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This paper investigates the applicability of thermodynamic concepts and principles to competitive systems. We show that Tsallis entropies are suitable for the characterisation of systems with transitive competition when mutations deviate from Gibbs mutations. Different types of equilibria in competitive systems are considered and analysed. As competition rules become more and more intransitive, thermodynamic analogies are eroded, and the behaviour of the system can become complex. This work analyses the phenomenon of punctuated evolution in the context of the competitive risk/benefit dilemma.

The question of whether systems involving competition can be characterised by quantities resembling conventional thermodynamic parameters does not have a simple unambiguous answer. This problem was investigated in [

Thermodynamics is strongly linked to the concept of equilibrium. Competitive systems allow the introduction of different types of equilibrium, possessing different degrees of similarity with the concept of equilibrium in conventional thermodynamics. The current work discusses possible cases of competitive equilibria and performs a detailed analysis based on the Tsallis entropy of the equilibrium through a point of contact, which is more similar to conventional thermodynamics than the other cases.

From the thermodynamic perspective, the present work is only an example of using Tsallis entropy. We do not attempt to draw any general thermodynamic conclusions, and the use of non-extensive entropy in other applications may well be different from our treatment of equilibria in completive systems. The problem of general consistency between physical equilibrium conditions and definitions of non-extensive entropies has been analysed by Abe [

The last section deals with intransitive cases when the thermodynamic analogy weakens, and the possibility of using entropy as a quantity that always tends to increase in time or remain constant is not assured. This section analyses the

Competitive systems involve the process of competition in its most generic form. The elements of competitive systems compete with each other according to preset rules. The rules define the winners and losers for each competition round based on properties of the elements, denoted here by _{A} ≺ _{B}) indicates that element B with properties _{B} is the winner in competition with element A with properties _{A}. If two elements have equivalent strength _{A} ≃ _{B}, the winner is to be determined at random. In computer simulations, the elements are also called Pope particles, and the exchange of properties is called mixing by analogy, with the conventions adopted in particle simulations of reacting flows. Two forms of mixing—conservative and competitive—can be distinguished. The former is predominantly used in the flow simulations, while the latter is associated with competitive systems. The rest of this section introduces the basic terms used in the characterisation of competitive systems; further details can be found in [

The competition rules are divided into two major categories: transitive and intransitive. In transitive competitions, the superiority of B over A and C over B inevitably demands the superiority of C over A, that is:

As illustrated in _{y}_{y}

The competition rules, however, are not necessarily transitive, and the competition is deemed intransitive when at least one intransitive triplet:
_{A}, _{B}) = −_{B}, _{A}). In the case of transitive competitions, the co-ranking function can be expressed in terms of absolute ranking by:
_{A}_{B}), it is useful to define sharp co-ranking:

The distributions of elements in the property space is characterised by the particle distribution function _{I}_{I}_{I}_{I}_{J}_{I}_{J}_{I}_{J}_{J}_{I}_{I}_{I}

Examples of systems using competitive mixing can be found in [

We consider transitive competition with elements possessing a scalar property _{m}_{m}_{m}_{m}_{m}

The Gibbs mutations [_{m}_{m}^{*} is the position of leading particle.

In this work, we are interested in the case when the pdf, ^{2}), provided the mutations are distributed according to:

_{m}

The cdf shapes presented in

Competitive systems are aimed at studying generic properties of systems with competition and mutations. Although we do not specifically intend to model distributions of biological mutations, these distributions are still of some interest here as real–world examples of complex competitive systems. Ohta [

Free Tsallis entropy in competitive systems is defined by:
_{y}_{y}^{γ}s_{y}_{y}_{y}_{y}

♦ _{y}_{q}q^{*}_{2} = 2 − ^{*} is arbitrary in this case, the distribution can be freely shifted along ^{*} is determined by Equation

♦ _{y}^{*}_{q}_{q}^{−1}/(2 − _{Q}^{*}_{q}k

In the case of physical thermodynamics, Tsallis ^{*}^{*}^{*}). Unlike the Tsallis entropy considered in the present work, the old treatment of the problem [^{*} (^{*}-independent definition of competitive entropy only for the Gibbs mutations).

A competitive system can be divided into subsystems, and the question of equilibrium conditions between these subsystems appears. If the system is subdivided into _{I}_{I}

♦

♦ _{I}_{I}_{I}_{I}_{J}_{I}

♦ _{I}_{J}_{I}_{I}_{I}

_{I}_{J}_{I}_{I}_{I}_{I}

_{0}(_{0}(

Among different types of equilibria in competitive systems, the equilibrium at a point of contact is most suitable for thermodynamic analysis, even when mutations substantially deviate from Gibbs mutations.

Connections through a point of contact can be given different interpretations. _{1} and _{2} are internal properties of the subsystems, generally not related to each other, while the point of contact is an agreement that establishes the correspondence of two locations,
_{I}

The case that is most interesting from the thermodynamic perspective is shown in _{I}_{I}_{I}

Let us consider how this equilibrium between _{I}

Maximisation of _{I}_{I}_{I}n_{I}_{I}_{I}_{I}_{I}_{I}_{J}

Assuming that _{I}_{I}_{I}

If competition becomes intransitive and intransitive triplets _{1} ≺ _{2} ≺ _{3} ≺ _{1} (consider the subsystems

In this section, we consider a different example that involves punctuated evolution: for most of the time, the system seems to behave transitively and escalate towards higher ranks and higher entropy. This escalation is nevertheless punctuated by occasional crisis events, where the state of the system collapses to (or near to) the ground state. The system then repeats the slow growth/sudden collapse cycle. Note that only the cyclic component of evolution is considered here, while competitive evolutions may also involve a translational component (or components) and become spiral [

The present example of punctuated evolution is based on the risk/benefit dilemma (RBD): when comparing the available strategies, we would like to have low risk and high benefits; hence the problem has two parameters: the risk is denoted by ^{(1)} and the benefit denoted by ^{(2)}. While high ^{(2)} and low ^{(1)} are most attractive, some compromises increasing the risk to increase the benefit or lowering the benefit to lower the risk may be necessary. When comparing two strategies, _{A} and _{B}, the choice is performed according to the following co-ranking:
_{A}_{B})

One can easily see that choice RBD2 is transitive, allowing for absolute ranking:

^{(2)} > (^{(1)})^{1/3} is prohibited, reflecting the fact that one cannot have large benefits without being exposed to significant risks. The strategies superior with respect to A are in the small dark area, causing the system to evolve to higher risks and higher benefits. In the transitive case, the system grows to reach the equilibrium point maximising the absolute ranking,

For the transitive case, the entropy is defined by Equation _{y}

If the underlying competition rules and long-term history of the evolution are unknown, determining how a system is going to behave in the future by analysing the current distributions may be very difficult.

The competitive mechanism represented by the risk/benefit dilemma can be one of the forces enacting economic cycles in the real world. From the economic perspective, the strategies reflected by RBD2 are seen as rational behaviours of individual players (say, investment agents). The benefit is represented by returns on investments, and ranking

In competitive systems with Gibbs mutations, the distributions tend to be exponential (assuming the isotropy of the property space). This case is described by the strongest similarity to conventional thermodynamics and the existence of detailed balance in the system. When the distribution of mutations deviates from that of Gibbs mutations, the

Unlike in conventional thermodynamics, competitive systems allow different types of equilibria possessing different degrees of similarity with the conventional thermodynamic equilibrium. Competition between subsystems without the exchange of mutations tends to be less stable than the connected equilibria, where subsystems exchange particles through both competition and mutations. Among connected equilibria, the case of Gibbs mutations bears the highest resemblance to conventional thermodynamics. When mutations are not of the Gibbs type, the point of contact equilibrium preserves this resemblance more than the other cases. The point of contact equilibrium has been analysed using Tsallis entropy. This analysis results in equilibrium conditions determined by the equivalence of competitive potentials. These potentials are linked to the introduced effective phase volumes of subsystems that depend on the location of the point of contact.

The thermodynamic analogy requires the transitivity of competition rules. In the case of intransitive competition rules, the system may behave anomalously when considered from the perspective of competitive thermodynamics. This involves the formation of structures, competitive degradations and cycles. The present work uses the example of the competitive risk/benefits dilemma and analyses the case of punctuated evolutions. For most of the time, the evolution of an intransitive competitive system, which represents the dilemma, closely resembles evolutions of transitive systems, which increase ranking and the associated entropy. At some moments, however, this evolution is punctuated and results in an abrupt collapse, which decreases ranking and the associated competitive entropy; this cannot possibly happen when the competition is transitive. Then, the system starts to grow and repeats the cycle again. While the consideration of competitive processes in this work is generic, similar behaviours can be found in biological, economic and other systems.

The author thanks Bruce Littleboy for insightful discussions of economic issues. The author acknowledges funding by the Australian Research Council.

The authors declare no conflict of interest.

Simulations of the cases, RBD1 and RBD2, involving 10,000 Pope particles are offered as video supplements to this article:

Competition is intransitive in RBD1 and transitive in RBD2. The cases are branched apart at 410 time steps with the same distribution of particles. The format of the videos is explained in

Simulations of the risk/benefit dilemma: notations used in the video files.

Examples of systems with (

Simulated long- and short-tailed distributions in comparison with q-exponents: solid curves, simulated for mutations I; dased curves, simulated for mutations II; solid curves with dots, q-exponents. The cdf are plotted for the values of

Distribution of the elements and exchanges between them in a transitive competition.

The cumulative distribution function (cdf) of the experimental [_{Q}

Equilibrium in competitive systems: (

Point of contact equilibrium when the subsystems have (

Intransitivity in the risk/benefit dilemma: (

Simulations of the risk/benefit dilemma. Solid curve, intransitive (RBD1) simulation; dashed curve, transitive (RBD2) simulation initiated at 410 time steps. (^{(1)}

The cdf for the rank distribution at 590 time steps (the same simulation as in _{max} ≈ 0.65). Solid curve, intransitive (RBD1) simulation; dashed curve, transitive (RBD2) simulation; solid curve with dots, approximation by the