This short chapter recapitulates the substantive advances made by the individual chapters in this volume before closing remarks on the difference between using probability to represent epistemic uncertainty and modelling variability, two exercises that are easily confused, and on the use of models to answer historical questions.

An intense debate has arisen among scholars concerning the financial sustainability of the grain funds that Greek and Roman cities used to cope with the instabilities of the grain market. In this paper, we apply a Monte Carlo simulation to model their financial dynamics. Due to the uncertainties pertaining to the scope of such funds (targeting urban dwellers only or including rural residents), our model takes into account two scenarios: ‘optimistic’ (urban only) and ‘pessimistic’ (both urban and rural). The analysis reaches several important findings: (1) For both scenarios, we witness a considerable rate of funds collapsing in their first 10 years of operation. After 10 years, however, the probability of failure displays very little change, as if there was a threshold over which the funds had accumulated enough capital to withstand shortages. (2) As expected, the survival rates are significantly higher for the optimistic scenario. (3) The withdrawals seem to have the most dramatic impact on the dynamic of the fund. Overall, while the grain funds do not appear to be sustainable in the urban-rural scenario, they show clear signs of sustainability in the urban-only scenario. The results invite reconsideration of the widespread view that grain funds were an inefficient and precarious response to food crisis.

Although we have made great strides in the last few years in our understanding of the size of the urban population of the Roman Empire, there is still some uncertainty about how to extrapolate from the sample of sites for which we have evidence to the total number of sites that we know existed, with obvious implications for our view of the urbanization rate. In this chapter, I investigate whether we can use probabilistic approaches not only to shed new light on the size of the urban population and urbanization rate (and how they changed over time), but also to assess our degree of confidence about them. This exercise suggests that, although the size of the urban population was reasonably large by historical standards, it grew extremely slowly in comparative terms, with a minimum doubling time of just over 600 years. This indicates a constant urbanization rate, with about a fifth of the inhabitants of the Roman Empire living in cities for most of the Imperial period.

This chapter uses an econometric, stochastic, model to answer questions on Roman Egypt. The questions posed are, how did the number of children impact on the financial position of ordinary families, and is it likely that parents were driven by debt to limit their family size by abandoning new-borns? The model examines the financial stresses and strains faced by typical families in rural Roman Egypt by tracking their net income, savings and debts over a generation to see how many families thrived or fell into financial difficulty. The plausibility of the model and its sensitivity to the key assumptions is assessed. It is concluded that whilst having children could certainly assist a family’s financial well-being over the long term, a large number of small children could act as driver to indebtedness, though the primary driver would still be the quality and variability of the harvest. Turning to the second question of family limitation, the model predicts that, as a minimum, there was a significant and socially visible chance that private and state tenants would have had to abandon a new-born from financial necessity. Some families would have had to resort to it on a number of occasions.

This chapter introduces the concepts and methods used by the other chapters in the volume, using the long-standing problem of estimating the land carrying capacity of classical Attica to illustrate the benefits of probabilistic modelling. We begin by surveying the development of techniques for managing uncertainty in ancient history (1.1) and past work on the specific problem of Attica’s land carrying capacity (1.2). The chapter then turns to theoretical questions about the nature of uncertainty and probability (1.3), introducing the ‘subjectivist’ conception of probability as degree of belief, a theoretical framework that makes probability a powerful tool for historians. We go on to discuss the procedure of using probability distributions to represent uncertainty about the actual value of a quantity such as average barley yield in ancient Attica (among other variables relevant to the problem of land carrying capacity) (1.4), the need to be aware of cognitive biases that distort our probability judgements (1.5), the use of Monte Carlo simulation to combine uncertainties (1.6), the potential problem of epistemic interdependence (1.7), the interpretation of the outputs of a Monte Carlo simulation (1.8), and the use of sensitivity analysis to identify the most important sources of uncertainty in a simulation. The appendix illustrates model code in R.

This chapter presents a new estimate of the value of coinage in circulation in the mid-second century CE Roman empire. More than 25 years ago, Richard Duncan-Jones revolutionized ancient economic history by offering a first projection through numismatic and statistical methods. At more than 20 bn sesterces, his estimate implied an anomalously high monetization ratio given past and current estimates of Roman GDP, an issue that economic historians have had to deal with ever since. In the first half, a review of the numismatic evidence points to a much smaller role for gold than posited by Duncan-Jones. The second half presents a new model of the money supply. It uses Monte Carlo simulation to estimate the value of centrally-minted precious metal coins produced annually under Hadrian, and then the total coinage in circulation ca 160 CE. Allowance for various uncertainties and other, minor components of the coinage suggests a money supply of around 16 bn sesterces, with less gold and more silver than expected. However, this is not quite low enough to explain away the monetization ratio, implying a higher GDP and more trade-oriented economy than currently thought. Two appendices contain lengthy but essential technical discussions of the assumptions in the estimate.

This paper utilizes Monte Carlo simulation to estimate the frequency with which private property was confiscated in the Greek world in the Classical period (circa 480–330 BCE). Confiscation is defined for the purposes of this paper as the uncompensated, sudden loss of landed property. The first step is to establish the demographic context within which confiscation occurred, and then to analyze four mechanisms for the confiscation of property: exile arising from stasis; the expulsion of a population by a foreign power; andrapodismos or mass enslavement; and the imposition of a cleruchy. Other mechanisms by which confiscation occurred, but which cannot be quantified, are briefly discussed. The model shows that, based on the beliefs I hold about the four mechanisms for confiscation that have been analyzed, the average estate owner faced a 2.6–18.6 % chance of experiencing confiscation in his lifetime, with a mean likelihood of 10.5 %. This finding is at odds with the prevailing view that Greek poleis ensured the security of private property, which in turn contributed to the remarkable economic growth of the period. The conclusion suggests some possible responses by the Greeks themselves to the imperfect property security they experienced.

This chapter argues that wealth was probably not the primary barrier for Pompeiians to enter the Roman senate. One oddity about the historical record of Pompeii is that it reveals not a single certain senator in the imperial period. A previous reconstruction of the local distribution of income offers a possible explanation: it predicts that there were no Pompeian households with a senatorial income, suggesting that a lack of wealth kept the Pompeiians outside the senate. However, a new reconstruction of the top part of the Pompeian wealth distribution suggests the opposite. This reconstruction is based on combining the archaeological remains of the intramural housing stock with an econometric model which assumes that the distribution of elite wealth follows a distinct mathematical function – a power law. Even though this type of cliometric modelling is pervaded by uncertainties, with the help of probabilistic calculations it is possible to conclude that at least several Pompeian households held enough wealth to satisfy the senatorial census qualification, implying that wealth may not have been the primary barrier preventing Pompeiians from embarking on a senatorial career.