## 1. Introduction

- Transitivity
- Linearity (in thermodynamics: extensivity) implying that$$U={P}_{\mathrm{A}}{u}_{\mathrm{A}}+{P}_{\mathrm{B}}{u}_{\mathrm{B}}$$$$S={m}_{\mathrm{A}}{s}_{\mathrm{A}}+{m}_{\mathrm{B}}{s}_{\mathrm{B}}$$

_{A}and u

_{B}and probabilities P

_{A}and P

_{B}. In Equation (2), the overall system is a combination of two subsystems A an B with specific entropies s

_{A}and s

_{B}and masses m

_{A}and m

_{B}.

## 2. Preference, Ranking and Co-Ranking

#### 2.1. Co-Ranking

#### 2.2. Absolute Ranking

_{m}(…) of a sharp ranking is still an equivalent ranking f

_{m}(r(…)) ∼ r(…) with the same ordering. In the case of the graded ranking, the value r(B) − r(A) represents the magnitude of our preference of B over A. In many cases, a graded ranking corresponds to a physical quantity that can be directly determined or measured. In economics, graded rankings are called utility, in biology graded rankings are called fitness, in thermodynamics graded rankings correspond to entropy. Here, we follow the notation of economics and, when applicable, refer to graded rankings as utility. Practically, the line between sharp and graded rankings is blurred. It is often the case that even a nominally sharp ranking can give some indication of the magnitude of the preference, for example, in terms of the density of elements. Different rankings (or co-rankings) are called equivalent if they correspond to the same preference (but might still can have different preference magnitudes).

#### 2.3. Average Rankings

_{i}∈ $\mathrm{S}$ and the corresponding weights g

_{i}= g(C

_{i}) > 0, while g(C

_{j}) = 0 for C

_{j}∉ $\mathrm{S}$. We can also write C

_{i}∈ $\mathrm{G}$ implying that C

_{i}∈ $\mathrm{S}$. The average co-ranking of element A and group $\mathrm{G}$, is defined by the equation

_{i}= 1 within the group, then G is the number of elements in the group and the meanings of $\mathrm{S}$ and $\mathrm{G}$ are essentially the same while the terms “set” and “group” become interchangable (i.e., specification $\mathrm{G}$ as a set implies unit weights for the elements). The co-ranking ρ(A,C

_{i}) and corresponding preference are referred to as underlying co-ranking and preference of the ranking ${\rho}_{\mathrm{G}}$, while the preference

_{i}are called the reference group and reference weights.

**Proposition 1**All conditional rankings indicate the same magnitude of preference, i.e.,

**Proposition 2**If the preferences induced by all conditional rankings are equivalent, i.e.,

_{{B}}(A) ≤ ρ

_{{B}}(C) ≥ 0 but 0 < ρ

_{{C}}(A) > ρ

_{{C}}(C) = 0. Hence, the following conditional preferences

#### 2.4. Is Intransitivity Irrational?

_{i}= 1. The conditional co-ranking, which is a measure of conditional preference of A over B introduced by similarity with Equation (10), becomes

_{{C}}(A,B) = ρ(A,B), δ(C,B,A) = 0 and ρ

_{{A},

_{B,C}}(A,B) is the same as ρ(A,B). However, ρ

_{{A},

_{B,C}}(A,B) and ρ(A,B) are not necessarily equivalent in intransitive cases. The combination of intransitivity with a presumption of invariance of conditional rankings, which is incorrect in intransitive cases, may result in logical contradictions. In a consistent approach, the specification of preference between A and B for selection from the set of {A,B,C} should be ρ

_{{A},

_{B,C}}(A,B) and not ρ(A,B).

## 3. Intransitivity and Game Theory

#### 3.1. Games with Explicit Intransitivity

_{1},C

_{2}, … from the respective subsets ${\mathrm{S}}^{\prime}$ and ${\mathrm{S}}^{\u2033}$ of set $\mathrm{S}$ with respective probabilities ${{g}^{\prime}}_{i}\ge 0$ and ${{g}^{\u2033}}_{i}\ge 0$. The sets ${\mathrm{S}}^{\prime}$ and ${\mathrm{S}}^{\u2033}$ may overlap. Hence, the players’ selections are represented by groups ${\mathrm{G}}^{\prime}$ and ${\mathrm{G}}^{\u2033}$. Player ${\mathrm{P}}^{\prime}$ chooses group ${\mathrm{G}}^{\prime}\underset{\xaf}{\subset}{\mathrm{S}}^{\prime}$ while player ${\mathrm{P}}^{\u2033}$ chooses group ${\mathrm{G}}^{\u2033}\underset{\xaf}{\subset}{\mathrm{S}}^{\u2033}$ but the players are not allowed to change the subsets ${\mathrm{S}}^{\prime}$ and ${\mathrm{S}}^{\u2033}$. Any C

_{i}∈ ${\mathrm{S}}^{\prime}$ is called available to player ${\mathrm{P}}^{\prime}$, while any C

_{i}∈ ${\mathrm{G}}^{\prime}$ is called selected by player ${\mathrm{P}}^{\prime}$. The relative strength of pure strategies C

_{i}and C

_{j}, which is called payoff in game theory, is determined by co-ranking ρ(C

_{i},C

_{j}). This defines a general zero-sum game for two players, ${\mathrm{P}}^{\prime}$ and ${\mathrm{P}}^{\u2033}$, while the groups ${\mathrm{G}}^{\prime}$ and ${\mathrm{G}}^{\u2033}$ represent mixed strategies of the players. It is easy to see that the overall payoff of the game is determined by

**Proposition 3**(Nash [24]) Nash equilibrium is achieved when and only when all options in a mixed strategy selected by each player have maximal (and the same within each mixed strategy) ranking conditioned on the mixed strategy of the opposition:

^{0}(A) = 0 and eliminating A from G

^{0}. In this proposition, ${\mathrm{P}}^{\prime}$ is understood as any of the two players and ${\mathrm{P}}^{\u2033}$ represents his opposition. Note that, generally, ${{\overline{\rho}}^{\u2033}}_{\mathrm{max}}\ne {{\overline{\rho}}^{\prime}}_{\mathrm{max}}$, where

#### 3.2. Games with Potential Intransitivity

## 4. Example: Intransitivity of Justice

- (A) Doing nothing;
- (B) Calling the fire service;
- (C) Trying to rescue people from the fire and/or extinguish the fire

#### 4.1. Popular View

#### 4.2. Treatment by Law: Can Do More but not Less

#### 4.3. Strict Intransitivity in Manager’s Choice

**Leadership:**if the manager is on site, he/she is expected to lead and organise the site personnel, deploying staff as necessary to actively contain or liquidate the cause of emergency.**II. Safety:**- (a) the manager and personnel stay on site during emergency if there is no immediate danger to personnel but
- (b) personnel evacuation must be promptly enacted whenever there is a significant danger to personnel.

- (A) Evacuating personnel and abandoning the site;
- (B) Organising personnel to monitor the situation on site;
- (C) Organising personnel to contain and extinguish the fire.

## 5. Potential Intransitivity of the Original Lotka-Volterra Model

_{R}represents the population of rabbits (R) and N

_{F}is the population of foxes (F). The model coefficients a

_{R}, a

_{R}, b and b′ are assumed to be positive indicating that the foxes win resources from the rabbits due to presence of the term bN

_{R}N

_{F}(b and b′ are generally different since N

_{F}and N

_{R}are measured in different resource units: in foxes and in rabbits). The intransitivity of this model is not apparent since this model does not explicitly mention the environment (E), but the presence of the environment is important. Indeed if N

_{R}= 1 and N

_{F}= 0 at t = 0, then N

_{R}→ ∞ as t → ∞, which is physically impossible. In real life, this growth of rabbit population would be terminated by environmental restrictions: a

_{R}= a

_{R}(N

_{E}) → 0 and N

_{E}→ 0 as N

_{R}→ (N

_{R})

_{max}. Here, N

_{E}represents free environmental resource, which is approximately treated as constant in the original Lotka-Volterra model. Figure 3 illustrates the intransitivity of this case: rabbits win from the environment, while foxes lose to the environment.

## 6. Fractional Ranking

^{(}

^{α}

^{)}(A). The notations which are used here are similar to those used in the previous sections. For example all of the following statements

^{(}

^{α}

^{)}is a fractional ranking, ρ

^{(}

^{α}

^{)}is a fractional co-ranking, etc. (Another equivalent term is “partial”. This term is commonly used in calculus but this might be in conflict with “partial orders”, where “partial” is interpreted as “incomplete”—in partial orders, preferences are not necessarily defined for all pairs of elements). Of course, fractional ranking r

^{(}

^{α}

^{)}exists only if elements can be transitively ordered with respect to criterion α. No criterion alone determines the overall preference: for example, we might have r

^{(1)}(A) > r

^{(1)}(B) but r

^{(2)}(A) < r

^{(2)}(B). The fractional ranking can be either graded or sharp; the former can be called fractional utility. In this section, we deploy the results of the social choice theory, where different criteria represent judgment of different individuals.

#### 6.1. Commensurable Fractional Rankings

^{(}

^{α}

^{)}are used to rescale units as needed. In principle, it is possible to consider cases when fractional utilities are combined in a non-linear manner but this would not change the main conclusion of this subsection:

**Proposition 4**Commensurable fractional utilities correspond to an overall preference that can be expressed by the absolute utility r(…) and, hence, is transitive.

#### 6.2. Commensurable Fractional Co-Rankings

^{(1)}(A) − r

^{(1)}(B) and r

^{(2)}(A) − r

^{(2)}(B) but cannot compare the absolute magnitude, say, r

^{(2)}(A) and r

^{(2)}(A). This is a common situation since, as discussed in the next chapter, human preferences are inherently relativistic). The fractional preferences can always be expressed by fractional co-rankings, which are treated in this subsection as graded and commensurable. (The relative character of real-world preferences, which is reflected by co-rankings, is discussed further in the paper. The case of completely incomparable partial preferences is considered in the following subsection). The overall co-ranking is expressed in terms of the fractional co-rankings by the equation

^{(}

^{α}

^{)}represent the scaling coefficients, whose physical meaning is similar to that of the weights in Equation (48). Depending on the functional form of the fractional co-rankings, three cases are possible (1) fractional and overall co-rankings are transitive (in this case the fractional and overall utilities exist); (2) fractional co-rankings are transitive but the overall co-ranking is intransitive and (3) all co-rankings are intransitive. As discussed further in the paper, the second case is common when fractional co-rankings have non-linear functional forms, which can appear due to imperfect discrimination or for other reasons.

#### 6.3. Incommensurable Fractional Preferences

^{(1)}(A,B) and ρ

^{(2)}(A,B) can not be rescaled to produce commensurable quantities. Grading of fractional rankings or co-rankings becomes useless if different gradings are incommensurable. The information that can be used in this case is limited to (1) sharp fractional ranking, if the fractional preferences are transitive or (2) indicator co-ranking (or sharp fractional co-ranking), if the fractional preferences are intransitive. The first case in considered first. If sharp (or incommensurable) fractional absolute rankings are strict, they represent an ordering as discussed in Section 2.2.

**Theorem 1**(Arrow [37]) For more than two elements, a set of K fractional orderings cannot be universally converted into an overall ordering in a way that is:

- (a) Non-trivial (non-dictatorial): absolute ranking does not simply replicate one of the fractional rankings: r(…)≁ r
^{(}^{α}^{)}(…) for all α; - (b) Pairwise independent: preference between any two elements does not depend on fractional rankings of the other elements, i.e., R(A,B) depends only on all R
^{(}^{α}^{)}(A,B), α = 1, …, K; - (c) Pareto-efficient: A≻B when r
^{(}^{α}^{)}(A) > r^{(}^{α}^{)}(B) for all α.

**Proposition 5**Strict fractional preferences (represented by fractional rankings if transitive or by fractional co-rankings if intransitive) can always be converted into an overall strict preference in an Arrow-consistent way, which is (1) non-trivial (for K > 2), (2) pairwise independent and (3) Pareto-efficient.

^{(}

^{α}

^{)}(A,B) for any α (we assume K > 2), (2) the formula for ρ(A,B) does not involve any characteristics of any third element (say C) and (3) ρ(A,B) = 1 when all R

^{(}

^{α}

^{)}(A,B) = 1. Here we put w

^{(}

^{α}

^{)}= 1 + ε

^{(}

^{α}

^{)}, where ε

^{(1)}, …, ε

^{(}

^{K}

^{)}are small random values, which ensure that ρ(A,B) = 0 only when A=B.

**Proposition 6**Any Arrow-consistent conversion of fractional orderings into an overall strict preference is potentially intransitive.

^{(}

^{α}

^{)}(A) > r

^{(}

^{α}

^{)}(B) > r

^{(}

^{α}

^{)}(C) for all α.

## 7. The Subscription Example

#### 7.1. Ariely’s Subscription Example

- (A) Web (W) subscription, $60;
- (B) Print & Web (P+W) subscription, $120;
- (C) Print (P) subscription, $120

#### 7.2. Evaluating Co-Rankings

^{(}

^{v}

^{)}and ρ

^{(}

^{p}

^{)}with equal weights and W = 2. When only two options, A and B, are available the conditional ranking with the reference set of $\mathrm{G}=\{\mathrm{A},\mathrm{B}\}$ is given by

_{i}set to unity. Hence, we would chose B from {A,B,C} but will have difficulty of selecting between A and B from {A,B}. (For the sake of our argument, it is sufficient to put ρ(A,B) = 0 and treat the preference between A and B as being close to 50% each. Ariely [38] indicates a marginal preference of A over B, which can be accommodated by introducing another grade of a preference—“marginally better” quantified by, say, 1/3 or 1/2. The co-ranking ρ(A,B) is thus redefined while the remaining co-rankings in Equation (56) are kept the without change. If 2ρ(A,B) = 1/3, then $\mathrm{A}{\prec}_{\mathrm{G}}\mathrm{B}$ since $2{\rho}_{\mathrm{G}}(\mathrm{A})=4/3$ and $2{\rho}_{\mathrm{G}}(\mathrm{B})=5/3$. If 2ρ(A,B) = 1/2, then $\mathrm{A}{~}_{\mathrm{G}}\mathrm{B}$ since $2{\rho}_{\mathrm{G}}(\mathrm{A})=2{\rho}_{\mathrm{G}}(\mathrm{B})=3/2$. Here, $\mathrm{G}=\{\mathrm{A},\mathrm{B},\mathrm{C}\}$. The author has repeated Ariely’s experiment in class of 60 students with half of the class selecting between A and B, while the other half selecting between A, B and C. The results {85%, 15%} and {35%, 62%, 3%} clearly confirm the effect discovered by Ariely, although indicate a higher level of acceptance of electronic communications than a decade ago).

#### 7.3. Potential Intransitivity of the Subscription Values

_{A}= $60, p

_{B}= $130 and p

_{C}= $80 correspond to the utilities of 3, 2/3 and 7/3 as specified by Equation (55). Our assessment of the subscription values remains the same as in Figure 4a. With the new price utilities, the overall co-ranking becomes

^{(}

^{v}

^{)}can be introduced and then the overall utility r = r

^{(}

^{v}

^{)}+ r

^{(}

^{p}

^{)}ensures transitivity of our preferences. Let r

^{(}

^{v}

^{)}(A) = 1, r

^{(}

^{v}

^{)}(B) = 3 and r

^{(}

^{v}

^{)}(C) = 2. This corresponds to replacing the last preference in Equation (53) by P+W≻P. This transitive correction does not necessarily represent human preferences better (in fact P+W≻P is not accurate for me, since I think that P+W is clearly better than P) but it removes potential intransitivity.

#### 7.4. Discussion of the Choices

_{C}from $80 to $130 to lure his customers into subscribing for option B. In this case the presence of C in the subscription list is not information but disinformation. How can the buyers protect themselves against such manipulations?

_{i}. Hence, a reasonable choice relies on a good selection of the perspective. Artificially or unscrupulously selected elements may distort the picture. In the subscription example, it might be desirable to weight the options by their estimated market shares. In this case the seller’s manipulations with option C would not have a significant effect on our choice.

## 8. Intransitivity Due to Imperfect Discrimination

#### 8.1. Discrimination Threshold

^{(}

^{α}

^{)}defined by

**Proposition 7**(Ng [39]) The overall preferences that correspond to threshold coarsening of K independent fractional utilities are

- (a) weakly intransitive (existence of A∼B∼C≻A) if K = 1,
- (b) semi-weakly intransitive (existence of A≻B∼C≻A) if K = 2 and w
^{(1)}ε^{(1)}= w^{(2)}ε^{(2)}, - (c) strictly intransitive (existence of A≻B≻C≻A) if K = 2 and w
^{(1)}ε^{(1)}≠ w^{(2)}ε^{(2)}or K ≥ 3.

^{(2)}, can be ignored in this case). The semi-weakly intransitive triplet A≻B∼C″~A in the same figure does not depend on the direction of the red line of constant fine utility. Finally, the points A≻B≻C≻A form a strictly intransitive triplet (note that C must be above the red line). In the three-dimensional case shown in Figure 5b, we select ε

^{(}

^{α}

^{)}= 1.5, α = 1, 2, 3, hence 1∼

^{(}

^{α}

^{)}2∼

^{(}

^{α}

^{)}3≻

^{(}

^{α}

^{)}1. Most values in the table are equivalent, while the three strict preferences of 3 over 1 are shown by the arrows. It is easy to see that the listed points form a strictly intransitive triplet A≻B≻C≻A.

#### 8.2. Imperfect Discrimination Due to the Presence of Noise

^{1}

^{/}

^{2}σ/β. The function $\overline{F}(\Delta y)$ represents Δy multiplied by a factor representing reliability of Δy giving a satisfactory estimate for Δr, i.e., $\overline{F}(\Delta y)$ is the reliable fraction of Δy. This models our inclination to ignore small Δy = y(A) − y(B) and accept large Δy while comparing A and B.

^{(}

^{α}

^{)}of the fractional utilities r

^{(}

^{α}

^{)}are different from the true values due to presence of some random noise, we are now compelled to define the fractional co-ranking by

^{(}

^{α}

^{)}is determined by the intensity of noise in direction α. The fine co-ranking and two coarse co-rankings that correspond to threshold coarsening Equation (62) and Gaussian coarsening Equation (67) with ε

^{(}

^{α}

^{)}= 1 are shown in Figure 7a. The effect of smooth coarse grading on intransitivity is qualitatively similar to the threshold case:

**Proposition 8**Gaussian coarsening in multiple dimensions K > 1 leads to strict intransitivity provided that not all w

^{(}

^{α}

^{)}ε

^{(}

^{α}

^{)}are the same.

^{(1)}= 1 and ε

^{(2)}= 1/10).

## 9. Risks and Benefits

#### 9.1. Hidden Degradation

#### 9.2. Competitive Simulations for Risk-Benefit Dilemma

^{(1)}and (desirable) benefit y

^{(2)}. In a simple transitive model, there exists a 1:1 trade off between the risk and the benefit according to co-ranking defined by

_{0}(A) indicates the transitivity of this case.

^{(1)}= y

^{(2)}= 0. The details can be found in previous publications [5–7,43]. The gray area in Figure 8 indicates prohibited space. The boundary is Pareto-efficient: it is impossible to increase the benefit without increasing the associated risk.

## 10. Thermodynamics and Intransitivity

_{A}> T

_{B}and T

_{B}> T

_{C}require that T

_{A}> T

_{C}. The concept of negative temperatures is compliant with the laws of thermodynamics and does not alter transitivity. Negative temperatures are placed above positive temperatures (for example T = −300 K is hotter than T = 300 K) but all temperatures are still linearly ordered according to −1/T. That is T = +0 K is the lowest possible temperature, while T = −0 K is the highest possible temperature (see [44]). If +0 K were identical to −0 K (which is not the case), then the thermodynamic temperatures would be intransitive). Hence, the constraints of physical thermodynamics allow for cyclic or intransitive behaviour only in thermodynamically open systems. There is no evidence of any kind that the laws of thermodynamics are violated in complex evolutionary processes (biological or social). Increase of order in a system is always compensated by dispersing much larger quantities of entropy. The question that is often discussed in relevant publications [45] is not the letter but the spirit of the thermodynamic laws—the possibility of explaining complex evolutions using thermodynamic principles. Such explanations can be referred to as apparent thermodynamics (i.e., thermodynamics-like behaviour explained with the use of the theoretical machinery of thermodynamics).

#### 10.1. Transitive Competitive Thermodynamics

#### 10.2. Nearly Transitive Systems

_{1}B

_{1}and C

_{1}as equivalent and distinguish only transitive preferences

_{1}, …,A

_{3}is transitive and unique ranking is possible within the region around element A. Under conditions specified in [5], this local competition can be characterised by competitive potentials χ, which are analogous to chemical potentials of conventional thermodynamics taken with a negative sign. However, the competition is intransitive if we look at larger scales: A≺B≺C≺A. If three identical systems have elements A, B and C as leading particles, the competitive potentials are related to each other by

_{A}< χ

_{B}< χ

_{C}< χ

_{A}since this does not make mathematical sense. Competitive potentials are numbers in transitive systems but they must be treated as more complex objects in intransitive systems.

#### 10.3. Strong Intransitivity

- rapid relaxation to a quasi-steady state (which, possibly, can be approximately treated as transitive) with subsequent slow evolution;
- the possibility of alternative directions of evolution (i.e., competitive degradation or competitive escalation);
- violation of Boltzmann’s Stosszahlansatz (the hypothesis of stochastic independence of the system elements) and formation of cooperative structures.

## 11. Discussion and Conclusions

- ♦ relative comparison criteria or
- ♦ multiple comparison criteria that are incommensurable or
- ♦ multiple comparison criteria that are known approximately or
- ♦ comparisons of groups of comparable elements.

- relativity of strength and dependence of preference on perspective
- cyclic behaviour instead of relaxation to a unique equilibrium
- relatively slow evolutions punctuated by sudden collapses and changes
- complex patterns of behaviour (e.g., cooperative structures).

## Acknowledgments

## Conflicts of Interest

## Appendix

## A. Potential Intransitivity

^{(1)}(C,A), then fractional co-ranking Equation (A2) can be represented by Equation (10) in terms of the following absolute fractional ranking

_{1}≠ 0, this representation is impossible. In this case setting ρ

^{(1)}(C,A) ≥ 0 to ${\rho}_{t}^{(1)}(\mathrm{C},\mathrm{A})\ge 0$ and ensuring that δ

_{1}= 0 would remove potential intransitivity but alter the magnitude of our preference. Note that a triplet Equation (A1) with δ

_{1}≠ 0 can always be found in an arbitrary set with a potentially intransitive co-ranking (otherwise we may define r(B) = r(A) + ρ(B,A) for fixed A and arbitrary B).

^{(1)}+ ρ

^{(2)})/2 then we obtain

_{1}> 0, then selecting δ

_{1}/2 > ϵ> 0 determines an overall preference that is currently intransitive.

_{1}< 0 is treated in a similar manner. If δ

_{1}= 0, intransitivity cannot appear. This discussion can alternatively be summarised in form of a short proposition:

**Proposition A1**A potentially intransitive co-ranking can always be represented as a superposition of a currently intransitive co-ranking and an absolutely transitive co-ranking.

_{0}, where ρ

_{0}= ρ

^{(1)}in Equation (A3) with δ

_{1}≠ 0, can always be selected from a set with potentially intransitive co-rankings. Consider ρ

_{0}= ρ

_{1}+ ρ

_{2}where, ρ

_{1}= −ρ

^{(2)}defined by Equation (A7) is transitive, and ρ

_{2}= 2ρ specified by Equation (A8) is currently intransitive with a proper choice of ε. This decomposition is extended to the remaining elements so that ρ

_{1}remains absolutely transitive. The representation ρ

_{0}= ρ

_{1}+ ρ

_{2}is, obviously, not unique.

## B. Preference Properties and Indicator Co-Ranking

#### B.1. Coarsenings and Refinements

_{a}and ≺

_{b}. Rule b represents a coarsening of rule a if A≺

_{b}B demands that A≺

_{a}B for any A and B (although A∼

_{b}B may correspond to any of A∼

_{a}B, A≺

_{a}B or A≻

_{a}B). If rule b represents a coarsening of rule a then the same property is expressed by saying that rule a represents a refinement of rule b, implying that A⪯

_{a}B demands that A⪯

_{b}B for any A and B (if A≻

_{b}B was correct, then this would demand A≻

_{a}B and thus contradict A⪯

_{a}B). (Statement “relation ⪯

_{b}contains relation ⪯

_{a}” is another equivalent, which is often used in literature but can be confusing in the context of the present work). If rules a and b are refinements of one another, then these rules are obviously equivalent.

#### B.2. Transitive Preferences in Intransitive Systems

_{t}, ∼

_{t}, ⪯

_{t}, etc.) or by ≺≺, which is equivalent to ≺

_{t}, and by ≈, which is equivalent to ∼

_{t}. Hence the relation A≺≺B, which is expressed as “A is transitively preferred to B”, indicates that A≺B and A≺

_{t}B (where preference ≺ is generally intransitive). The relation A≺≺B implies that there should not exist any set {C

_{1},…,C

_{k}} such that A≺B⪯C

_{1}⪯…⪯C

_{k}⪯A and, in particular, there is no such C that can form an intransitive triplet C⪯A≺B⪯C.

#### B.3. Transitivity and the Indicator Co-Ranking

**Proposition B1**If the underlying preference is currently transitive, then the preference induced by the conditional indicator ranking (i.e., $A{\le}_{\mathrm{G}}B\iff {R}_{\mathrm{G}}(A)\le {R}_{\mathrm{G}}(B)$ represents a coarsening of the underlying preference for any reference group$\mathrm{G}$.

_{i}) ≤ R(B,C

_{i}) for all C

_{i}∈ $\mathrm{G}$. Note that the combination of ${R}_{\mathrm{G}}(\mathrm{A})={R}_{\mathrm{G}}(\mathrm{B})$ and A≺B is possible for some $\mathrm{G}$.

**Proposition B2**If conditional indicator rankings based on different reference groups are strictly nonequivalent (i.e., there exist at least two elements A and B and at least two groups${\mathbb{G}}^{\prime}$ and${\mathbb{G}}^{\u2033}$ so that$A{\prec}_{{\mathbb{G}}^{\prime}}B$ and$A{\prec}_{{\mathbb{G}}^{\u2033}}B$, then the underlying preference is currently intransitive.

**Proposition B3**Group preference based on the indicator co-ranking is not necessarily transitive even if the underlying element preference is currently transitive.

## C. Primary and Secondary Rankings

_{i}= g(C

_{i}) > 0 specified for every element C

_{i}∈ ${\mathrm{G}}_{0}$. In this section, weights g

_{i}are associated with the whole system ${\mathrm{G}}_{0}$ and thus is the same for all groups ${\mathrm{G}}_{q}\subseteq {\mathrm{G}}_{0}$, which are referred to in this section as sets ${\mathrm{S}}_{q}={\mathrm{G}}_{q}$, q = 1, 2, … to emphasise that the element weights are specified only for the whole system.

#### C.1. Current Rankings

^{*}(A) is the conditional ranking Equation (24) of element A with respect to all other elements in the system. In the same way the current ranking can be introduced for an arbitrary set ${\mathrm{G}}_{q}$

_{0}and G

_{q}are total weights of the system and of the set ${\mathrm{G}}_{q}$. The group co-ranking $\overline{R}({\mathrm{G}}_{q},{\mathrm{G}}_{0})$ is introduced according to Equation (25). The current ranking R

^{*}(…), the underlying preference ≺, and the corresponding indicator co-ranking R(A,C) are referred to as primary when we need to distinguish them from secondary characteristics.

#### C.2. Properties of Current Rankings

^{*}(A) is ${R}_{{\mathrm{G}}_{q}}(\mathrm{A})$ with set ${\mathrm{G}}_{q}$ expanded to the whole system ${\mathrm{G}}_{0}$. The following propositions characterise properties of

**Proposition C1**If the primary preference is currently transitive, the secondary preference is equivalent to the primary preference:$A{\underset{\xaf}{\prec}}^{\prime \prime}B\iff {R}^{*}(A)\le {R}^{*}(B)\iff A\underset{\xaf}{\prec}B$ for any A and B.

^{*}(A) = R

^{*}(B) (i.e., A∼″B) is now impossible when A≺B. Indeed, all elements are presumed to be connected and present in the reference set in definition of the current ranking. Hence, the terms R(A,B) = −R(B,A) < 0 are present in the sums Equation (C1) evaluated for R

^{*}(A) while including C

_{i}=B, and R

^{*}(B) while including C

_{i}=A.

**Proposition C2**If B is transitively preferred to A in a generally intransitive preference, then the preference of B over A is preserved by the current ranking: A≺≺B =⇒ R

^{*}(A) < R

^{*}(B) ⇔A≺″B

_{i}that satisfies C

_{i}A≺B C

_{i}. Hence, R(A,C

_{i}) ≤ R(B,C

_{i}) and R(A,B) = −R(B,A) < 0 so that R

^{*}(A) < R

^{*}(B) as defined by Equation (C1). Note that the inverse statements R

^{*}(A) < R

^{*}(B) =⇒ A≺≺B and R

^{*}(A) < R

^{*}(B) =⇒ A≺B are incorrect in intransitive systems.

^{*}(A) ≠ R

^{**}(A). Deviations of secondary ranking from primary ranking indicate intransitivity in evolutions of competitive systems as determined by the following theorem:

**Theorem C1**The following statements are correct for a system${\mathrm{G}}_{0}$ of connected elements (i.e., g(C

_{i}) > 0 for any C

_{i}∈ ${\mathrm{G}}_{0}$):

- (a) If the primary preference is currently transitive, the primary and secondary current rankings coincide (i.e., R
^{*}(C_{i}) = R^{**}(Ci) for all C_{i}∈ ${\mathrm{G}}_{0}$). - (b) If the primary and secondary rankings coincide (i.e., R
^{*}(C_{i}) = R^{**}(Ci) for all elements C_{i}∈ ${\mathrm{G}}_{0}$), then the secondary preference is a coarsening of the primary preference (i.e., C_{i}≺″C_{j}=⇒ C_{i}≺C_{j}). - (c) In particular, if the primary and secondary rankings coincide and are strict (i.e., current rankings of different elements are different: R
^{*}(C_{i}) ≠ R^{*}(C_{i}) for any C_{i}≁ C_{j}), then the primary preference is currently transitive.

**Proof**. Statement (a) immediately follows from the equivalence of primary and secondary preferences, as stated in Proposition C1, while statements (b) and (c) require detailed consideration.

^{th}set ${\mathrm{G}}_{q}$ contains n

_{q}≥ 1 elements that have primary current ranking ${R}_{q}^{*}$. The elements C

_{1}, …,C

_{i}, …,C

_{n}are thus ordered according to decreasing primary ranking. Figure C1 shows the structure of the n × n matrices R

_{ij}= R(C

_{i},C

_{j}) and ${{R}^{\u2033}}_{ij}={R}^{\u2033}({\mathrm{C}}_{i},{\mathrm{C}}_{j})$, which correspond to the primary and secondary co-rankings. The co-ranking of different sets is denoted by ${\overline{R}}_{qp}=\overline{R}({\mathrm{G}}_{q},{\mathrm{G}}_{p})$ for primary preferences and by ${{\overline{R}}^{\u2033}}_{qp}={\overline{R}}^{\u2033}({\mathrm{G}}_{q},{\mathrm{G}}_{p})$ for the secondary preferences. The average primary current ranking of a set is denoted by ${\overline{R}}^{*}{}_{q}={\overline{R}}^{*}({\mathrm{G}}_{q})=\overline{R}({\mathrm{G}}_{q},{\mathrm{G}}_{0})$, while the average secondary current ranking is denoted by ${\overline{R}}^{**}{}_{q}={\overline{R}}^{**}({\mathrm{G}}_{q})={\overline{R}}^{\u2033}({\mathrm{G}}_{q},{\mathrm{G}}_{0})$. These quantities are specified by Equations (24), (25) and (C1)–(C4). Obviously ${\overline{R}}^{*}{}_{q}={\overline{R}}_{q}^{**}$ for all q due to equivalence of the primary and secondary current rankings as stated in the theorem (i.e., R

^{*}(C

_{i})= R

^{**}(C

_{i}) for all C

_{i}∈ ${\mathrm{G}}_{0}$ ).

_{ij}and ${{\overline{R}}^{\u2033}}_{ij}$ are antisymmetric. Since ${\overline{R}}_{1}^{*}=\overline{R}({\mathrm{G}}_{1},{\mathrm{G}}_{0})$ is the same as ${\overline{R}}_{1}^{**}={\overline{R}}^{\u2033}({\mathrm{G}}_{1},{\mathrm{G}}_{0})$ while $\overline{R}({\mathrm{G}}_{1},{\mathrm{G}}_{1})={\overline{R}}^{\u2033}({\mathrm{G}}_{1},{\mathrm{G}}_{1})=0$, the co-rankings $\overline{R}({\mathrm{G}}_{1},{\mathrm{G}}_{0}-{\mathrm{G}}_{1})$ and ${\overline{R}}^{\u2033}({\mathrm{G}}_{1},{\mathrm{G}}_{0}-{\mathrm{G}}_{1})$ evaluated in terms the corresponding sums over the rectangle a2-b2-b4-a4 must be the same. Hence R

_{ij}= 1 in this rectangle since in any other case the sums $\overline{R}({\mathrm{G}}_{1},{\mathrm{G}}_{0}-{\mathrm{G}}_{1})$ cannot coincide with ${\overline{R}}^{\u2033}({\mathrm{G}}_{1},{\mathrm{G}}_{0}-{\mathrm{G}}_{1})$.

_{ij}= −1 and ${{R}^{\u2033}}_{ij}=-1$ in the rectangle b1-c1-c2-b2. We take into account that $\overline{R}({\mathrm{G}}_{2},{\mathrm{G}}_{2})={\overline{R}}^{\u2033}({\mathrm{G}}_{2},{\mathrm{G}}_{2})=0$ (the sums over the dark squares are zeros) and reiterate our previous consideration for rectangle b3-c3-c4-b4, where ${{R}^{\u2033}}_{ij}=1$ and the sums $\overline{R}({\mathrm{G}}_{2},{\mathrm{G}}_{0}-{\mathrm{G}}_{1}-{\mathrm{G}}_{2})$ and ${\overline{R}}^{\u2033}({\mathrm{G}}_{2},{\mathrm{G}}_{0}-{\mathrm{G}}_{1}-{\mathrm{G}}_{2})$ must be the same. Hence, R

_{ij}= 1 in this rectangle. Continuing this consideration for the remaining sets q = 3, 4…, k proves that ${R}_{ij}={{R}^{\u2033}}_{ij}$ provided i and j belong to different sets.

_{ij}= 0 or competition is intransitive within the set (if the primary preferences within the set q were transitive then, according to Proposition C2, C

_{i}≻C

_{j}demands R

^{*}(C

_{i}) > R

^{*}(C

_{j}), which contradicts C

_{i},C

_{j}∈ ${\mathrm{G}}_{q}$ ). The secondary preference represents a coarsening of the primary preference since R

_{ij}

^{00}> 0 demands R

_{ij}> 0 (when j and i belong to different sets), while ${{R}^{\u2033}}_{ij}=0$ may correspond to R

_{ij}> 0, R

_{ij}= 0 or R

_{ij}< 0 (when j and i belong to a common set).

_{q}= 1 for all q, then ${R}_{ij}={{R}^{\u2033}}_{ij}$ since all i ≠ j always belong to different sets. This means that the primary preference coincides with the secondary preference based on primary current ranking and is transitive. ■

#### C.3. Maps of Current Rankings

^{*}of the elements against their secondary current ranking R

^{**}), indicates the presence, intensity, extent and localisation of intransitivity by deviations from the line R

^{*}= R

^{**}. We wish to stay under the conditions of Theorem C1(c), and avoid complexities related to statement b of this theorem since, in this case, equivalence between the primary and secondary rankings ensures transitivity. However, in the case of g(C

_{i}) = 1, which perhaps is most common in practice, coincidences of primary rankings for different elements R

^{*}(C

_{i}) = R

^{*}(C

_{j}) for i ≠ j are likely due to the limited number of values spaced by 1/n that these rankings can take. The practical solution for this problems is simple—consider g(C

_{i}) = (1 + ε

_{i})g

_{0}(C

_{i}), where ε

_{i}represent small random values and g

_{0}(C

_{i}) are the original weights. Presence of these small values does not significantly alter the maps but makes coincidences R

^{*}(C

_{i}) = R

^{*}(C

_{j}) impossible unless all properties of the elements C

_{i}and C

_{j}are identical, which is sufficient for our purposes.

_{i}

_{(}

_{q}

_{)}≻≻C

_{j}

_{(}

_{p}

_{)}for any C

_{i}

_{(}

_{q}

_{)}∈ ${\mathrm{G}}_{q}$ and any C

_{j}

_{(}

_{p}

_{)}∈ ${\mathrm{G}}_{p}$. According to these notations, the index i(q) runs over all elements of set ${\mathrm{G}}_{q}$. In this subsection, we consider partition of all elements in the system into range sets, where each set is represented by a range of primary current ranking (and consequently by a range of the secondary current ranking). The range sets are non-overlapping and jointly cover all elements.

^{*}versus R

^{**}maps. Consider an intransitive preference and its transitive closure. In this closure, the elements C

_{1}, …,C

_{n}are divided into k transitively ordered sets ${\mathrm{G}}_{1}\succ \succ {\mathrm{G}}_{2}\succ \succ \dots \succ \succ {\mathrm{G}}_{k}$ of elements that are transitively equivalent within each set. That is for any C

_{i}

_{(}

_{q}

_{)}∈ ${\mathrm{G}}_{q}$ and any C

_{j}

_{(}

_{p}

_{)}∈ ${\mathrm{G}}_{p}$

^{*}(C

_{i}

_{(}

_{q}

_{)}) > R

^{*}(C

_{j}

_{(}

_{p}

_{)}) and R

^{**}(C

_{i}

_{(}

_{q}

_{)}) > R

^{**}(C

_{j}

_{(}

_{p}

_{)}) when q < p, that is the ranges of current rankings of different sets do not overlap. Hence sets ${\mathrm{G}}_{1},\dots ,{\mathrm{G}}_{k}$ represent a set of range sets. This implies equivalence of primary and secondary set co-rankings

**Proposition C3**The primary and secondary average current rankings of range sets coincide if and only if these sets are subject to a transitive primary preference (although this preference may remain intransitive within each set).

_{i}

_{(1)}≻″C

_{i}

_{(}

_{q}

_{)}for all q > 1, the equality ${\overline{R}}^{*}({\mathrm{G}}_{1})={\overline{R}}^{**}({\mathrm{G}}_{1})$ is achieved if and only if C

_{i}

_{(1)}≻C

_{i}

_{(}

_{q}

_{)}for all q > 1. After applying this consideration sequentially to sets ${\mathrm{G}}_{2},\dots ,{\mathrm{G}}_{k}$, we conclude that C

_{i}

_{(}

_{p}

_{)}≻C

_{i}

_{(}

_{q}

_{)}for p < q. Finally we note that ${\mathrm{G}}_{1}\succ {\mathrm{G}}_{2}\succ \dots \succ {\mathrm{G}}_{K}$ requires ${\mathrm{G}}_{1}\succ \succ {\mathrm{G}}_{2}\succ \succ \dots \succ \succ {\mathrm{G}}_{K}$ since the preference ≻′ defined by C

_{i}

_{(}

_{p}

_{)}≻′C

_{i}

_{(}

_{q}

_{)}for p < q and C

_{i}

_{(}

_{p}

_{)}∼′C

_{i}

_{(}

_{q}

_{)}for p = q is transitive and is a coarsening of the primary preference and therefore is a coarsening of its transitive closure.

^{**}). The range sets, shown in the figure, are transitively ordered so that ${\mathrm{G}}_{1}\succ \succ {\mathrm{G}}_{2}\succ \succ {\mathrm{G}}_{3}\succ \succ {\mathrm{G}}_{4}\succ \succ {\mathrm{G}}_{5}$. The large dots indicate average set ranking, which is compliant with Equation (C9). The preferences are transitive within ${\mathrm{G}}_{3}$ and ${\mathrm{G}}_{5}$ and intransitive within ${\mathrm{G}}_{1},{\mathrm{G}}_{2}$ and ${\mathrm{G}}_{4}$. Small deviation from the line specified by R* = R

^{**}within set ${\mathrm{G}}_{4}$ indicate that intransitivity is present but not frequent within this set. Two subsets ${\mathrm{G}}_{1a}$ and ${\mathrm{G}}_{1b}$ are distinguished within set ${\mathrm{G}}_{1}$. The preferences between these sets are close to be transitive but some intransitive interference between subsets is present as indicated by angle γ > 0. The small dots show current set rankings of ${\mathrm{G}}_{1a}$ and ${\mathrm{G}}_{1b}$, which are not compliant with Equation (C9).

**Figure C2.**Current ranking map (the thick red line shows primary vs secondary current ranking). The map is divided into range sets (groups) ${\mathrm{G}}_{1},\dots ,{\mathrm{G}}_{5}$ (group current rankings of the sets are shown by large dots) and subsets ${\mathrm{G}}_{1a}$ and ${\mathrm{G}}_{1b}$ (group current rankings of the subsets are shown by small dots). The black dashed line corresponds to R

^{∗}= R

^{∗∗}.

## D. Evolutionary Intransitivity

_{i}) > 0 is proportional to the probability of selecting element C

_{i}for competition.

#### D.1. Competitive Evolution

^{∗}(A) is the primary current ranking defined with the weight g(B) = f(B)ψ(B), which is proportional to the probability ψ(B) of selecting element B for competitive mixing. (Generally, this selection weight can depend on both A and B, that is ψ = ψ(A,B) as considered in [5] but this is not needed in the present work). Since selection of elements for mixing is stochastic, the process of competition requires a probabilistic description (the fluctuations with respect to the averages are not considered). This is achieved with the use of probability f, which can be initially set according to the location of n stochastic particles, which numerically represent f by a large set of delta-functions, before the competition steps. Mutation step, which is not considered here, would make this distribution continuous. As noted above, isolated elements g(…) = 0 are not of interest and all elements are presumed to be connected by competition.

**Proposition D1**Competition (transitive or intransitive) results in improved competitiveness with respect to the current distribution f, that is for t

_{2}= t

_{1}+ dt

^{2}and (R

^{∗})

^{2}are non-negative. Fluctuations that can be present in the system due to random particle selection are not considered here. Note that, in intransitive systems, competitiveness increases only with respect to the current distribution and improvement is not necessarily achieved when viewed from a different perspective, that is $\overline{R}({\mathrm{G}}_{t},{\mathrm{G}}_{r})$ may decrease in time for some reference groups ${\mathrm{G}}_{r}$. It is common to have a view attached to current distribution—this, according to Proposition D1, produces the impression that competitiveness is improved by the competition but, in intransitive systems, the actual result might be different if viewed from a different perspective (i.e., using a different reference set). The effect of mutation steps (which is not considered here) on competitiveness is typically negative.

#### D.2. Evolutionary Intransitivity in Simulations of the Risk/benefit Dilemma

## E. Summary of the Terms Characterising Intransitivity

- By strictness of the preference:
- – strict intransitivity: there exists A≻B≻C≻A
- – semi-strict intransitivity: there exists A≻C
_{1}≻…≻C_{k}≻A but intransitivity is not strict - – semi-weak intransitivity: there exists A≻B~C≻A but intransitivity is not semi-strict
- – weak intransitivity: there exists A∼B∼C≻A but intransitivity is not semi-weak

- By localisation:
- – local intransitivity (combined with global transitivity)
- – global intransitivity (combined with local transitivity)
- – strong intransitivity (strict and both local and global)

- By explicit presence:
- – absolute transitivity: any kind of intransitivity is impossible under the given preference rules, implying existence of absolute utility or ranking
- – potential intransitivity: intransitivity does not necessarily show on the current set of elements but may appear when conditions are changed
- – current intransitivity or current transitivity: indicate the properties of preferences on the current set of elements
- – near-transitive evolution: evolution of an intransitive system that, within a fixed time interval, can be reasonably approximated by evolution of a transitive system

## F. Note on Quantum Preferences

#### F.1. Quantum Preferences and Co-Rankings

_{j}= e

^{−iωj}where ω

_{j}(j = 1, …) are random angles, which are uniformly and independently distributed between −π and π. A mixture of quantum states is interpreted here as an entanglement with special quantum states |θ

_{j}] representing the random phases. These states have only one operation applicable to these states and resulting in a physically measurable quantities—the scalar product:

_{2}) − arg(c

_{1}) of complex amplitudes c

_{1}and c

_{2}affect the quantum state. In the case of a mixed state, the phases are not important as they are randomised by θ

_{1}and θ

_{2}but the relative magnitudes $|{c}_{2}|/|{c}_{1}|$ determine the state: the effect of a mixed state is similar to classical probabilities P

_{2}= |c

_{2}|

^{2}and P

_{1}= |c

_{1}|

^{2}, P

_{1}+ P

_{2}= 1.

_{1}= |c

_{1}|

^{2}and P

_{2}= |c

_{2}|

^{2}. These probabilities are the same for superposition states Equation (F1) and mixed states Equation (F2).

_{1}and P

_{2}. The bra/ket notations are abbreviated here to |A〉 = |A ≻ B〉 and |B〉 = |B≻A〉. In the superposition state Equation (F1), the additional terms with interferences $\langle \mathrm{A}|\mathbb{H}|\mathrm{B}\rangle $ and $\langle \mathrm{B}|\mathbb{H}|\mathrm{A}\rangle $ appear in the sum: conditional alternatives do interfere in quantum mechanics. However, it is not completely clear what measurable physical quantities operator $\mathbb{H}$ might represent, when the two basic states |A〉 and |B〉 represent alternative preferences, and how experiments distinguishing mixed cognitive states from superimposed cognitive states in decision-making can be carried out.

#### F.2. Absolute Ranking in Quantum Case

_{1}, …, r

_{k}. If these ranks are sharp, we can assume that r

_{i}= i without loss of generality. The state of a single quantum element is specified by the wave function

_{i}(A)〉 |r

_{j}(B)〉.

_{1}= (k

^{2}− k)/2 states i > j are followed by k states i = j and then by k

_{1}states states i < j, then the matrix $\langle {i}^{\prime},{j}^{\prime}|\mathbb{R}|i,j\rangle $ of the preference co-ranking operator $\mathbb{R}$ takes the following form

**Î**is the k

_{1}× k

_{1}unit matrix while the remaining elements are zeros. The preference co-ranking value is given by

_{1}, …,C

_{n}are simultaneously considered, then the wave function takes the form

_{q}and i

_{p}. The probabilities P

_{ij}(C

_{q},C

_{p}) are joint probability distributions of ranks for two elements C

_{q}and C

_{p}. The case of interest when these distributions are independent (separable in quantum terminology) that is for any two elements A and B

_{i}= P

_{i}.

**Proposition F1**A quantum (or probabilistic) preference can be intransitive even if it has an absolute ranking.

_{i}(A), P

_{i}(B) and P

_{i}(C) over k = 9 ranks i = 1, …, 9. This case follows the dice example shown in Figure 1b. It can be seen that

**Figure F1.**Intransitive A ≺ B ≺ C ≺ A preference for quantum/probabilistic elements with absolute ranking r

_{i}, where P

_{i}indicates probability of state i.

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**Figure 1.**(

**a**) The rock-paper-scissors game is the best-known example of intransitivity in games; (

**b**) Intransitive dice game where the die thrown by a player wins when it has a higher score than the die thrown by the opposition. The opposite sides of the dice have the same numbers. With the probability of 5/9, die B wins over A, C wins over B, and A wins over C. This dice game is a simple version of Efron’s dice (see article “nontransitive dice” in Wikipedia).

**Figure 2.**Comparison of choices A, B and C in case of a fire emergency: (

**a**) “can do more but not less” (dashed line—naive perspective; solid arrows—legal perspective) and (

**b**) “manager’s choice” (arrows indicate the regulation requirements).

**Figure 4.**Choices A, B and C for The Economist subscription [38]. Dashed blue arrows—value preferences, dotted red arrows—price preference, thick black arrows— overall preferences (shown as the sum of the fractional preferences); the price utility shown in the circles. (

**a**) Original prices; (

**b**) adjusted prices leading to explicit intransitivity

**Figure 5.**Intransitivity of fractional (partial) selection criteria when these criteria are known approximately: (

**a**) case of two criteria (

**b**) case of three criteria.

**Figure 7.**(

**a**) Co-ranking functions vs. fractional (partial) utility: dotted line—original fine-graded; dashed line—threshold-coarsened; solid line—Gauss-coarsened; (

**b**) Intransitivity due to coarsening in two dimensions: solid line (blue) represents elements X that X∼A and dashed line (red) represents elements X that X∼B for Gauss-coarsened co-rankings with ε

^{(1)}= 1 and ε

^{(2)}= 1/10. Dotted (blue) line represents elements X that X∼A for fine-graded co-ranking.

**Figure 9.**Simulations of the risk-benefit dilemma. Scale on the left: solid line (blue)—average benefit, dots (black)—equilibrium state in the corresponding transitive competition. Scale on the right: dashed line (red)—evolutionary intrasitivity parameter Ω (see Appendix D). Vertical lines: cases shown in Figure 10.

**Figure 10.**The case as in Figure 9: (

**a,d,g**) at 8 time steps; (

**b,e,h**) at 285 time steps; (

**c,f,i**) at 500 time steps. Plots a,b,c: Primary current ranking R

^{*}vs. secondary current ranking R

^{**}(solid red line). The cases shown are the same as in Figure 10. The solid blue line with dots corresponds to transitive or effectively transitive case where primary and secondary rankings coincide (see Appendix C). Plots d,e,f: Intransitive primary current ranking R

^{*}evaluated for co-ranking ρ specified by Equation (70) vs. transitive primary current ranking ${R}_{0}^{*}$ evaluated for co-ranking ρ

_{0}specified by Equation (68). Deviations from the solid line with dots indicate differences between current rankings based on ρ and on ρ

_{0}. Plots g,h,i: Domain snapshots. Colour shows the primary current ranking R

^{*}for each particle according to the colour bar. Plot i: blue line is a 500-step history of mean values; enlarged box is shown as insert.

**Figure 11.**Intransitive systems that display a degree of similarity with transitive systems: (a) locally intransitive globally transitive system and (b) locally transitive globally intransitive system.

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).