this paper appeared in the Proceedings of the 2002 IEEE Information Theory Workshop [see Grnwald and Dawid (2002)]

It is suggested that thermodynamical equilibrium equals game theoretical equilibrium. Aspects of this thesis are discussed. The philosophy is consistent with maximum entropy thinking of Jaynes, but goes one step deeper by deriving the maximum entropy principle from an underlying game theoretical principle. The games introduced are based on measures of complexity. Entropy is viewed as minimal complexity. It is demonstrated that Tsallis entropy (q-entropy) and Kaniadakis entropy (κ-entropy) can be obtained in this way, based on suitable complexity measures. A certain unifying effect is obtained by embedding these measures in a two-parameter family of entropy functions.

The article presents a new interpretation for Zipf’s law in natural language which relies on two areas of information theory. We reformulate the problem of grammar-based compression and investigate properties of strongly nonergodic stationary processes. The motivation for the joint discussion is to prove a proposition with a simple informal statement: If an n-letter long text describes n β independent facts in a random but consistent way then the text contains at least n β /log n different words. In the formal statement, two specific postulates are adopted. Firstly, the words are understood as the nonterminal symbols of the shortest grammar-based encoding of the text. Secondly, the texts are assumed to be emitted by a nonergodic source, with the described facts being binary IID variables that are asymptotically predictable in a shift-invariant way. The proof of the formal proposition applies several new tools. These

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We continue the development of an abstract, though quantitative, theory of cognition which is rooted in philosophical considerations. Applications include classical Shannon theory and results from geometry. Special attention is payed to inference which is treated as the outcome of a situation of conflict between Nature and Observer, “you”. 1.