Abstract
In recent years, judgement aggregation has emerged as an important area of social choice theory. Judgement aggregation is concerned with aggregating sets of individual judgements over logically connected propositions into a set of collective judgements. It has been shown that even seemingly weak conditions on the aggregation function render it impossible to find functions that produce rational collective judgements from all possible rational individual judgements. This implies that the step from individual judgements to collective judgements requires trade-offs between different desiderata, such as universal domain, rationality, epistemological quality, and unbiasedness. These dilemmas challenge us to decide which conditions we should relax. The typical application for judgement aggregation is the problem of group decision making, with juries and expert committees as stock examples. However, the relevance of judgement aggregation goes beyond these cases. In this survey I review some core results in the field of judgement aggregation and social epistemology and discuss their implications for the analysis of distributed thinking.
Keywords
- Aggregation Function
- Aggregation Procedure
- Impossibility Result
- Social Choice Theory
- Individual Judgement
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Notes
- 1.
For the example discussed here we can take → as the material conditional.
- 2.
The literature on judgement aggregation has produced many refinements and extensions to List and Pettit’s (2002) result, which cannot be described in detail here. Most important is perhaps Pauly and van Hees’s (2006) generalizations, and further more general results in Dietrich and List (2007). The general structure of these additions is to discuss other, often weaker or differently constructed desiderata and prove impossibility (and sometimes possibility) results for aggregation functions. A very clear framework for judgement aggregation in general logic is provided by Dietrich (2007).
- 3.
It is not always the case that the propositions can be neatly divided into premises and conclusions. In addition, the premises do not necessarily determine the truth value(s) of the conclusion(s). For instance, if the votes on the premises had resulted in {¬P,P→Q}, the conclusion Q would not be determined by deductive closure because both Q and not-Q are consistent with the judgements on the premises.
- 4.
Except for those cases described in note 3.
- 5.
Also, relaxing anonymity does not yield particularly attractive aggregation procedures. In a very closely related setup, Pauly and van Hees (2006) show that the only aggregation procedure that meets all other desiderata is a dictatorship.
- 6.
More precisely, if the votes are probabilistically independent, conditional on the truth value of Φ.
- 7.
The joint assumption of competence and independence rarely holds in practice. Weaker versions of the theorem have been proved (e.g. Dietrich and Spiekermann 2013). Dietrich (2008) points out that it is not possible to (statistically) justify both the independence and the competence assumptions and discusses less demanding assumptions and their implications.
- 8.
Assuming that the normative proposition R refers to a fact.
- 9.
This feature of the premise-based procedure has been overlooked by Bovens and Rabinowicz (see Fig. 6, where this dip is missing). The reason for this dip is quite easy to grasp intuitively: For very low p, the premise-based procedure is reliably wrong on both premises. If the world is in state S1 or S4, it will produce the wrong judgement on R, but if the world is in S2 or S3, it will produce the right outcome (though for the wrong reason, swapping the true and the false premise). As p approaches the watershed of 0.5, the procedure is less reliable false. It is still very unlikely that it is correct about both premises, but it is occasionally correct on one of them. Being sometimes right on one conclusion produces better results if the world is in S4, but worse results if the world is in either S2 or S3 (and it does not matter for S1). Since the world is more often in either S2 or S3 than in S4, the performance of the premise-based procedure dips for p close to but lower than 0.5.
- 10.
List also operates with asymmetrical individual competence, that is individuals have different competence for correctly judging true and false propositions.
- 11.
For judgement aggregation with regard to judgement change see List (2011).
- 12.
In addition, Bonnefon (2007) reports that individuals change their preference for the conclusion- and premise-based procedure with the nature of the decision.
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Acknowledgements
An earlier version of this chapter was presented at the “Distributed Thinking” workshop at the University of Hertfordshire, organized by the Distributed Language Group. I am grateful for many constructive comments. I owe special thanks to Hemdat Lerman for several helpful discussions and extensive comments. The chapter benefited from the reviews provided by three anonymous referees and helpful comments from the editors of AI & Society. All remaining errors are my own.
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Appendix
Appendix
Bovens and Rabinowicz (2006) calculate the probabilities of the group being correct, conditional on the state. They define M pbp as the proposition ‘The premise-based procedure yields the correct result’ and calculate probabilities conditional on all 4 states:
Note that one can arrive at the correct result even though some or even both collective judgements on the premises are wrong. Given the logical dependency between the propositions, we know that π(R)=π(P)π(Q). Summing up the conditional probabilities of being correct with the premise-based procedure, weighted by the probabilities that the different states obtain yields:
Following Bovens and Rabinowicz’s exposition for the conclusion-based procedure, let V be the proposition that a single voter determines the conclusion correctly, and P(V) the probability the voter does so. Since each single voter applies deductive closure, we obtain the following probabilities for each single voter to be correct on the conclusion, based on their competence p:
Each individual can reach the correct conclusion by being correct on both premises (probability p 2) but one can also be correct, even if one is wrong on one or even both of the premises. Let M cbp denote the proposition that the conclusion-based procedure yields the correct result. Conditional on the state, we can apply Eq. (3.1) to calculate the probability of a correct majority vote on the conclusion:
Summing up the probabilities weighted by the prior probabilities of the different states yields:
The results for the premise-based procedure in (3.3) and the conclusion-based procedure in (3.6) are based on the assumption that it does not matter whether the correct result is deduced from correct or incorrect judgements on the premises. If we want to be right “for the right reasons”, the cases where incorrect judgements lead to correct outcomes need to be removed. Let M pbp−rr denote the proposition that the group has arrived at the right judgement for the right reasons. This yields:
Similarly, for the conclusion-based procedure one want to consider the probability that a majority of voters is correct for the right reasons:
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Spiekermann, K. (2013). Judgement Aggregation and Distributed Thinking. In: Cowley, S., Vallée-Tourangeau, F. (eds) Cognition Beyond the Brain. Springer, London. https://doi.org/10.1007/978-1-4471-5125-8_3
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