We modify the Rényi (1961) axioms for entropy to apply to negative (“signed”) measures as arise, for example, in phase-space representations of quantum mechanics. We obtain two new measures of (lack of) information about a system – which we propose as signed analogs to classical Shannon entropy and classical Rényi entropy, respectively. We show that signed Rényi entropy witnesses non-classicality of a system. Specifically, a measure has at least one negative component if and only if signed Rényi α-entropy is negative for some α > 1. The corresponding non-classicality test does not work with signed Shannon entropy. We next show that signed Rényi 2k-entropy, when k is a positive integer, is Schurconcave. (An example shows that signed Shannon entropy is not Schur-concave.) We then establish an abstract quantum H-theorem for signed measures. We prove that signed Rényi 2k-entropy is nondecreasing under classical (“decohering”) evolution of a signed measure, where the latter could be a Wigner function or other phase-space representation of a quantum system. (An example shows that signed Shannon entropy may be non-monotonic.) We also provide a characterization of the Second Law for signed Rényi 2-entropy in terms of what we call eventual classicalization of evolution of a system. We conclude with an argument that signed Rényi 2-entropy of the Wigner function is constant under Moyal bracket evolution.

Any quasi-probability representation of a no-signaling system – including quantum systems – can be simulated via a purely classical scheme by allowing signed events and a cancellation procedure. This raises a fundamental question: What properties of the non-classical system does such a classical simulation fail to replicate? We answer by using large deviation theory to show that the probability of a large fluctuation under the classical simulation can be strictly greater than under the actual non-classical system. The key finding driving our result is that negativity in probability relaxes the data processing inequality of information theory. We propose this potential large deviation stability of quantum (and no-signaling) systems as a novel form of quantum advantage.

Interactions involving multiple parties and necessitating their agreement are pervasive in both market and non-market settings. As the number of participants increases, these situations become progressively complex to describe and analyze. Despite the prolific nature of such scenarios, a comprehensive conceptual framework addressing such settings is often lacking. The focus of this dissertation lies in a distinct type of multilateral interaction, where a commitment of a group, or a coalition, of participants is required for achieving a positive surplus. The analysis encompasses three scenarios, namely, government formation in parliamentary democracies, bilateral trading on a market with multiple buyers and sellers, and resource allocation in the US presidential campaign. This dissertation proposes an approach that provides axiomatic foundations for a theory of coalition formation in these settings, and, for two of these scenarios, simultaneously provides an empirically accurate forecast methodology.

The Agreement Theorem Aumann (1976 Ann. Stat. 4, 1236–1239. (doi:10.1214/aos/1176343654)) states that if two Bayesian agents start with a common prior, then they cannot have common knowledge that they hold different posterior probabilities of some underlying event of interest. In short, the two agents cannot ‘agree to disagree’. This result applies in the classical domain where classical probability theory applies. But in non-classical domains, such as the quantum world, classical probability theory does not apply. Inspired principally by their use in quantum mechanics, we employ signed probabilities to investigate the epistemics of the non-classical world. We find that here, too, it cannot be common knowledge that two agents assign different probabilities to an event of interest. However, in a non-classical domain, unlike the classical case, it can be common certainty that two agents assign different probabilities to an event of interest. Finally, in a non-classical domain, it cannot be common certainty that two agents assign different probabilities, if communication of their common certainty is possible—even if communication does not take place. This article is part of the theme issue ‘Quantum contextuality, causality and freedom of choice’.

We suggest a new component efficient solution for monotonic TU games with a coalition structure, the conditional Shapley value. In contrast to other such solutions, it satisfies the null player property. Nevertheless, it accounts for the players’ outside options in productive components of coalition structures. For all monotonic games, there exist coalition structures that are stable under the conditional Shapley value. For voting games, such stable coalition structures support Gamson’s theory of coalition formation (Gamson, 1961).