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.
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Funding was provided by Australian Research Council (DP130100766) and Australian Research Council (AU) (DP160104292).
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The authors thank the referee team for providing critical and constructive comments, leading to this much improved version of this paper.
Appendix
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,
and
We therefore have
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.,
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
This also indicates that \(n^*+1\), and thus, \(n^*+2\), \(n^*+3,\ldots \) are all optimal.
From (15), we have
Similarly if \(n^*+1\) is also optimal, we have
which implies
This contradicts our assumption. Therefore, \(n^* = \infty \) is the only optimal solution for n.
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Zhu, M., Liu, C. & Wang, YG. A comment on Koh’s “The optimal design of fallible organizations: invariance of optimal decision threshold and uniqueness of hierarchy and polyarchy structures”. Soc Choice Welf 48, 385–392 (2017). https://doi.org/10.1007/s00355-016-1009-5
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DOI: https://doi.org/10.1007/s00355-016-1009-5