A comment on Koh’s “The optimal design of fallible organizations: invariance of optimal decision threshold and uniqueness of hierarchy and polyarchy structures”

Abstract

Koh (Soc Choice Welf 25:207–220, 2005) studied the project evaluation problem when decision-makers in an organization are fallible and showed that in the absence of evaluation costs the optimal organization size and the optimal majority rule are not unique. We show that, in the absence of evaluation costs, the optimal organization size is $\infty$, which is also conventional because more evaluations can always lead to better judgment. Thus, more evaluations are desirable when there are no additional costs. Koh (Soc Choice Welf 25:207–220, 2005) also claimed that, in the presence of evaluation costs, polyarchy and hierarchy are the only possible optimal structures. We disprove this conclusion using a simple numerical example.

Appendix

1.1 Proof of Lemma 2

Let $p_Q$ be the probability of an evaluator recommending a project when $Q=G$ or B. $p_G= 1-H(\theta |G)$ and $p_B = 1-H(\theta |B)$. Thus, $P(\theta , n, k , Q) \equiv P(p, n, k)$. By definition, for any n and k,

$$\begin{aligned} P(\theta , n+1, k,Q) - P(\theta ,n, k,Q) = \begin{pmatrix} n\\ k-1 \end{pmatrix} p_Q^k (1-p_Q)^{n-k+1}, \end{aligned}$$

and

$$\begin{aligned}&P(\theta ,n+1, k + 1,Q) - P(\theta ,n,k,Q) \\&\quad = P(\theta ,n+1, k+1,Q) - P(\theta ,n+1,k,Q) + P(\theta ,n+1,k,Q) - P(\theta ,n,k,Q) \\&\quad = -\begin{pmatrix} n+1\\ k \end{pmatrix} p_Q^k (1-p_Q)^{n+1-k} + \begin{pmatrix} n\\ k-1 \end{pmatrix}p_Q^k (1-p_Q)^{n+1-k}\\&\quad =- \frac{n+1-k}{k} \begin{pmatrix} n\\ k-1 \end{pmatrix}p_Q^k (1-p_Q)^{n+1-k} . \end{aligned}$$

We therefore have

$$\begin{aligned} V(n+1, k) - V(n, k)&= \alpha \pi (G) \begin{pmatrix} n\\ k-1 \end{pmatrix}p_G^{k} (1-p_G)^{n+1-{k}} \\&\quad -(1-\alpha )\pi (B) \begin{pmatrix} n\\ k-1 \end{pmatrix}p_B^{k} (1-p_B)^{n+1-{k}}, \\ V(n+1, k + 1) - V(n, k)&= - \frac{n+1-k}{k} [\alpha \pi (G) \begin{pmatrix} n\\ k-1 \end{pmatrix}p_G^{k} (1-p_G)^{n+1-{k}}\\&\quad -(1-\alpha )\pi (B) \begin{pmatrix} n\\ k-1 \end{pmatrix}p_B^{k} (1-p_B)^{n+1-{k}}]\\&= - \frac{n+1-k}{k} [V(n+1, k) - V(n, k)]. \end{aligned}$$

Because $- (n+1-k)/k < 0$, this shows $V(n+1, k + 1) - V(n, k)$ and $[V(n+1, k) - V(n, k)]$ must always have different signs, implying that V(n, k) is between $V(n+1, k + 1)$ and $V(n+1, k)$, i.e.,

$$\begin{aligned} \min (V(n+1, k + 1), V(n+1, k)) \le V(n, k) \le \max (V(n+1, k + 1), V(n+1, k)). \end{aligned}$$

1.2 Proof of Theorem 1

We will show that no finite number $n^*$ exists. Suppose there is a finite optimal organization size, $n^*$, the optimal majority rule $k^* (\le n^*)$ also finite. From Lemmas 1 and 2, it is easy to see that

$$\begin{aligned} V(\theta ^*, n^*, k^*) = V(\theta ^*, n^*+1, k^*) = V(\theta ^*, n^*+1, k^* + 1). \end{aligned}$$

(15)

This also indicates that $n^*+1$, and thus, $n^*+2$, $n^*+3,\ldots$ are all optimal.

From (15), we have

$$\begin{aligned} \alpha [1-H(\theta ^*|G)]^{k^*}H(\theta ^*|G)^{n^*+1-k^*} = (1-\alpha )[1-H(\theta ^*|B)]^{k^*}H(\theta ^*|B)^{n^*+1-k^*} \end{aligned}$$

(16)

Similarly if $n^*+1$ is also optimal, we have

$$\begin{aligned} V(\theta ^*, n^*+2, k^*) = V(\theta ^*, n^*+2, k^* + 1), \end{aligned}$$

(17)

which implies

$$\begin{aligned} \alpha [1-H(\theta ^*|G)]^{k^*}H(\theta ^*|G)^{n^*+2-k^*} = (1-\alpha )[1-H(\theta ^*|B)]^{k^*}H(\theta ^*|B)^{n^*+2-k^*} \end{aligned}$$

(18)

Equations (16) and (18) imply

$$\begin{aligned} H(\theta ^*|G) = H(\theta ^*|B). \end{aligned}$$

(19)

This contradicts our assumption. Therefore, $n^* = \infty$ is the only optimal solution for n.