Abstract
The current note clarifies why, in committees, the prior probability of a correct collective choice might be of particular significance and possibly should sometimes even be the sole appropriate basis for making the collective decision. In particular, we present sufficient conditions for the superiority of a rule based solely on the prior relative to the simple majority rule, even when the decisional skills of the committee members are assumed to be homogeneous.
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Notes
The general uncertain dichotomous choice model can be used to study optimal informative voting in any voting body, Baharad et al. (2011, 2012); Ben-Yashar and Danziger (2011); Ben-Yashar and Kraus (2002); Ben-Yashar and Nitzan (1997); Nitzan (2010); Nitzan and Paroush (1982, 1985); Nurmi (2002); Shapley and Grofman (1984); Young (1995). Here both decisional skills and the prior of a correct collective decision are explicitly taken into account and, of course, the optimal rule need not be the SMR.
Proof: As noted above,
$$\begin{aligned} f^{*}=sign\left( \sum _{i=1}^n w x_i +\gamma \right) \end{aligned}$$The SMR is defined as follows:
$$\begin{aligned} f^{SMR}=sign\left( \sum _{i=1}^n x_i\right) \end{aligned}$$Therefore, the SMR is optimal if
$$\begin{aligned} \forall x\left( {\sum _{i=1}^n w x_i +\gamma >0\Leftrightarrow \sum _{i=1}^n {x_i } >0} \right) \end{aligned}$$Notice that for \(x\) such that the number of supporters in alternative 1 is \(\frac{n-1}{2}\),
$$\begin{aligned}&f^{*}=sign\left( (\frac{n-1}{2}-\frac{n+1}{2})\ln \frac{p}{1-p})+\ln \frac{\alpha }{1-\alpha }\right) =sign\left( -\ln \frac{p}{1-p}+\ln \frac{\alpha }{1-\alpha }\right) =1, \\&\qquad \quad \text{ since, } \text{ by } \text{ assumption }, \alpha >p.\\&f^{\mathrm{SMR}}=sign\left( \frac{n-1}{2}-\frac{n+1}{2}\right) =-1. \text{ That } \text{ is }, f^{*}\ne f^{\mathrm{SMR}}. \end{aligned}$$\(\square \)
Proof: Note that when \(\alpha >1/2\) , \(f^{\mathrm{PBR}}=1.\)
$$\begin{aligned}&f^{*}=f^{\mathrm{PBR}}=1, if \forall x, \end{aligned}$$\(\mathop {\sum }\nolimits _{i=1}^n w x_i +\gamma >0 \quad \), and this is true for every \(x\), in particular, even for \(x=(-1,-1,-1,\ldots ,-1).\) This means that \(\gamma -nw>0\Leftrightarrow \ln \frac{\alpha }{1-\alpha }>n\ln \frac{p}{1-p}\Leftrightarrow \frac{\alpha }{1-\alpha }>(\frac{p}{1-p})^{n}\). \(\square \)
For larger committees, more individuals are required to vote non-informatively to increase the probability of making the correct collective decision. In the extreme case where the PBR is the optimal rule, non-informative voting by the majority of the committee members can lead to the attainment of the maximal performance (the highest possible probability of making the correct collective choice). Clearly, strategic voting by an individual committee member is less effective than strategic voting by a subgroup of the committee members.
This advantage is decreasing with \(k\) because
$$\begin{aligned}&\left( {{\begin{array}{l} {2(k+1)} \\ {k+1} \\ \end{array} }} \right) p^{k+1}\left( {1-p} \right) ^{k+1}(\alpha -p)-\left( {{\begin{array}{l} {2k} \\ k \\ \end{array} }} \right) p^{k}\left( {1-p} \right) ^{k}(\alpha -p)<0\Leftrightarrow \\&\begin{array}{l} \\ \left( {{\begin{array}{l} {2k} \\ k \\ \end{array} }} \right) p^{k}\left( {1-p} \right) ^{k}(\alpha -p)(\frac{(2k+1)(2k+2)}{(k+1)(k+1)} p\left( {1-p} \right) -1)<0\Leftrightarrow \\ \left( {{\begin{array}{l} {2k} \\ k \\ \end{array} }} \right) p^{k}\left( {1-p} \right) ^{k}(\alpha -p)(\frac{(2k+1)2p\left( {1-p} \right) -(k+1)}{(k+1)} <0\Leftrightarrow \\ 2p\left( {1-p} \right) < \frac{k+1}{2k+1}. \\ \\ \end{array}\end{aligned}$$
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Ben-Yashar, R., Nitzan, S. On the significance of the prior of a correct decision in committees. Theory Decis 76, 317–327 (2014). https://doi.org/10.1007/s11238-013-9362-7
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DOI: https://doi.org/10.1007/s11238-013-9362-7