SOS for bounded rationality

TL;DR

This paper develops a computationally feasible framework for evaluating desirable gambles under bounded rationality, focusing on efficiently checkable nonnegativity conditions like SOS polynomials, to address the NP-hardness of the problem.

Contribution

It introduces a new bounded rationality criterion based on sum-of-squares polynomials, enabling practical assessment of desirability in infinite spaces.

Findings

Provides a computable approach to desirability evaluation.

Addresses NP-hardness by restricting to SOS polynomials.

Offers a framework for bounded rational decision-making.

Abstract

In the gambling foundation of probability theory, rationality requires that a subject should always (never) find desirable all nonnegative (negative) gambles, because no matter the result of the experiment the subject never (always) decreases her money. Evaluating the nonnegativity of a gamble in infinite spaces is a difficult task. In fact, even if we restrict the gambles to be polynomials in R^n , the problem of determining nonnegativity is NP-hard. The aim of this paper is to develop a computable theory of desirable gambles. Instead of requiring the subject to accept all nonnegative gambles, we only require her to accept gambles for which she can efficiently determine the nonnegativity (in particular SOS polynomials). We refer to this new criterion as bounded rationality.