Abstract
This paper presents a natural extension of Bayesian decision theory from the domain of individual decisions to the domain of group decisions. We assume that each group member accepts the assumptions of subjective expected utility theory with respect to the alternatives from which they must choose, but we do not assume, a priori, that the group as a whole accepts those assumptions. Instead, we impose a multiattribute utility independence condition on the preferences of the group with respect to the expected utilities of its actions as appraised by its members. The result is that the expected utility of an alternative for the group is a weighted average of the expected utilities of that alternative for its members. The weights must be determined collectively by the group. Pareto optimality is not assumed, though the result is consistent with Pareto optimality.
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Notes
This does not mean that Bayes’ rule should be applied retrospectively in an automatic fashion after the event is observed, i.e., it is not a model of actual learning over time. Rather, it is a model of expected learning that can be used for developing and comparing hypothetical contingency plans and for evaluating sources of information.
Finiteness of the sets of states and consequences will be assumed here partly to simplify the exposition and partly because the problem is usually framed this way in applied decision analysis.
When utilities are state-dependent, the unique determination of subjective probabilities requires either that they should be measured verbally rather than inferred from preferences (DeGroot 1970) or that they should be measured by assessing preferences among larger sets of acts involving higher degrees of counterfactualism, for example, acts in which states of nature are imagined to have objective probabilities or lotteries in which each outcome is a combination of a prize and a state (Karni et al. 1983; Karni 1985; Schervish et al. 1990; Karni 2007). However, models of the latter kind, which invoke preferences that do not correspond to the sort of choices that are really available, give rise to their own impossibility theorems.
From the first four axioms, it follows by a separating hyperplane argument that preferences have (at least) a state-dependent expected utility representation, and the fifth axiom allows it to be decomposed into a product of a unique probability distribution and a state-independent utility function, although the uniqueness of the probabilities depends on a conventional implicit assumption that utilities which are state-independent in relative terms are also state-independent in absolute terms.
Mongin (1998) observes that the additive model is a “mathematically trivial resolution” of the group decision problem insofar as the Pareto condition is obviously satisfied by any positive weighted sum of the individual utility functions, although the uniqueness of the weights and the “only if” part of the theorem are not quite so trivial, requiring an appeal to the separating hyperplane argument.
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Acknowledgment
Helpful comments on various drafts were provided by David Bell, Jay Kadane, Howard Raiffa, and Richard Zeckhauser.
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Keeney, R.L., Nau, R. A theorem for Bayesian group decisions. J Risk Uncertain 43, 1–17 (2011). https://doi.org/10.1007/s11166-011-9121-5
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DOI: https://doi.org/10.1007/s11166-011-9121-5
Keywords
- Group decisions
- Utility theory
- Probability aggregation
- Preference aggregation
- Impossibility theorems
- Pareto assumption
- Constructing values
- Interpersonal comparison of preferences