Note on a classical case of self-organization: The Persian Immortals Open Access

In Ref. 1, Kanellopoulos et al. proposed a dynamical system in which the changes in the distribution of the fighting ability (“fitness”) of the warriors forming the army of the Persian Immortals are tracked. Changes in the army's fitness occur when warriors who fall in battle are replaced by others from a reserve unit. The fighting ability of a warrior is represented by a random variable x, which follows a cumulative distribution function (CDF) $Fk(x)$ at time k of the fight. Let us call $Q$ the rules that determine which warriors leave the Persian Immortals' army (warriors fallen in combat) and which warriors replace them. To analyze this dynamical system, it is necessary to know how the rules $Q$ give rise to $Fk+1(x)$ from $Fk(x)$⁠, i.e., we need to find the relation $Fk+1=DQ(Fk)$⁠.

Kanellopoulos et al. used three replacement rules to simulate the evolution of the fitness of the army of Persian Immortals with N warriors, which are as follows:

1. Remove the weakest warrior,

2. Then, remove m randomly chosen warriors, and

3. Add $m+1$ new recruits from the corps's reserves.

Unfortunately, the authors' derivation of Eq. (1) is somewhat complex because it relies on a relation [their Eq. (2)] whose justification is difficult to follow for those unfamiliar with how the CDF of a set changes when some of its elements are replaced. The existence of this difficulty in following the derivation of Eq. (1) is a pity because it may lead the reader to underestimate or overlook this excellent, original, and interesting article. Moreover, the derivation of $DQ(F)$ for other $Q$-rules then appears to be an equally complicated task, which limits the usefulness of the article. The purpose of this comment is to show that the derivation of Eq. (1) can be done in an alternative and simpler way. As an aside, we will see that obtaining the dynamic law $DQ(F)$ for other replacement rules would also be straightforward.

In the new derivation, the key is to know how the exit and/or entry of a certain number warriors changes the CDF $F(x)$ of the fighting ability x of the army of Persian Immortals. Although the probabilistic reasoning will be done with this compelling example, this is of course irrelevant, since it does not matter whether x is a random variable characterizing a warrior' fitness or any other property of a given set of interest.

By the very definition of cumulative distribution, we know that $F(x)$ is the probability that a Persian Immortal has a fighting ability less than or equal to x. Then, in this army of N warriors, we will have, on average, $NF(x)$ warriors with fighting ability less than or equal to x. A simple, didactic, and colorful way to visualize the following arguments is to imagine that warriors with a fighting ability less than or equal to our arbitrary chosen value of x are, say, blue or, alternatively, that we put a blue badge on them.

Let us look at the effect that rules of the type used by Kanellopoulos et al. has on the CDF of the Persian Immortals' fighting ability (a sketch of these rules and how they work is shown in Fig. 1).

• Suppose that within the unit of Persian Immortals, there is a subset of warriors whose fighting ability is given by the CDF $S(x)$⁠. For example, this subset could consist of warriors who are sick, or could consist of the first w weakest warriors (Kanellopoulos et al. considered the monocomponent subset of the weakest warrior, i.e., $w=1$⁠). If we select s warriors from this subset, the number of warriors with fighting ability less than or equal to x (i.e., blue warriors) will be, on average, $sS(x)$⁠. [To simplify the language, the phrase “on average” will be omitted in the following when referring to the number of blue warriors.] If we remove these s warriors from the army of N Persian Immortals, the number of blue warriors in the resulting army of $N−s$ members will be equal to the number of blue warriors originally there, $NF(x)$⁠, minus the number $sS(x)$ of blue warriors removed. The proportion of blue warriors in this new army, i.e., the CFD $G(x)$ of this new army, is then given by the following relation:

  $(N−s)G(x)=NF(x)−sS(x).$

  (2)
• If we randomly remove m warriors from the former Persian Immortal army, the CDF of the new army of $N−s−m$ warriors remains $G(x)$⁠. This rather obvious result can be derived as a trivial exercise by counting blue warriors. In the army of $N−s$ warriors, the number of blue warriors is $(N−s)G(x)$⁠. If we randomly take m warriors from this army, $mG(x)$ will be blue. Then, the number of blue warriors in the army of $N−s−m$ warriors is $(N−s)G(x)−mG(x)$ and, therefore, the resulting CDF (the proportion of blue warriors) is

  $(N−s)G(x)−mG(x)N−s−m=G(x),$

  (3)

  which is the initial CDF, as expected.

• Suppose now that a reserve unit with CDF equal to $R(x)$ is available. If we randomly choose r warriors from this reserve unit, we know that $rR(x)$ soldiers will be blue, i.e., $rR(x)$ will have fighting ability less than or equal to x. Then, when we add these r warriors from the reserve army to the former army of $N−s−m$ warriors (of which $(N−s−m)G(x)$ were blue), the total number of blue warriors in the combined army of $N−s−m+r$ warriors will be

  $(N−s−m+r)H(x)=(N−s−m)G(x)+rR(x),$

  (4)
  where $H(x)$ is the proportion (CDF) of blue warriors in the new army. Inserting Eq. (2) in this equation, we get the CDF $H(x)$ in terms of the initial CDF $F(x)$⁠,

  $H(x)=N−s−m(N−s−m+r)(N−s) [NF(x)−sS(x)]+rN−s−m+rR(x).$

  (5)

In summary, $H(x)$ is the resulting CDF when, from the army of N Persian Immortals with CFD $F(x)$⁠, (i) we remove s warriors with CDF $S(x)$⁠, next (ii) we remove m warriors at random, and finally, (iii) we add r warriors from a reserve army with CDF $R(x)$⁠. It is immediately seen that Eq. (5) reduces to Eq. (1) [Eq. (5) in Ref. 1] if the initial CDF $F(x)$ is called $Fk(x)$⁠, the resulting CDF $H(x)$ is called $Fk+1$⁠, and take the particular values and CDFs used by Kanellopoulos et al. in Ref. 1: $s=1$⁠, $r=m+s=m+1$⁠, $S(x)=Fweakest=1−[1−Fk]N$⁠, and $R=Funiform$⁠.

The above-mentioned simple arguments for finding the CDF after applying each replacement rule, and in particular for finding a multi-step relation like Eq. (5), show how easy it is to obtain the dynamics $DQ$ of the CDF of a set (army) when this set is modified by removing and/or replacing its elements (warriors).

The author acknowledges financial support from Grant No. PID2020-112936GB-I00 funded by the Ministerio de Ciencia y Tecnología, Spain (MCIN/AEI/10.13039/501100011033).

The author has no conflicts to disclose.