### 2019

Benavoli, Alessio; Facchini, Alessandro; Zaffalon, Marco

Bernstein's socks, polynomial-time provable coherence and entanglement Inproceedings

In: Bock, J De; de Campos, C; de Cooman, G; Quaeghebeur, E; Wheeler, G (Ed.): ISIPTA ;'19: Proceedings of the Eleventh International Symposium on Imprecise Probability: Theories and Applications, JMLR, 2019.

Abstract | Links | BibTeX | Tags: bounded rationality, Sum-of-squares polynomials

@inproceedings{zaffalon2019b,

title = {Bernstein's socks, polynomial-time provable coherence and entanglement},

author = {Benavoli, Alessio and Facchini, Alessandro and Zaffalon, Marco},

editor = {J De Bock and C de Campos and G de Cooman and E Quaeghebeur and G Wheeler},

url = {http://alessiobenavoli.com/wp-content/uploads/2019/05/benavoli19-1.pdf},

year = {2019},

date = {2019-07-07},

booktitle = {ISIPTA ;'19: Proceedings of the Eleventh International Symposium on Imprecise Probability: Theories and Applications},

publisher = {JMLR},

series = {PJMLR},

abstract = {We recently introduced a bounded rationality ap-

proach 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, Sum-of-squares polynomials},

pubstate = {published},

tppubtype = {inproceedings}

}

We recently introduced a bounded rationality ap-

proach 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.

proach 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.

Benavoli, Alessio; Facchini, Alessandro; Piga, Dario; Zaffalon, Marco

Sum-of-squares for bounded rationality Journal Article

In: International Journal of Approximate Reasoning, vol. 105, pp. 130 - 152, 2019, ISSN: 0888-613X.

Abstract | Links | BibTeX | Tags: Sum-of-squares polynomials

@article{Benavoli2019b,

title = {Sum-of-squares for bounded rationality},

author = {Alessio Benavoli and Alessandro Facchini and Dario Piga and Marco Zaffalon},

url = {https://arxiv.org/abs/1705.02663},

doi = {10.1016/j.ijar.2018.11.012},

issn = {0888-613X},

year = {2019},

date = {2019-01-01},

journal = {International Journal of Approximate Reasoning},

volume = {105},

pages = {130 - 152},

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 Rn, 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 desire all nonnegative gambles, we only require her to desire gambles for which she can efficiently determine the nonnegativity (in particular sum-of-squares polynomials). We refer to this new criterion as bounded rationality.},

keywords = {Sum-of-squares polynomials},

pubstate = {published},

tppubtype = {article}

}

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 Rn, 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 desire all nonnegative gambles, we only require her to desire gambles for which she can efficiently determine the nonnegativity (in particular sum-of-squares polynomials). We refer to this new criterion as bounded rationality.