Black-Box Strategies and Equilibrium for Games with Cumulative Prospect Theoretic Players

TL;DR

This paper investigates how cumulative prospect theory (CPT) preferences relate to the betweenness property, explores implications for game equilibria, and introduces new equilibrium concepts for CPT players.

Contribution

It proves CPT preferences satisfy betweenness only when they align with expected utility theory and develops four types of equilibrium notions considering CPT's deviations.

Findings

CPT preferences satisfy betweenness iff they conform to EUT.

Four different equilibrium concepts are defined for CPT players.

Existence of these equilibria is established and compared.

Abstract

The betweenness property of preference relations states that a probability mixture of two lotteries should lie between them in preference. It is a weakened form of the independence property and hence satisfied in expected utility theory (EUT). Experimental violations of betweenness are well-documented and several preference theories, notably cumulative prospect theory (CPT), do not satisfy betweenness. We prove that CPT preferences satisfy betweenness if and only if they conform with EUT preferences. In game theory, lack of betweenness in the players' preference relations makes it essential to distinguish between the two interpretations of a mixed action by a player - conscious randomizations by the player and the uncertainty in the beliefs of the opponents. We elaborate on this distinction and study its implication for the definition of Nash equilibrium. This results in four different notions of equilibrium, with pure and mixed action Nash equilibrium being two of them. We dub the other two pure and mixed black-box strategy Nash equilibrium respectively. We resolve the issue of existence of such equilibria and examine how these different notions of equilibrium compare with each other.