Stochastic expected utility for binary choice: a 'modular' axiomatic foundation

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 independenceaxiom into two parts: one (which we callweak 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 callpreference-bounded domains) stochastic symmetry can be replaced withweak transparent dominance (WTD). For choice between lotteries, WTD only restricts behaviour when choosing between probability mixtures of a "best" and a "worst" possible outcome.