@techreport{Benavoli2019d,
title = {Bernstein's socks and polynomial-time provable coherence},
author = {Benavoli, Alessio and Facchini, Alessandro and Zaffalon, Marco},
url = {https://arxiv.org/abs/1903.04406},
year = {2019},
date = {2019-03-12},
abstract = {We recently introduced a bounded rationality approach for the theory of desirable gambles. It is based on the unique requirement that being non-negative for a gamble has to be defined so that it can be provable in polynomial-time. In this paper we continue to investigate properties of this class of models. In particular we verify that the space of Bernstein polynomials in which non-negativity is specified by the Krivine-Vasilescu certificate is yet another instance of this theory. As a consequence, we show how it is possible to construct in it a thought experiment uncovering entanglement with classical (hence non quantum) coins. },
keywords = {bounded rationality, desirability, Quantum mechanics},
pubstate = {published},
tppubtype = {techreport}
}

We recently introduced a bounded rationality approach for the theory of desirable gambles. It is based on the unique requirement that being non-negative for a gamble has to be defined so that it can be provable in polynomial-time. In this paper we continue to investigate properties of this class of models. In particular we verify that the space of Bernstein polynomials in which non-negativity is specified by the Krivine-Vasilescu certificate is yet another instance of this theory. As a consequence, we show how it is possible to construct in it a thought experiment uncovering entanglement with classical (hence non quantum) coins.

@conference{Benavoli2017b,
title = {SOS for bounded rationality},
author = { Benavoli, Alessio and Facchini, Alessandro and Piga, Dario and Zaffalon, Marco},
url = {https://arxiv.org/abs/1705.02663},
year = {2017},
date = {2017-05-07},
booktitle = {Proc. ISIPTA'17 Int. Symposium on Imprecise Probability: Theories and Applications, },
pages = {1--12},
publisher = {PJMLR},
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 exper-
iment 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 call this new criterion bounded rationality.},
keywords = {bounded rationality, SOS},
pubstate = {published},
tppubtype = {conference}
}

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 exper-
iment 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 call this new criterion bounded rationality.