \alpha}{{\sqrt {\beta \cdot \left({n - 1} \right) \cdot \left({\sigma_{\theta}^{2} - 1} \right)}}}} - {\frac{{\sqrt {\beta \cdot \left({n - 1} \right) + \alpha \cdot \left({\sigma_{\theta}^{2} - 1} \right)} - \sqrt {\beta \cdot \left({n - 1} \right)}}}{{\sqrt {\beta \cdot \left({n - 1} \right)} \cdot \left({\sigma_{\theta}^{2} - 1} \right)}}} > 0. $$

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By combining ‘local’ and ‘global’ incentives, i.e., by weighting the opposite cases of goal incongruence (as in the main body) and perfect goal alignment (as in this appendix), one could easily achieve varying degrees of goal alignment. The results then evidently generalize: The higher the degree of goal congruency, the less private information is used for determining primary actions, the more coordination via common knowledge is crowded out by coordination via sticking to the status quo, and the greater is the optimal fraction of idiosyncratic knowledge.