Decision Theory

Do Large Language Models Decision Makers Integrate Assets?

Abstract

Expected utility theory (EUT) has long been the benchmark model for decisions under risk, yet persistent experimental anomalies have motivated competing nonexpected utility approaches, particularly rank-dependent utility (RDU) and cumulative prospect theory (CPT). This paper develops a matrix-valued measure of risk attitudes that nests EUT, RDU, and CPT, and uses large language model (LLM) decision makers to generate risky-choice data under wealth-quantile treatments. The framework tests whether decision makers integrate assets into terminal wealth or evaluate gains and losses relative to reference wealth. The theory links utility-based loss aversion to probabilistic loss aversion in the Marschak--Machina triangle and embeds utility loss aversion in a multivariate risk matrix over wealth and lottery payoffs. Model selection is conducted through nested Kullback--Leibler likelihood-ratio tests obtained by imposing identifying restrictions on the parameter space. This design identifies whether rejection of a model is concentrated in particular wealth treatments. The results reject EUT in favor of richer nonexpected utility specifications. They support gain--loss separability and additive risk representation in wealth and lottery payoffs in gain frames, but indicate asset integration, or covariance aversion, in loss frames. Both utility-based and probabilistic loss aversion vary across wealth treatments. Overall, our tests produce a softer Bernheim-Sprenger result: the CPT gain in fit is not driven  by frame-specific probability weighting as much as it is driven by loss aversion and wealth-treatment heterogeneity. 

Incoherent preferences

Under Bruno De Finetti’s coherence theory of additive probability, the expected value of a sequence of mutually exclusive bets should not expose the bettor to certain loss for any of the bets in the sequence (i.e. no formation of Dutch books). However, decision makers (DMs) are known to have non-additive probability preferences represented in the frequency domain. This conundrum of choice implies that DMs are incoherent. If so, then preference reversal (PR) is more likely to occur. That is, DMs response to choice and valuation procedures (with similar expected value) are more likely to be dissimilar or their preferences may appear to be intransitive. We prove that even when the true states of choice experiments are procedure invariance and transitive preferences, PR will still be observed because of: (1) phase incoherence between paired gambles with the same expected value–when probability cycles are incomplete, and (2) experimenter interference in probability measurement. We introduce a utility coherence ratio for paired gambles, and estimates from simulated phase transition from incoherent states to coherent states in binary choice to illustrate the theory. We find that coherence measures are very sensitive to measurement error, coherent states have higher frequency phase transition, and incoherent states represent momentary lapse in judgment that eventually disappear. So, Dutch books and PR are prevented. 

Keywords: preference reversal; transitivity axiom; probability phase incoherence; wavelets; probability weighting

Utility representation in Abstract Wiener Space

We extend Machina’s (1982) preference functional to abstract Wiener space. This has the advantage of extending utility functions to infinite dimensional spaces, providing estimates for Machina’s (1982) nonlinear utility functional, and establishing a nexus between microfoundations of local utility, behavioural probability, prospect theory, and elements of quantum decision theory without complex valued Hilbert spaces. For example, the class of nonconvex Markowitz utility functions (for which prospect theory’s value function is a special case) are vector valued functions in abstract Wiener space. Instead of preferences over probability distributions, the problem is transformed into one of preferences over states. Under Arzela-Ascoli Theorem, Wiener measure is the limiting measure and hence unique prior in Wiener space. By a change of measure local subjective (posterior) probability is a Wiener integral in that space.