POTENTIAL, VALUE AND PROBABILITY

This paper focuses on the probabilistic point of view and proposes a extremely simple probabilistic model that provides a single and simple story to account for several extensions of the Shapley value, as weighted Shapley values, semivalues, and weak (weighted or not) semivalues, and the Shapley value itself. Moreover, some of the most interesting conditions or notions that have been introduced in the search of alternatives to Shapley's seminal characterization, as 'balanced contributions' and the 'potential', are reinterpreted from this same point of view. In this new light these notions and some results lose their 'mystery' and acquire a clear and simple meaning. These illuminating reinterpretations strongly vindicate the complementariness of the probabilistic and the axiomatic approaches, and shed serious doubts about the achievements of the axiomatic approach since Nash's and Shapley's seminal papers in connection with the genuine notion of value.Coalition games, value, potential

Fuzzy Topology, Quantization and Gauge Invariance

Dodson-Zeeman fuzzy topology considered as the possible mathematical framework of quantum geometric formalism. In such formalism the states of massive particle m correspond to elements of fuzzy manifold called fuzzy points. Due to their weak (partial) ordering, m space coordinate x acquires principal uncertainty dx. It's shown that m evolution with minimal number of additional assumptions obeys to schroedinger and dirac formalisms in norelativistic and relativistic cases correspondingly. It's argued that particle's interactions on such fuzzy manifold should be gauge invariant.Comment: 12 pages, Talk given on 'Geometry and Field Theory' conference, Porto, July 2012. To be published in Int. J. Theor. Phys. (2015

Incompatible Multiple Consistent Sets of Histories and Measures of Quantumness

In the consistent histories (CH) approach to quantum theory probabilities are assigned to histories subject to a consistency condition of negligible interference. The approach has the feature that a given physical situation admits multiple sets of consistent histories that cannot in general be united into a single consistent set, leading to a number of counter-intuitive or contrary properties if propositions from different consistent sets are combined indiscriminately. An alternative viewpoint is proposed in which multiple consistent sets are classified according to whether or not there exists any unifying probability for combinations of incompatible sets which replicates the consistent histories result when restricted to a single consistent set. A number of examples are exhibited in which this classification can be made, in some cases with the assistance of the Bell, CHSH or Leggett-Garg inequalities together with Fine's theorem. When a unifying probability exists logical deductions in different consistent sets can in fact be combined, an extension of the "single framework rule". It is argued that this classification coincides with intuitive notions of the boundary between classical and quantum regimes and in particular, the absence of a unifying probability for certain combinations of consistent sets is regarded as a measure of the "quantumness" of the system. The proposed approach and results are closely related to recent work on the classification of quasi-probabilities and this connection is discussed.Comment: 29 pages. Second revised version with discussion of the sample space and non-uniqueness of the unifying probability and small errors correcte