Abstract
In the basic model of Condorcet’s jury theorem and in the literature that follows, an odd-numbered group of voters is assumed so that the simple majority rule can be used. We show that this assumption is not necessary either in Condorcet’s basic model or in the general framework of dichotomous choice. We first apply simple majority rule to an even-numbered homogeneous fixed-size committee. We then provide a justification for using simple majority rule for an even-numbered heterogeneous fixed-size committee when the competence structure of the committee members is not common knowledge.
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Notes
Condorcet (1785).
CJT has been generalized in several ways. Early generalizations were proposed by Feld and Grofman (1984), Nitzan and Paroush (1982), and Young (1988). Ladha (1995) relaxed the independence assumption, Baharad and Ben-Yashar (2009) studied the validity of CJT under subjective probabilities, and Dietrich and List (2013) presented a general analysis of proposition-wise judgment aggregation.
Sah and Stiglitz (1988) relaxed the symmetry assumption on the states of nature and allowed the decision-making skills of each voter to depend on the state of nature. Ben-Yashar and Nitzan (1997) derived the optimal group decision rule under such asymmetric settings. Ben-Yashar (2014) reassessed the validity of Condorcet’s jury theorem when voters are homogeneous and they each know the correct decision with an average probability of more than one half.
CJT can be generalized to the case of heterogeneous voters. Nitzan and Paroush (1982) find the condition that SMR is still the optimal rule in the absence of identical competence.Kanazawa (1998) showed that heterogeneous groups perform better than homogeneous groups. Berend and Paroush (1998) formulated necessary and sufficient conditions for outcomes in heterogeneous groups. For an overview of decision theory for which CJT is central, see Gerling et al. (2005).
This result (in the basic, symmetric model) can be derived from Feld and Grofman (1984) who discuss the probability of an enlarged group reaching the correct decision when two groups are combined.
In this case, the optimal rule is a supermajority rule; see Ben-Yashar and Nitzan (1997).
This implies that \({p}^{\text I} > 1 - p^{\text {II}}\); that is, a voter is more likely to decide 1 in state 1 than in state − 1.
If \(x > p^{{\text{I}}} { }\) and \(y < p^{{{\text{II}}}}\) then \(\pi_{{{\text{SMRE}}}} \left( {\underline{{\left( {p^{{\text{I}}} ,p^{{{\text{II}}}} } \right)}}_{h}^{n} ,\left( {x,y} \right)} \right)\frac{ > }{ < }\pi_{{{\text{SMR}}}} \left( {\underline{{\left( {p^{{\text{I}}} ,p^{{{\text{II}}}} } \right)}}_{h}^{n} } \right) \Leftrightarrow \left( {\frac{{p^{{\text{I}}} \left( {1 - p^{{\text{I}}} } \right)}}{{p^{{{\text{II}}}} \left( {1 - p^{{{\text{II}}}} } \right)}}} \right)^{k} \frac{ > }{ < }\frac{{p^{{{\text{II}}}} - y}}{{x - p^{{\text{I}}} }}.\)
If \(x < p^{{\text{I}}}\) and \(y > p^{{{\text{II}}}}\) then \(\pi_{{{\text{SMRE}}}} \left( {\underline{{\left( {p^{{\text{I}}} ,p^{{{\text{II}}}} } \right)}}_{h}^{n} ,\left( {x,y} \right)} \right){ \gtreqless }\pi_{{{\text{SMR}}}} \left( {\underline{{\left( {p^{{\text{I}}} ,p^{{{\text{II}}}} } \right)}}_{h}^{n} } \right) \Leftrightarrow \left( {\frac{{p^{{\text{I}}} \left( {1 - p^{{\text{I}}} } \right)}}{{p^{{{\text{II}}}} \left( {1 - p^{{{\text{II}}}} } \right)}}} \right)^{k} { \lesseqgtr }\frac{{y - p^{{{\text{II}}}} }}{{p^{{\text{I}}} - x}}.\)
For more details see Ben-Yashar and Nitzan (1997).
The notations \({x}^{S}\) and \({\overline{x} }^{S}\) are used in many papers. See, for example Ben-Yashar et al. (2021).
The last equation is equal to zero since \(\forall\) \(s \in S_{k + 1}^{{NN{ \setminus }\left\{ l \right\}}}\) and \(l \in NN\) there exists \(s^{\prime} \in S_{k}^{{NN{ \setminus }\left\{ {l^{\prime}} \right\}}}\) and \(l^{\prime} \in NN\)\(,\) such that \(l^{\prime} \ne l\) and \(g\left( {x^{S} } \right)\left( {1 - p_{l} } \right) = g\left( {x^{{s^{\prime}}} } \right)p_{{l^{\prime}}} .\)
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Ben-Yashar, R. An application of simple majority rule to a group with an even number of voters. Theory Decis 94, 83–95 (2023). https://doi.org/10.1007/s11238-022-09872-1
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DOI: https://doi.org/10.1007/s11238-022-09872-1