Abstract
When the benefit of making a correct decision is sufficiently high, even a slight increase in the probability of making such a decision justifies an increase in the number of decision makers. Applying a standard uncertain dichotomous choice benchmark setting, this study focuses on the relative desirability of two alternatives: adding individuals with capabilities identical to the existing ones and adding identical individuals with mean-preserving capabilities that depend on the states of nature. Our main result establishes that when the group applies the simple majority rule, variability in the capabilities of the new decision makers under the two states of nature, which is commonly observed in various decision-making settings, is less desirable in terms of the probability of making the correct decision.
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Sah (1991), Sah and Stiglitz (1988) relaxed the symmetry assumption with respect to the states of nature and allowed the decisional 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 setting. Ben-Yashar (2014) reassesses the validity of the Condorcet Jury Theorem when voters are homogeneous and each knows the correct decision with an average probability of more than a half. This paper shows that larger groups, in most cases, are less likely to reach a correct collective decision, even if the average individual probability of making a correct decision exceeds one half.
CJT has previously been generalized in several other ways. Early expositions and generalizations were proposed by Grofman et al. (1983), Feld and Grofman (1984), Nitzan and Paroush (1982), Young (1988) and Owen et al. (1989). Ladha (1995) relaxed the independence assumption. Austen-Smith and Banks (1996) and Ben-Yashar and Milchtaich (2007) generalized the setting to a strategic one. CJT can be generalized to the case of heterogeneous voters. See, for example, Ben-Yashar and Zahavi (2011), Ben-Yashar and Danziger (2011) and Berend and Paroush (1998). Baharad and Ben-Yashar (2009) studied the validity of CJT under subjective probabilities. Dietrich and List (2013) presented a general analysis of proposition-wise judgment aggregation. For a recent survey of the literature inspired by CJT, see Nitzan and Paroush (2016).
Since we consider a choice between two alternatives using the simple majority rule, we require an odd number of decision makers. The rule is not defined otherwise.
Within our framework, the objective is to make the right decision, which is equivalent to maximizing the expected payoff when symmetric payoffs are assumed.
Note that in this case the probabilities \(p_1 \)and \( p_2 \) are of different individuals, whereas in (2) the different probabilities \(p_1\) and \(p_2 \) represent the skills of a single individual under two states of nature.
Notice that this economic intuition is not valid for explaining the former result of Ben-Yashar and Paroush (2000). The reason is that in their case, in each state of nature the added decisional capabilities of the two individuals are different. Since the positive marginal effect requires the use of both of these different capabilities, it is meaningless to resort to the notion of declining marginal productivity.
Formally, let a be the a priori probability of state 1 . In this case
$$\begin{aligned} \Delta _\mathrm{has} =A\left( {ap_1^2 +(1-a)p_2^2 } \right) -B\left( {a\left( {1-p_1 } \right) ^{2 }+(1-a)\left( {1-p_2 } \right) ^{2 }} \right) . \end{aligned}$$If a is close to 1 and \(p_1\) is higher than \(p_2 \), then \(\Delta _\mathrm{hs} <\Delta _\mathrm{has}\). Alternatively, one can add a parameter q such that \(p_1 =p+q\) and \( p_2 =p-q\),and look for the optimal value of q. When the prior is completely symmetrical, one can obtain \(q = 0\) ,i.e, symmetric skills. As far as the a priori probability stays away from one half, the optimum value of q is considerably greater than 0.
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The authors are indebted to two anonymous referees and an associate editor for their most useful comments and suggestions.
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Ben-Yashar, R., Nitzan, S. Is diversity in capabilities desirable when adding decision makers?. Theory Decis 82, 395–402 (2017). https://doi.org/10.1007/s11238-016-9570-z
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DOI: https://doi.org/10.1007/s11238-016-9570-z