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Stochastic expected utility for binary choice: a ‘modular’ axiomatic foundation

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Abstract

We present new axiomatisations for various models of binary stochastic choice that may be characterised as “expected utility maximisation with noise”. These include axiomatisations of simple scalability (Tversky and Russo in J Math Psychol 6:1–12, 1969) with respect to a scale having the expected utility (EU) form, and strong utility (Debreu in Econometrica 26(3):440–444, 1958) of the EU form. The latter model features Fechnerian “noise”: choice probabilities depend on EU differences. Our axiomatisations complement the important contributions of Blavatskyy (J Math Econ 44:1049–1056, 2008) and Dagsvik (Math Soc Sci 55:341–370, 2008). Our representation theorems set all models on a common axiomatic foundation, with additional axioms added in modular fashion to characterise successively more restrictive models. The key is a decomposition of Blavatskyy’s (2008) common consequence independence axiom into two parts: one (which we call weak independence) that underwrites the EU form of utility and another (stochastic symmetry) than underwrites the Fechnerian structure of noise. We also show that in many cases of interest (which we call preference-bounded domains) stochastic symmetry can be replaced with weak transparent dominance (WTD). For choice between lotteries, WTD only restricts behaviour when choosing between probability mixtures of a “best” and a “worst” possible outcome.

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Acknowledgements

My thanks to Aurélien Baillon and two anonymous referees for numerous suggestions which have materially improved the paper. I have also benefitted from the comments of audiences at the University of Queensland, Queen Mary University of London, the 37th Australasian Economic Theory Workshop (University of Technology Sydney) and the DECIDE Workshop on Experimental Economics (University of Auckland).

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    Matthew Ryan

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Previous drafts have circulated under the title“Stochastic Expected Utility for Binary Choice: New Representations”.

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Ryan, M. Stochastic expected utility for binary choice: a ‘modular’ axiomatic foundation. Econ Theory 72, 641–669 (2021). https://doi.org/10.1007/s00199-020-01307-8

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