Commutative Prospect Theory and Confident Behaviour Under Risk and Uncertainty in Psychological Space

Abstract

This paper contributes to the literature on decision making under risk and uncertainty by attaching a weighted probability space to outcome space. Thereby inducing a commutative map of behavior on prospect theory's function space. We endow that space with a psychological metric space, and a time dependent probability density function with kurtosis controlled by a subject's strength of preference. Several new results are derived on that behavioral topological apparatus. First, we prove that gambles are random fields over outcome space. In which case, an uncertain prospect or act is akin to an unobserved configuration of a random field. Second, we introduce a priority heuristic result by proving that a subject's confidence evolves like a stopped behavioral stochastic process depicted by behavior mimicking epsilon-homotopy of a fair gamble, i.e. a martingale. There, we use Dudley-Talagrand metric to characterize large deviation probabilities for the stopped process. Third, we introduce an impossibility theorem for equivalent martingale measures on psychological space -- which explains why subjects gamble with over or under confidence almost surely. We use that to construct a confidence index, and a "term structure of confidence'' from simulated probability distributions. Fourth, we show that even when subjects have Von Neuman Morgenstern preferences, and know ex-ante that the gamble is fair, they still exhibit confident behavior due to the common consequence of probability leakage arising from measurement error -- a de facto priority heuristic. Fifth, our model mitigates critique of constructive choice models which allege that expected-utility models, and prospect theory, are unable to explain anomalous results that deviate from actuarially fair gambles.