Intransitivity in Theory and in the Real World

Abstract

This work considers reasons for and implications of discarding the assumption of transitivity-the fundamental postulate in the utility theory of von Neumann and Morgenstern, the adiabatic accessibility principle of Caratheodory and most other theories related to preferences or competition. The examples of intransitivity are drawn from different fields, such as law, biology and economics. This work is intended as a common platform that allows us to discuss intransitivity in the context of different disciplines. The basic concepts and terms that are needed for consistent treatment of intransitivity in various applications are presented and analysed in a unified manner. The analysis points out conditions that necessitate appearance of intransitivity, such as multiplicity of preference criteria and imperfect (i.e., approximate) discrimination of different cases. The present work observes that with increasing presence and strength of intransitivity, thermodynamics gradually fades away leaving space for more general kinetic considerations. Intransitivity in competitive systems is linked to complex phenomena that would be difficult or impossible to explain on the basis of transitive assumptions. Human preferences that seem irrational from the perspective of the conventional utility theory, become perfectly logical in the intransitive and relativistic framework suggested here. The example of competitive simulations for the risk/benefit dilemma demonstrates the significance of intransitivity in cyclic behaviour and abrupt changes in the system. The evolutionary intransitivity parameter, which is introduced in the Appendix, is a general measure of intransitivity, which is particularly useful in evolving competitive systems.

A Survey on the Mathematical Foundations of Axiomatic Entropy: Representability and Orderings

“…These entropies have similarities with but are not the same as the molecular entropy, which characterizes disorder of molecular movements and is subject to the laws of thermodynamics. We use the term apparent entropy to distinguish entropy-like quantities from the molecular entropy.>> This key idea, namely the use of more than one entropy seems to support also the observation made in [38], already mentioned before, namely:…”

“…Nevertheless, neither in that study nor in the more recent [38], the key notion of an interval order just defined and analyzed in Sections 2 and 5 of the present manuscript has been used as a tool.…”

“…Those additional axioms lean on additivity properties, and the aim here is finding entropy functions that are unique up to linear transformations. In a similar way, a battery of papers by Klimenko (see [38,[73][74][75][76][77]) deals with concepts as mixing (see also [78]) and competition of systems. It is not our intention here to make a criticism on those papers.…”

“…Intransitivities are common in the real world and are responsible for common behavior in competitive systems.>> With similar ideas in mind, in [38], it is claimed that: <<While physical thermodynamics is always transitive, the same cannot be stated a priori about apparent thermodynamics.>> Therefore, it is interesting at this stage to find a suitable mathematical approach in order to interpret intransitivities arising in state spaces.…”

“…However, as far as we know, it is uncommon in the analysis of axiomatics of entropy. Although several recent papers (see, e.g., [38]) have analyzed in depth intransitivities encountered in various physical or chemical processes, no use of the concept of an interval order as the main tool is observed there, as already commented in [1].…”