SOS for bounded rationality

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.