Relative entropy and learning rules

Periodic orbits in deterministic discrete-time evolutionary game dynamics: An information-theoretic perspective

Even though the existence of nonconvergent evolution of the states of populations in ecological and evolutionary contexts is an undeniable fact, insightful game-theoretic interpretations of such outcomes are scarce in the literature of evolutionary game theory. As a proof-of-concept, we tap into the information-theoretic concept of relative entropy in order to construct a game-theoretic interpretation for periodic orbits in a wide class of deterministic discrete-time evolutionary game dynamics, primarily investigating the two-player two-strategy case. Effectively, we present a consistent generalization of the evolutionarily stable strategy-the cornerstone of the evolutionary game theory-and aptly term the generalized concept "information stable orbit." The information stable orbit captures the essence of the evolutionarily stable strategy in that it compares the total payoff obtained against an evolving mutant with the total payoff that the mutant gets while playing against itself. Furthermore, we discuss the connection of the information stable orbit with the dynamical stability of the corresponding periodic orbit.

A DEMATEL-Based Method for Linguistic Multiple Attributes Group Decision Making Using Strict t-Norms and t-Conorms

To evaluate fuzzy information precisely, researchers and practitioners are apt to use linguistic variables to model vague or uncertain contexts in natural language. In this paper, some new operation laws for continuous linguistic terms using strict t-norms and t-conorms are defined. Significantly, these operation laws have some desirable properties and are closed on the restricted continuous linguistic term set. On the basis of these new linguistic operation laws, a series of triangular t-(co)norm-based linguistic generalized power geometric operators are developed. In order to consider the interactive influence and interrelationship of decision makers (DMs) and attributes, a decision-making trial and evaluation laboratory (DEMATEL)-based method for linguistic multiple attributes group decision making (MAGDM) is proposed. In the method, the weighting information for DMs and attributes are dependent on the initial direct-relation matrices among DMs and attributes, respectively. Finally, a numerical example is provided. In comparison with the existing methods, two aspects of the DEMATEL-based method for linguistic MAGDM in the work can be highlighted: the underlying operators for linguistic terms using strict t-norms and t-conorms that are closed on the set of the restricted continuous linguistic term set; and the techniques in determining the weighting information, with which the weighting information for DMs and attributes are determined by the interactive influence and interrelationship among DMs or attributes.

Higher-order neural networks, Polyà polynomials, and Fermi cluster diagrams

The problem of controlling higher-order interactions in neural networks is addressed with techniques commonly applied in the cluster analysis of quantum many-particle systems. For multineuron synaptic weights chosen according to a straightforward extension of the standard Hebbian learning rule, we show that higher-order contributions to the stimulus felt by a given neuron can be readily evaluated via Polyà's combinatoric group-theoretical approach or equivalently by exploiting a precise formal analogy with fermion diagrammatics.