Decision Theory

Representation theory for risk with applications to rare disasters in financial markets

We introduce a representation theory for risk based on the tensor between a safe and risky asset. In particular, we show that symmetric risk seeking and risk averse behaviors, assumed by neoclassical economics for expected utility theory (EUT) , belong to the classic group SO(n). However, the structure constant is distorted by loss aversion in behavioural economics under cumulative prospect theory (CPT). We use that distortion to construct an infinitesimal risk transformation matrix operator (RTMO). The RTMO has complex eigenvalues so its elements can be quantized. We prove that the local loss aversion (LLA) index has a trace class distribution i.e. its sum is invariant to matrix transformations We estimate local (LLA) and global (GLA) loss aversion indexes for rare economic disasters like the Great Recession of 2008 (GR2008), and the more recent COVID19 pandemic to evaluate the robustness of our theory.

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.