**1. Introduction**

In 1999, one of the authors of this paper (PR) arrived in Ireland to spend what became a decade in Trinity College. During that year he had the opportunity to attend the first ever European Physical Society sponsored conference on econophysics in Dublin. During the meeting he obtained a copy of the book 'An Introduction to Econophysics' by Rosario N Mantegna and H Eugene Stanley. As for many other physicists, that meeting and the book inspired new research directions. This paper is the latest in a series that have emerged from that initial revelation over two decades ago.

In a series of recent papers [1–3] the present authors have studied human mortality demonstrating how the shape of the mortality function has a bathtub type of shape where the infant mortality decreases with age whereas in old age it increases (Figure 1). In medical terminology infancy refers to new born under one year of age. However, in reality the decrease of the death rate continues until the age of 10, For humans, the increase of the death rate is described by the well-known law of Gompertz [4]. This law can be summarized for by saying that the death rate doubles approximately every 10 years of age. Even the mortality of small animals such as rotifers [3] exhibit similar behaviour as is shown within the inset in Figure 1.

It has been suggested in the literature that non-biological systems obey a similar law however evidence of such behaviour in non-biological systems is not easy to find. Very recently Richmond et al. [5] studied the mortality of systems consisting of soap films and confirmed the bathtub nature of such systems. However, the systems were relatively small and towards the end of life, whilst the mortality increased there was no clear evidence of Gompertz behaviour. In this paper we present evidence for company mortality which mirrors the behaviour shown in Figure 1. The mortality of start-up companies decreases according to a hyperbolic law whereas the mortality of mature companies increases and the long-term trend is in accordance with the Gompertz law. This is shown in the next section. In Section 3, we present a simple model with offers and explanation as to why such behaviour can be expected for complex systems. We close with comments and thoughts for further studies.

**Citation:** Richmond, P.; Roehner, B.M. On the Mortality of Companies. *Entropy* **2022**, *24*, 208. https:// doi.org/10.3390/e24020208

Academic Editors: Ryszard Kutner, Christophe Schinckus and H. Eugene Stanley

Received: 21 December 2021 Accepted: 26 January 2022 Published: 28 January 2022

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**Figure 1.** Infant versus old age human mortality. The data are for the US over the period 1999–2016. Between birth and the age of 10 (note the log-log scale) the infant mortality rate falls off as a power law: *μ*(*x*) = A/*x*<sup>γ</sup> where the exponent γ is 0.99 and usually of the order of 1. After the infant phase comes the aging phase (note the linear-log scale) during which the death rate increases exponentially: *μ*(x) = *μ*(0) exp(α*x*) in agreemen<sup>t</sup> with Gompertz's law and for humans α = 0.079. Source: Wonder-CDC data base for detailed mortality data.

#### **2. The Mortality of Companies**
