Abstract
We propose a model of an agent’s probability and utility that is a compromise between Savage (The foundations of statistics, Wiley, 1954) and Jeffrey (The Logic of Decision, McGraw Hill, 1965). In Savage’s model the probability–utility pair is associated with preferences over acts which are assignments of consequences to states. The probability is defined on the state space, and the utility function on consequences. Jeffrey’s model has no consequences, and both probability and utility are defined on the same set of propositions. The probability–utility pair is associated with a desirability relation on propositions. Like Savage we assume a set of consequences and a state space. However, we assume that states are comprehensive, that is, each state describes a consequence, as in Aumann (Econometrica 55:1–18, 1987). Like Jeffrey, we assume that the agent has a preference relation, which we call desirability, over events, which by definition involves uncertainty about consequences. For a given probability and utility of consequences, the desirability relation is presented by conditional expected utility, given an event. We axiomatically characterize desirability relations that are represented by a probability–utility pair . We characterize the family of all the probability–utility pairs that represent a given desirability relation.
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Notes
This is done with almost no loss of generality as by the Stone Theorem, Stone (1936), every Boolean algebra is isomorphic to the Boolean algebra of a family of subsets.
Axiom P6\('\) is imposed on a qualitative probability relation on events. Here, it is imposed on the desirability relation on events. Axiom P6 is a translation of P6\('\) for a preference relation on acts.
The symmetric difference of two events consists of all the states in these events that do not belong to both, that is, \(A\Delta B=(A\cup B)\setminus (A\cap B)=(A\setminus B)\cup (B\setminus A)\).
A probability measure P is non-atomic if for each event E and \(\alpha \) in [0, 1], there exists an event \(F\subseteq E\) such that \(P(F)=\alpha P(E)\). This condition appeared first in Savage (1954) and was described as non-atomicity by Machina and Schmeidler (1992). Gilboa (1987) defined an extension of this condition to non-additive measures, and referred to a measure that satisfies it as convex ranged.
It is straightforward to see that Likelihood-ratio dominance implies stochastic dominance.
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Acknowledgements
Dov Samet acknowledges financial support of ISF Grant #722/18.
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Samet, D., Schmeidler, D. Desirability relations in Savage’s model of decision making. Theory Decis 94, 1–33 (2023). https://doi.org/10.1007/s11238-022-09883-y
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DOI: https://doi.org/10.1007/s11238-022-09883-y