## FORESEEING THE PAST: PROBABILITY AND ANCIENT GREEK DECISION-MAKING

By Paul V ă dan

'Probability does not exist.' - Bruno de Finetti

Summary: The article explores the concept of probability in ancient Greece from a nonscientific perspective and shows how ancient decision-makers used historical data to make calculated decisions  and  speculate  about  the  future.  First,  the  paper  considers  how quantitative data was used by ancient Greek communities to make economic projections. It then shows how ancient Greek generals used the same conceptual tools to determine their odds of victory by tallying up and comparing the number and composition of armies and resources available to them and their enemy. In the third section, the paper examines how qualitative probability was articulated through the language of hope and likelihood to formulate chances of success in moments of crisis. Finally, the paper shows that ancient decision-makers implemented 'power laws' to adapt to changing circumstances and the flow of new information, as they sought to improve their odds of success relative to their rivals.

## INTRODUCTION

In  his  published  conversations  with  Christopher  Pierson,  sociologist Anthony Giddens contends that human history turned 'modern' when the mathematical discoveries of the 16th century set the foundations of the sciences of statistics and probability. 1 For  the  first  time  in  human

* I am grateful to Alain Bresson and Paul T. Keyser for their time and generous comments at various stages in the paper's development. Caitlin Miller's proofreading and suggestions have improved the clarity and structure of this paper. I would also like to thank the organizers and participants of the Ancient Societies Workshop at the University of Chicago, whose questions and suggestions have helped me strengthen

history, human actors could quantify uncertainty and think about the future through the modern concept of risk. Giddens could thus announce that 'risk has replaced fortuna ,' 2 confident that the 'modern' enlightened world has finally managed to shed the old superstitions of 'traditional' societies by relying on the rational and 'novel' sciences of statistics and probability to quantify, predict, and control an otherwise dangerous future.

Giddens' thesis has been influential in the study of sociology, and has also  impacted  the  way  classicists  approach  ancient  decision-making. Notably, Mary Beard has characterized the Graeco-Roman world as an 'aleatory society,' where the model of gambling luck governed the way the ancients approached danger. For Beard, 'Rome was a culture that looked danger in the eye. It did not attempt to avert or calculate danger, but  rather  to  assert  (almost  celebrate)  the  uncertainties,  chances  and dangers of human existence.' 3 Her  interpretation  takes  the  form  of  a verdict that relegates ancient considerations of danger to the narrow and morally-charged concern with daring, '[facing] danger head-on,' since 'anything like a calculation of the probability of danger, let alone a recognisable risk agenda,' 4 was absent in antiquity.

In fact, recent scholarly attempts to talk about ancient 'risk-taking' have been countered by the same scientific argument pointing to the absence of mathematical probability in antiquity. For instance, Esther Eidinow's (2007) non-quantitative approach to ancient Greek

my argument. Finally,  I  benefited  from  conversations  on  ancient  risk  with  Anna Francesca Bonnell-Freidin, Esther Eidinow, Stephen Kidd, and Brent Shaw.

- 1 Giddens describes the modern world as 'vastly more dynamic than any previous type of social order. It is a society - more technically, a complex of institutions which unlike any preceding culture lives in the future rather than in the past.' Giddens & Pierson 1998: 94. Similar arguments for a conceptual divide between antiquity and modernity have been promoted by Christian Meier (1990), who argued that the modern concept of 'the State' was absent in antiquity. Also, Reinhart Koselleck (2006) has argued that the modern idea of 'crisis' referring to a political and economic event was not found in the ancient notion of κρίσις ; the argument has been disputed by Kuin & Klooster 2020: 3-14.
- 2 Giddens 1990: 30.
- 3 Beard 2011: 98.
- 4 Beard 2011: 91, 98.

perceptions of risk has been critiqued by classically-trained sociologist of religions Kim Beerden (2013), who questions Eidinow's use of the term 'risk'  as  a  modern  imposition  upon  antiquity.  Referencing  Giddens' work, Beerden considers risk as intimately linked to the modern sciences of statistics and probability, whereas 'all [ancient Greek] expressions of thinking about the future differ crucially from modern conceptions of risk: there was no calculation of the chances or probability of disaster or success'; the ancients had to content themselves only with divination. 5

However, this modernist sociological attitude to risk  and  decisionmaking  carries  with  it  an  insidious  claim:  namely,  that  the  modern world's response to crisis is, in some sense, original, where contemporary  issues  have  contemporary  solutions.  It  follows  that ancient  experiences  and  crisis-solving  mechanisms  are  no  longer helpful, being relegated to the categories of superstition and credulity. What is more, this sociological insistence on technological progress as a marker of cognitive ability is less than truly explanatory, if not morally dubious,  and  attracts  simplistic  value  judgments.  Take,  for  instance, sociologist  and  philosopher Niklas Luhmann's (1991) comment on the seeming absence of mathematical probability in antiquity, that Greek ingenuity  had  finally  reached  its  limits,  unable  to  explore  futurity beyond the use of cosmology and a passive acceptance of divine agency.

Likewise, economist Peter Bernstein (1996) deemed the failure of 'the Greeks' to engage with probability theory as 'astonishing,' concluding that 'despite the emphasis that the Greeks placed on theory, they had little interest in applying it to any kind of technology that would have changed their views of the manageability of the future.' Bernstein adds that  only  after  the  mathematical  revolution  sparked  by  Pascal  and Fermat did views about gambling move beyond ancient and outdated conceptions of chance: 'The act of risk-taking floated free, untrammeled by the theory of risk management.' 6 Consequently, ancient societies are denied a fundamental  level of cognitive rationality, institutional complexity,  and  individual  agency  to  anticipate,  assess,  and  mitigate potential dangers.

5 Beerden 2013: 202.

6

Bernstein 1996: 11, 16.

And yet, as I will show in this article, the extant evidence pertaining to  ancient  decision-making  counteracts  these  primitivist  sociological attitudes towards the ancient world. Indeed, if we expand our enquiry beyond the narrow constraints of ancient mathematics, we find different strategies  by  which  ancient  decision-makers  formulated  and  applied probabilistic  thinking  to  quantify  uncertainty  and  inform  collective economic, social, and military decisions. To do so, I specifically focus on literary  evidence  from  the  4th  century  BCE  onwards  when  ancient thinkers started theorizing about probabilistic thinking in a systematic way  by  prescribing  codes  of  behavior  and  systems  of  knowledge  to calculate  the  future.  I  first  show  that  in  the  absence  of  conclusive evidence pertaining to ancient mathematical probability, we nevertheless have instances where ancient decision-makers used abstract numbers to express odds of success about economic and military risks. I then assess the qualitative language of likelihood used by ancient decision-makers to assign gradations of risk to dangerous events. Finally, I  turn  to  the  use  of  the  past  by  military  leaders  to  imagine  historical precedents to present circumstances as a way to generate statistical data and shape collective expectations about the future. In doing so, I bridge the conceptual divide between antiquity and modernity by highlighting the culturally-specific character of the concept of probability.

## 1. ANCIENT PROBABILITY AS A METHODOLOGICAL PROBLEM

The  narrative  that  probability  is  an  inherently  modern  product  of Enlightnement mathematicians has recently been dismissed by statistician Glenn Shafer (2018) as mere  legend. Schafer assigns responsibility  to  the  work  of  Ian  Hacking  (1975  and  1990)  for  further popularizing  the  notion  that  these  polymaths  were  responsible  for combining,  for  the  first  time,  the  philosophical  ideas  of  belief  and frequency. 7 He points to pre-existing evidence collected by Marie-France Bru  and  Bernard  Bru  (2018),  some  from  Arabic  texts,  discussing  dice games  and  contracts  that  express  probabilistic  logic.  An  imporant instance is the famous 13th century CE poem De Vetula , 8 whose author is clearly  aware that some arrangements in dice games do not have the same force or frequency, whose complexity has been deemed by the Brus (2018) a veritable 'calcul de chances.' 9 For Shafer, such texts indicate an understanding of the character of probabiliy as the union between belief (betting) and frequency (outcome). 10

Nevertheless,  Hacking  has  often  defended  his  thesis  from  such criticisms by stating that 'what is important is not the occurrence of a few probability ideas in antique texts but a use for them, a use that spans morals, politics, economics and social affairs, and which engenders a new era of conjecturing on the one hand and a new mode of representing reality on the other.' 11 This statement, however, is unfair to both ancient and modern thinkers alike because the act of choosing a 'birth moment' for a concept is a misleading exercise. 12 For the sake of argument, one could  just  as  easily  claim  that  the  real  revolution  in  statistics  and probability theory came not in the 17th century, but much later in 1933, when Andrei Kolmogorov laid the axiomatic foundations of probability theory by publishing his Grundbegriffe der Wahrscheinlichkeitsrechnung . 13 Kolmogorov's achievement eventually allowed the application of probability theory to solve economic problems, but only after World War II, when it was gradually employed in the modern financial system. That, however, would deny Cardano, Pascal, Fermat and all of Kolmogorov's predecessors the cognitive capacity to think axiomatically about probability theory, which would be both unfair and misleading.

And  yet,  while  medievalists  have  been  quick  to  take  up  Hacking's challenge  by  highlighting  the  complex  probabilistic  character  of  so-

- 8 For a discussion on calculating permutations in ps.-Ovid's De Vetula, see Kidd 2020: 19-20.
- 9 Bur & Bru 2018: 306.
- 10  Shafer 2018: 280.
- 11  Hacking 1975: 108.
- 12  Similar arguments have been made against the presumed modern origin of concepts like 'crisis' (Kuin & Klooster 2020), 'intuition' (Struck 2016), 'landscape' (Zientek 2014) and 'risk' (V ă dan 2018).
- 13  Shafer & Vovk 2001: 39: 'Among mathematicians, its simplicity, clarity, and power made it the easy victor in the spirited debate on the foundations of probability that took place in the 1930s.'

called 'aleatory contracts,' 14 classicists have instead limited themselves to justifying the ostensibly rudimentary character of ancient theoretical mathematics.  In  a  recent  article  on  ancient  gambling,  Stephen  Kidd (2020) has sought to account for the apparent absence of mathematical probability in antiquity by looking at the character of games that ancient gamblers  played:  whereas  modern  gamblers  play  games  that  require them to take individual risky bets based on personal calculations, ancient gamblers, by contrast, played games with previously agreed-upon group wagers,  which  rendered  risk  a  communal  affair.  The  result,  Kidd explains,  is  that  'the  incentives  to  calculate  such  probable  outcomes were  not  at  all  glaring,  since  there  was  simply  no  gambling  game  to which such calculations would have been applicable.' 15 According to this argument, we would have to wait until the 16th century when gamblers finally had the incentive to calculate their individual gambling risks for profit,  which  would  eventually  lead  them  to  ponder  the  theory  of probability: 'with new games to play, people began to think in a new way. That new form of thinking gave rise to mathematical probability and the related field of statistics.' 16

Kidd's  analysis  of  ancient  games  is  impressive  and  highlights  the importance of incentives to finding new solutions to old problems. He is also  right  to  point  out  the  cumulative,  rather  than  individualistic, character  of  technological  progress.  But  as  is  often  the  case,  the presumed absence of a certain kind of technology does not necessarily

- 14  Hald 2003: 32: 'The basis of such contracts became the specification of conditions for the equity of the parties involved, which required assessment of risks combined with the possible gains and losses.' For the theological and legal aspects of risk-taking in the development of the concept of expectation in probability theory, see also Coumet (1970), Daston (1980), and Schneider (1980).
- 15  Kidd 2020: 3, 5. It is worth noting that while Kidd (n. 16) acknowledges that ancients tried to get an advantage in dicing through cheating, he does not connect this phenomenon  with  the  possibility  of  probabilistic  thinking.  However,  Jerzy  Neyman (1976: 152) has interpreted tampered dice as an awareness by the cheater of the important phenomenon of long-run frequency. He mentions loaded dice found in Egyptian Pharaonic burial chambers, suggesting that such an understanding of dice is as old as dicing itself.
- 16  Kidd 2020: 22.

entail the absence of ideas about it. 17 In fact, scholars have recently made the case that even modern technological industrial discoveries generally rely  on  non-scientific  rather  than  scientific  processes. 18 Likewise,  the absence  of  evidence  poses  a  methodological  challenge  to  historians because  it  does  not  automatically  discard  the  possibility  that  such evidence  did  -  or  does  -  exist.  Kidd's  argument  thus  needs  to  be considered  with  caution  because  it  relies  in  part  on  the  (supposed) silence  of  the  evidence.  To  this  point,  Shafer  is  confident  that  there remains the real possibility of discovering ancient manuscripts detailing probabilistic thinking, especially in the oft-ignored Arabic manuscripts. Indeed,  Reviel  Netz  (2016)  has  estimated  that  'we  have  attested something like 20% of ancient mathematical authors, and have extant something  like  3%-5%  of  ancient  mathematical  texts.' 19 There  thus remains  the  real  possibility  that  some  of  them  may  have  explored mathematical  probability,  as  hinted  at  by  a  variety  of  philosophical works that touch on the subject, if only in a rudimentary way. 20 In fact, it has  been  argued  that  the  rise  and  rule  of  Rome  negatively  impacted scientific innovation, with the number of mathematicians and scientists regularly decreasing during the Roman empire, until finally becoming

- 17  One notable example is the development of the abstract principles of thermodynamics by Nicolas Carnot in the 1820s, one century after the implantation of Newcomen's steam  engine  (Mokyr  2009:  124-44).  From  a  different  perspective  on  the  ancient Greek world, John K. Davies (2003) approached Athenian democracy through systems analysis to explain its development in the 5th century BCE in the absence of a general Athenian political theory. Likewise, Josiah Ober (2008) showed how Athenian institutions allowed for the spread of knowledge needed by novice office holders to govern the state through 'demotic clusters' of administrative memory, despite the absence of complex information networks.
- 18  Bresson 2014: 67. See also Clark 2012 for the 'idealist' model, and Allen 2009.
- 19  Netz 2016: 85. Bru & Bru 2018: 302 also agree that we may yet unearth ancient Babylonian tablets or Egyptian papyri that explore the concept of probability.
- 20  Keyser & Scarborough 2018; Keyser & Irby-Massie 2002. For instance, we know of Xenokrates of Chalkedon's now-lost work on combinatorics, entitled On Numbers. In contrast, most claims on ancient statistics focus almost exclusively on the rudimentary observations of Cicero and Aristotle on dice and numbers, without considering the historical implications that these writers were not known as mathematicians, whose observations on the topic were perhaps influenced by other works. Cic. Div. 1.23, 2.48, 2.121; Arist. Cael. 2.12 (292a30), PN 463b19-23, 3.4.1407b1.

negligible  in  the  fifth  century  CE, 21 leading  Alain  Bresson  (2014)  to describe the history of Greek science as an interrupted process. 22 We can thus  imagine  a  scenario  where  diminishing  interest  in  theoretical science would have discouraged further innovation, which then compounded the problem of manuscript preservation, some of whom still surviving in Arabic texts that have yet been discovered, read, or even translated.

Even  so,  I  suggest  that  we  can  bypass  the  problem  of  missing mathematical evidence by looking at instances of probabilistic thinking beyond mathematics. To do so, we need to expand our understanding of probability beyond the notion of a closed system governed by symmetry and abstract logic where numerical odds can be objectively calculated. This  so-called  'classical  symmetry'  model  implies  that  probability theory could have only developed in a very specific historical context like gambling, whereas statistician David Spiegelhalter (2011) explains that  'classical  symmetry'  is  but  one  way  to  think  about  assigning probabilities  to  events.  When  it  comes  to  real-world  circumstances, Spiegelhalter points out that another means to quantify uncertainty is to use  historical  data,  the  so-called  'frequentist'  method:  'If  the  future follows  the  same  pattern  as  the  past,  then  frequencies  of  events  in history should reflect reasonable probabilities for events in the future,' thus rendering potential responses and outcomes to present circumstance rather predictable. 23

Mathematician and philosopher James Franklin (2001) concurs that probability in the modern form did not develop earlier in part for the simple reason that dice and other 'classical symmetry' tools are not a reliable  model  to  tackle  real-world  situations. 24 Richard  Thaler  (2015) illustrates this problem succintly by distinguishing between 'Econs' and 'Humans,' where Econs are fully 'rational' optimizers when it comes to economic theory. By contrast, Humans 'misbehave' according to beliefs, instincts, and patterns of thought that are decidedly non-optimal. And since we do not live in a world of Econs but in a world of Humans, Thaler asserts that culturally-specific Human behaviors and experiences need to  be  considered when building theoretical models. For our purposes, Thaler's approach is important because it also implies that one does not necessarily need to be versed in economic theory to behave economically by counting on historical data and personal experience, if not  always  precisely. 25 Accordingly,  I  will  show  that  the  nature  and contents of our sources speak to a Hellenistic interest in using historical data in social, political, and economic contexts other than gambling, to make calculated decisions and speculate about the future. As such, in the presumed absence of an ancient theory of probability, we can interpret the ancient evidence through a frequentist approach to identify clear instances of the philosophical concept of probability.

21  Keyser 2010.

22  Bresson 2014: 68-69.

23  Spiegelhalter 2011: 19.

24  Franklin 2001: 334.

## 2. QUANTITATIVE PROBABILITY AND THE ANCIENT ECONOMY

Ancient  economic  practices  offer  several  illuminating  instances  of quantitative probability based on experience and historical data. At a fundamental  level,  agricultural  production  relied  on  risk-mitigating strategies meant to offset periods of wide climatic variation, with rainfall alternating  sharply  between  wet  and  dry  phases,  which  would  have otherwise made it difficult to estimate yields and plan for the future. 26 As ancient  economists  have  already  pointed  out,  diversification  of  crops and polyculture, the building of waterworks, together with sharecropping contracts, were some of the ways in which landowners sought to control the uncertainty of an irregular climate. 27 These efforts

25  Thaler 2015: 2-12.

- 26  Sallares 1991: 393-95, building on the work of Peter Garnsey (1988), who has shown that despite regular crop failure, poleis generally did not experience famine due to various social and economic measures implemented. These included setting up reserves, price moderation, and patronage.
- 27  Thomas Gallant (1991) provides a general overview of the resource strategies that Greek households would implement to deal with shortfalls in production. He illustrates how an agricultural system could adapt to the pressures of land life cycles

were aimed at absorbing the potential risk of resource scarcity in the chora , which in turn helped a community make better predictions about future agricultural yields.

Following the same logic, ancient communities implemented various financial  schemes  to  regulate  public  funds  in  an  economic  crisis.  A somewhat  morbid  case  study  is  offered  by  a  Milesian  inscription recording  a  public  decision  to  create  an  annuity  fund  sometime  in 211/210  BCE  to  incentivize  wealthy  individuals  to  facilitate  public investment:

The Milesians have voted that those male or female citizens who wish to give 3,600 drachmas on behalf of themselves or on behalf of others […]. In return for the money given to the city, each of the donors shall receive thirty drachmas per month from the city, for as long as they live. This money  shall  be  given  each  year  by  the  treasurers, withdrawing  and  distributing  the  money,  in  the  same  way  as  is prescribed  in  the  laws  for  the  priests  and  those  who  have  won contests in games with a prize of crowns. 28

The  initiative  attracted  no  less  than  thirty-nine  contributions  from thirty-four  individuals,  who  could  recuperate  their  money  within  ten years. 29 There  was,  however,  a  catch:  'If  any  of  those  who  gave  the proposed amount to the city depart from life, the people shall be released from repaying the donation and the reserved annuity, but one hundred

through strategies of crop diversification, intercropping, irrigation, and fragmentation of land holdings. On more detailed examples from the Roman world, see the more recent work of Bruce Frier (2007) and Dennis Kehoe (2007).

- 28  I.  Milet.  I.3  147,  ll.  7-9,  18-22 ἐψηφίσ̣θαι Μιλησίοις· | τοὺς μὲν βουλομένους τῶμ πολιτῶν ἢ πολιτίδων δοῦνα [ ι ] | τῆι πόλει δραχμὰς τρισχιλίας ἑξακοσίας ὑπὲρ αὑτῶν ἢ ὑπὲρ ἄλλων |  […] ἀντὶ δὲ τοῦ δοθέντος τῶι | δήμωι λαμβάνειν παρὰ τῆς πόλεως δραχμὰς τριάκοντα κατὰ μῆν [ α ]  | τῶν δόντων ἕκαστον , ἕως ἂν ζῆι . δίδοσθαι δὲ τοῦτο καθ ' ἕκαστον ἔτος̣ | ὑπὸ τῶν ταμιῶν , γινομένης τῆς ἐξαιρέσεως καὶ δόσεως τοῦ ἀργυ | ρίου , καθότι καὶ τοῖς ἱερεῦσι καὶ νενικηκόσι τοὺς στεφανίτας ἀγῶνας | ἐν τοῖς νόμοις συντέτακται . Trans. Sosin. According to the decree, it had not been possible for the city to collect an eisphora due to lack of funds and revenues; contra Franklin 2001: 259.
- 29  Recorded at Milet I.3 147, ll. 87-104.

and fifty drachmas shall be given to the relatives of each of them for their burial.' 30 Joshua  Sosin  (2014)  interprets  this  'death  clause'  in  the annuity contract as an attempt at financial speculation: if the state bank were to invest the collected money at a common rate of 12% per year, 'the fund would have yielded Miletos a meager 2,808 drachmas annually, until the beneficiaries started to die out; every death tipped the scale in the state's favor.' 31 The demos ,  therefore, made a long-term bet whose value  was  directly  correlated  to  the  probability  that  older  wealthier individuals would die before they would collect all the money they had made available for the public.

And while the study of ancient demography and mortality rates is beyond the scope of this article, it is worth noting that we have evidence from the Roman Empire of early attempts to calculate annuities based on life expectancy. The so-called 'Ulpian's Life Table' has been interpreted to represent the calculation of annuity premiums with an interest rate of about 1.5% based on age, which has helped scholars approximate a life expectancy of 40 years for someone aged 20. 32 These numbers have been disputed, but scholars agree that we are looking at a crude annuity table. Returning  to  the  Miletos  decree,  it  is  clear  from  its  clauses  that  rich Milesians were themselves aware of the unavoidable mortality problem. They took advantage of a special representation clause, which perhaps they  themselves  maneuvered  to  have  included  in  the  decree,  which stipulated that:

if  anyone registers the name of another male or female citizen, he shall be given the resulting annuity for as long as those registered are

- 30  I. Milet. I.3 147, ll. 48-51 ἐὰν δέ τινες τῶν δόντων τῆι πόλει τὸ ἐκκείμενον πλῆθος ἐγλ̣ [ ί ]| π̣ωσι τὸμ βίον , τοῦ μὲν δοθέντος καὶ τοῦ ἐξαιρουμένου σιτηρεσίου | ἀπολελύσθαι τὸν δῆμον , δίδοσθαι δὲ εἰς ταφὴν τοῖς προσήκουσιν ὑ | πὲρ ἑκάστου δραχμὰς ἑκατὸν πεντήκοντα . Trans. Sosin.
- 31  Sosin 2014: 80.
- 32  Pflaumer  2015:  2677-78;  though  his  numbers  are  slightly  different  from  DuncanJones (1990: 94, 100-1), who suggests a life expectancy of 32 years from someone aged 25, and that the beneficiaries of the life-annuities were slaves or ex-slaves, and not Roman elites. See also Frier 1982 and 2018 for a close analysis of the Ulpian Life Table and the projected life expectancy for both Roman men and women. See also Cicero's observations on different mortality rates between youths and adults (Cic. Sen. 19).

alive.  If  the  one  who  registered  the  name  dies  first,  then  the  one whose name was registered shall receive the reserved annuity for the remainder of the time. 33

The clause was a way for rich families to bypass the 'death clause' and recoup their investment within ten years and, furthermore, to continue to make a profit beyond that point. Sosin's analysis of the names of the benefactors and their beneficiaries highlights the point that 'Milesians were not demographers, but they could do the math,' explaining that 'of all of the donations, roughly two thirds were made on behalf of a younger beneficiary or else by a young beneficiary on his or her own behalf.' 34 The inscription, therefore, is a classic example of the rich getting richer at the expense of the state during times of general financial hardship. But the greater point is that both the state and its wealthy families used their  understanding  of  life  expectancy  to  make  more  predictable financial speculations. 35

Financial incentives to quantify uncertainty also defined how ancient trade was conducted. Alain Bresson (2004 and 2016) has analyzed ancient insurance  practices  to  show  how  investors  quantified  danger.  He concludes  that  interest  rates  were  directly  correlated  to  the  risk  of

- 33  I.  Milet  I.3  147,  ll.  72-75 ἐὰν δέ τις ἕτερον ἀπογράψῃ ὄνομα τῶμ πολιτῶν ἢ πολιτ [ ί ]| δων , δίδοσθαι αὐτῶι τὸ γινόμενον σιτηρέσιον ζώντων τῶν ἀπογεγραμ | μένων . ἐὰν δὲ προεγλίπῃ ὁ ἀπογράψας , λαμβανέτω τῶν ἐφεξῆς | χρόνων τὸ ἐξαιρούμενον ὁ ἀπογραφείς . Trans. Sosin.
- 34  Of  the  39  donations,  22  were  made  on  behalf  of  others,  most  probably  sons  and daughters, and of the 17 who contributed in their own names, two were females under the kyrieia of men not said to be their husbands, and so perhaps orphaned minors, and two were male minors. Sosin 2014: 81.
- 35  Other epigraphic examples of financial speculation include Austin 115, where the Olbians honor their benefactor Protogenes for, among other things, helping them purchase  grain  at  a  decent  price,  after  correctly  speculating  that  the  price  of  a medimnos would increase exponentially: 'Again in the priesthood of Plistarchus, when there was a severe shortage of corn and / grain was being sold at a medimnos and 60 two thirds for a gold coin, and it was clear that the price would rise further, and in fact the medimnos immediately reached the price of one gold coin and two thirds' (ll. 58-64). For an analysis of the financial crisis at Olbia, see Müller 2011. See also, Austin 118 where we get a glimpse into the public budget of Halikarnassos that includes a debt repayment plan and future funds to be earmarked for public works.

shipwreck, which was known  to lenders from historical data. 36 Estimations were precise enough to not only evaluate total damages but also to distinguish the number of shipwrecks in connection to the time of the year: while at the beginning of the sailing season one could expect one ship in five to sink, at the end of the season the chances could be as high as one in three. 37 Literary evidence corroborates these finds, as we learn from the description of a maritime loan contract in Demosthenes' Against Lakritos that interest rates regularly changed in accordance with the  time  of  the  year:  during  the  high  sailing  season,  interest  was estimated at 22%, while after the rising of Arktouros the rate could go as high  as  30%. 38 Demosthenes thus provides  us  with  a  glimpse  into  the intricate  ancient  practice  of  putting  numbers  on  uncertainty  that determined  the  future  behavior  of  traders  and  investors.  Indeed,  as Edward  Cohen  has  shown,  'maritime  yields'  were  determined  by contractual agreements that took into account the degree of risk and anticipated profitability of a trading venture. 39 Contracts thus anticipated a variety of circumstances and contingencies pertaining to the  itinerary  and  the  inter-personal  trust  involved  in  the  trading venture. 40 Given  that  the  entire  maritime  commercial  infrastructure relied  on  credit,  creditors  made  profits  from  transactions  where  high

- 36  On the economic and insurance information that can be teased out from shipwrecks,
- see Gibbins 2001; Bresson 2016: 89-90, 283-84.
- 37  The economic system was based on acquired experience and shared knowledge of everyone involved in maritime trading, making it possible to stimulate trade while also diminishing the inevitable risks of seafaring for everyone involved; a business
- practice now known as 'risk pooling' (Bresson 2016: 280-83).
- 38  Bresson and Bresson 2004: 8-9. By also looking at grain trade prices, the Bressons further explain how the leverage investment system made borrowing preferable for the trader because he did not have to put his whole fortune at stake. We may also note the treatise De Contractibus by the Franciscan monk Olivi in the 13th century, that assigns numerical values to the perceived risk of maritime insurance contracts. For  Marie-France  and  Bernard  Bru,  Olivi's  calculations  of  gains  and  risk  are comparable to those of the founders of the insurance science in the 20th century. Bru & Bru 2018: 320.
- 39  Cohen 1992: 53-55.
- 40  Using  New  Institutional  Economics,  Vincent  Gabrielsen  shows  how  the  ancient Greek  state  promoted  systems  of  trust  and  information  sharing  that  resulted  in lower transaction costs (Gabrielsen 2016: 87).

risks yielded high rewards, which in turn allowed them to absorb the risk of an individual disaster like the sinking of a ship. 41

These trading ventures thus speak to the complex interplay between collective and individual risk. Like in a game of dice, the risk was indeed common to all investors but each investor still had to decide whether the venture was personally worthwhile in the first place. See, for instance, the investment plan recorded in one of the papyri in the Zenon Archive 42 dating from ca. 256-248 BCE, where a certain lender offers Zenon three investment  propositions  for  the  exploitation  of  a  trade  ship  for  the period  of  a  year,  where  each  option  contained  different  financial obligations with respect to crew and taxes, as well as distinct opportunities for profit. Whereas it was in the interest of all parties that the ship be utilized to make a profit, it was left to Zenon to decide his preferred course of action and the financial risk he was willing to expose himself to use the ship. 43 Such instances further explain  why Demosthenes accuses Lakritos of 'not sharing in the danger because you put nothing on board [the ship],' as per the clause stipulating that any kind of payment is only made 'upon the ship arriving safely.' 44 Lakritos made a personal calculation and decided that he was not prepared to invest,  but  still  tried  to  illicitly  make  a  profit  without  taking  on  the collective financial risk of the venture. Therefore, ancient traders and investors not only had a personal incentive to quantify danger, but did so using historical data to determine the risk of an investment over time.

## 3. QUANTIFIABLE DEGREES OF DANGER

Demosthenes' use of the expression 'sharing in the danger' is notable for its prevalence in political and military discourse. It pertains to the deliberative process of ancient decision-makers, 45 and in some cases is linked specifically to ancient efforts at quantifying danger. Polybios, for instance, describes the Roman practice of 'decimation,' where members of a cohort accused of cowardice were severely punished with public humiliation and even fatal beatings. Polybios considers the practice a good deterrent against cowardice because 'the danger and dread of drawing the lot hang over all equally, as the outcome is uncertain; and as the public disgrace of receiving barley rations falls on all alike.' 46 The passage expresses what is commonly referred to as 'relative perception of risk,' where people are willing to accept a probability of harm up to a certain threshold, beyond which the risk is considered unacceptable - in this case 10%, or one in ten. 47 Such use of numerical 'odds' is just one of the ways  in  which  probabilities  were  expressed,  and  we  have  examples where one's risk threshold could be swayed towards taking more risks based on the increased amount of coined money promised to them. 48 While  we  do  not  always  have  detailed  pay  information,  we  can  still glimpse into how individuals quantified danger in terms of money as they weighed the potential rewards against the risk of participating in collective action.

41  Oliver 2007: 40-41.

42  P. Cairo Zenon IV 59649.

43  For a detailed financial analysis of the three investment propositions, see Gachet 1990.

44  Dem. 34.33 ἡ συγγραφὴ σωθείσης τῆς νεὼς αὐτὸν ἀποδοῦναι κελεύει τὰ χρήματα […] οὐ γὰρ μετέσχηκας τοῦ κινδύνου διὰ τὸ μηδὲν ἐνθέσθαι .

Nor was Polybios describing 'decimation' as a uniquely Roman way of approaching danger pace Beard - as we find other instances in ancient

- 45  For other telling examples of personal calculations for getting involved in collective risky  initiatives,  see  Isoc.  4.97.7  on  the  Spartans'  decision  to  join  the  Athenians against the Persians. Also, Xen. Cyr .  5.5.20 where one of the Persian King's men is excused from sharing in the danger because he did not think it was personally safe to pursue the enemy. Conversely, the Macedonian general Parmenion was willing to gamble his life  over  his  plan  to  engage  the  Persians  by  sea,  saying  that  'he  was willing even to embark himself and share in the danger,' only to be dismissed by Alexander as flawed in his judgement. Arr. Anab. 1.18.6-7.
- 46  Polyb. 6.38.3-4 λοιπὸν τοῦ μὲν κινδύνου καὶ φόβου τοῦ κατὰ τὸν κλῆρον ἐπ᾿ ἴσον ἐπικρεμαμένου πᾶσιν , ὡς ἂν ἀδήλου τοῦ συμπτώματος ὑπάρχοντος .
- 47  For more on 'relative perception of risk,' see V ă dan 2018: 42 n. 59.
- 48  Alexander the Great, for instance, was able to convince his tired Macedonians to continue campaigning eastward by promising them to 'make them the objects of envy to those at home, and stir up the rest of the Macedonians to readiness for sharing  the  same  dangers and hardships.' Arr. 7.8.1 ἐπιδώσει δὲ < τοῖς > μένουσιν ὅσα αὐτούς τε ζηλωτοτέρους ποιήσει τοῖς οἴκοι καὶ τοὺς ἄλλους Μακεδόνας ἐξορμήσει ἐς τὸ ἐθέλειν τῶν αὐτῶν κινδύνων τε καὶ πόνων μετέχειν .

Greek literature of conveying degrees of danger in terms of numerical 'odds.' Diodorus Siculus tells us about the panicked call by Pancylus Paucus to his fellow Capuans to surrender to Hannibal during the Second Punic War:

He was driven out of his mind for fear of Hannibal, and he swore to his fellow citizens a peculiar oath: 'If', he said, 'there were still one chance in a hundred for the Romans, he would not go over to the Carthaginians; but since the superiority of the enemy was clear and danger was at their gates, it was necessary to yield to superiority.' 49

Pancylus' words are strongly rhetorical, highlighting fear as a driving force in shaping a group's risk calculations. But beyond the trope of emotions overcoming reason, the passage also suggests that the audience and,  by  extension,  Diodorus'  readers  -  would  have  understood  the strength of his message because they understood its probabilistic logic. Pancylus' calculation may not have been necessarily accurate but is nevertheless expressive of the cognitive ability to assign an abstract fraction to an outcome. Similarly, Xenophon in the Anabasis also uses fractions to underline the danger that his fellow Greeks were in during their journey back to Greece. He reports that the envoy of the Persian King snidely tells them that 'if you have one chance in ten thousand to save yourselves by continuing to fight against the King, I advise you not to give up your arms.' 50 Again, the odds given by the envoy were clearly rhetorical and were simply meant to suggest that in fact the Greeks had little chance of escape. Even so, for Spiegelhalter such basic expressions of numerical 'odds' are sufficient to identify one's cognitive ability to understand and

- 49  Diod. Sic. 26.10 ὁ δὲ ἐκτὸς τῶν φρενῶν γεγονὼς διὰ τὸν Ἀννίβου φόβον ὤμοσε τοῖς πολίταις ἰδιότροπον ὅρκον . ἔφησε γάρ , εἰ τῶν ἑκατὸν ἐλπίδα μίαν εἶχεν ἐν τοῖς Ῥωμαίοις , οὐκ ἂν μετέστη πρὸς Καρχηδονίους· νῦν δὲ φανερᾶς οὔσης τῆς τῶν πολεμίων ὑπεροχῆς καὶ τοῦ κινδύνου ταῖς πύλαις ἐφεστῶτος , ἀναγκαῖον εἶναι ταῖς ὑπεροχαῖς εἴκειν .
- 50  Xen. Anab. 2.1.18-19 Φαλῖνος δὲ ὑποστρέψας παρὰ τὴν δόξαν αὐτοῦ εἶπεν· ἐγώ , εἰ μὲν τῶν μυρίων ἐλπίδων μία τις ὑμῖν ἐστι σωθῆναι πολεμοῦντας βασιλεῖ , συμβουλεύω μὴ παραδιδόναι τὰ ὅπλα · εἰ δέ τοι μηδεμία σωτηρίας ἐστὶν ἐλπὶς ἄκοντος βασιλέως , συμβουλεύω σῴζεσθαι ὑμῖν ὅπῃ δυνατόν .

represent probabilities without the need of complex mathematical models. 51

Indeed, the probabilistic thinking expressed in these examples is underlined by the use of the term ἐλπίς to convey chances of success. Douglas Cairns (2016) reminds us that the word does not only mean 'hope,' as we are sometimes wont to translate it, but can also mean 'expectation' in relation to rational deliberation and endurance. 52 The term itself, therefore, signals the futurity inherent in probabilistic thinking, as protagonists formulate expectations by resorting to observation and deliberation to determine what actions are likely to have higher odds of success. It is in fact telling that early modern mathematicians also resorted to ἐλπίς , so to speak, to explain observable probabilities. Indeed, in the wake of Blaise Pascal's publication of his Usage du Triangle Arithmétique , French mathematicians began using the phrase 'espérance mathématique' to refer to quantifiable probabilities; literally 'mathematical hope.' That is not to say that the French word 'éspoir' and the Greek ἐλπίς are causally linked, but that the probabilistic concept behind their usage is fundamentally the same, despite different technologies.

I  would thus argue that the language of expectation is more useful than the metaphor of dice when accounting for probability in real sociopolitical circumstances. Crises and conflicts do not take place in a controlled environment but in a world of changing circumstances, and the language  of  expectation  shows  us  how  ancient  decision-makers  were able to communicate probabilistic variations. For instance, in one of the myriads of local conflicts that make the history of Hellenistic Anatolia a mire of confusion, the people of Pednelissos were being besieged by their neighbors the Selgians, during the summer of 218 BCE and were about to surrender. But after receiving positive news that the Seleukid general Achaios  would  send  the  help  that  they  had  earlier  asked  for,  'The Pednelissans undertook the siege boldly, relying on their hopes ( ἐλπίσι ) of salvation, and Achaios, appointing Garsyeris to command the expedition, dispatched him with six thousand foot and five hundred horse to the Pednelissans' assistance.' 53 The timely promise of assistance thus led the defenders to alter their risk calculations, feeling confident that their chances of success were increased, which in turn induced them to persevere under siege. In the absence of game theoretical scenarios and equations, the Pednelissans speak to the cognitive ability of Hellenistic communities to articulate probabilities in culturally specific terms. The difference between ancient and modern probability, therefore, appears as one of form rather than substance.

51  Spiegelhalter 2011: 21-22.

52  Cairns 2016: 43-44.

## 4. QUANTIFYING WAR AND PEACE

As our previous examples show, 'odds' of success were often correlated to concrete numbers. Since war was the most dangerous game to play, ancient  military  commanders  were  understandably  concerned  with determining their 'odds' of victory. They did so in part by tallying up and comparing the number and composition of armies and resources. 54 Indeed, ancient historians offer many examples of commanders deciding on a course of action based on their (in)sufficient forces compared to

- 53  Polyb.  5.72.1-3 κατὰ δὲ τὴν αὐτὴν θερείαν Πεδνηλισσεῖς , πολιορκούμενοι καὶ κινδυνεύοντες ὑπὸ Σελγέων , διεπέμψαντο περὶ βοηθείας πρὸς Ἀχαιόν . τοῦ δ᾿ ἀσμένως ὑπακούσαντος , οὗτοι μὲν εὐθαρσῶς ὑπέμενον τὴν πολιορκίαν , προσανέχοντες ταῖς ἐλπίσι τῆς βοηθείας , ὁ δ᾿ Ἀχαιός , προχειρισάμενος Γαρσύηριν μετὰ πεζῶν ἑξακισχιλίων , ἱππέων δὲ πεντακοσίων , ἐξαπέστειλε σπουδῇ παραβοηθήσοντα τοῖς Πεδνηλισσεῦσιν .
- 54  For instance, the Punic Wars are described by Polybios as an arms race, where both sides initially thought that the contest was even, which in turn spurred each of them to acquire more ships and manpower. Polyb. 1.25.5. We are also told that, desperate to increase their odds of victory, the Romans 'were so alarmed and anxious as to the future that they decided to bring into action not four legions but eight.' See Polyb. 3.107.9 προέθεντο δὲ στρατοπέδοις ὀκτὼ διακινδυνεύειν , ὃ πρότερον οὐδέποτ᾿ ἐγεγόνει παρὰ Ῥωμαίοις , ἑκάστου τῶν στρατοπέδων ἔχοντος ἄνδρας εἰς πεντακισχιλίους χωρὶς τῶν συμμάχων . On the numbers at the battle of Cannae, see also Polyb. 3.117. See also Polyb. 1.53.10 on the Carthaginians who considered themselves not strong enough to engage the Romans on account of their inferior numbers.

those of the enemy. 55 This has led me (V ă dan 2018) to argue that Hellenistic decision-makers understood the concept of risk as a deliberative expertise, a τέχνη expressed through verbs like κινδυνεύω or κρίνω , based on one's experience, knowledge, and sagacity. 56 In turn, Roel Konijnendijk (2020) has recently shown that 'Classical Greeks would not have accepted the gamble of battle in the open without careful deliberation.' 57 This  attitude,  in  turn,  explains  why  Xenophon  and  Aristotle

- 55  Xen. Anab . 3.4.14, 7.1.20; Polyb. 1.53.10, 1.25.5, Fr . 6 (Suda α 1312). In this light, the wars of the Athenians and Macedonians against the Persian Empire are the exceptions that strengthen the rule, so to speak, where local communities and potential allies 'had little respect for the small numbers of the [former] but were much impressed with the great size of the [latter], abandoned Alexander and came over to Dareios. They brought the Persians food and other materials with great goodwill, and based on their own decision they foretold the victory of the barbarians.' Diod. Sic. 17.32.4 οἱ δ᾿ ἐγχώριοι τῆς μὲν τῶν Μακεδόνων ὀλιγότητος καταφρονήσαντες , τὸ δὲ πλῆθος τῆς τῶν Περσῶν στρατιᾶς καταπεπληγμένοι καταλιπόντες τὸν Ἀλέξανδρον προσέθεντο τῷ Δαρείῳ καὶ τάς τε τροφὰς καὶ τὴν ἄλλην παρασκευὴν μετὰ πολλῆς προθυμίας ἐχορήγουν τοῖς Πέρσαις καὶ διὰ τῆς ἰδίας κρίσεως προεσήμαινον τοῖς βαρβάροις τὴν νίκην . On the size of the Persian Army, see also Hdt. 7.184-87 and later during the campaign of Alexander Arr. Anab. 3.8.3-6. At the same time, others sought to make their army seem larger so as to deter an enemy attack or to psychologically overwhelm the opponent to surrender, as in the case of the siege of Rhodes where Demetrios the Besieger made sure that 'the whole space between the island and the opposite shore was seen to be filled with his vessels, which brought great fear and panic to those who were watching from the city.' Diod Sic.  20.83.1 ὥστε πάντα τὸν ἀνὰ μέσον τόπον τῆς τε νήσου καὶ τῆς ἀντικειμένης παραλίας συμπεπληρωμένον φαίνεσθαι τοῖς πλοίοις καὶ πολὺν φόβον καὶ κατάπληξιν παρέχεσθαι τοῖς ἀπὸ τῆς πόλεως θεωροῦσιν . For this strategy, see also Xen. 3.4.13 and Plut. Eum. 15. We can think of such prognostications ( προσημασίαι ) as the 'odds' calculated with concrete numbers that in turn informed the decisions of the many smaller factions caught between the two main antagonists.
- 56  Polybios believed that one's deliberative expertise could lead one to make seemingly 'correct' decisions during a crisis; in other words, the κίνδυνος could be calculated and handled in any situation. V ă dan 2018: 27-40.
- 57  Battle was considered a risk that was not always worth taking if necessity did not demand it, while 'senseless' leaders were censured for 'playing dice with the whole city at stake.' Xen. Hell. Oxy . 1.2; Diod. Sic. 13.65.2. By contrast, someone like Phrynichos was praised by Thucydides for not 'running a risk senselessly,' but calculated carefully,  weighing  the  potential  rewards  versus  dangers  8.27.2-3  (Konijnendijk 2020: 183-84).

consider it imperative for a leader to know the resources and expenses of the state, its diplomatic standing, and the state's military capability to make  correct  estimations.  In  his Memorabilia, for  instance,  Xenophon offers  an  enlightening  conversation  between  Sokrates  and  a  young, ignorant Glaukon:

- S. In order to advise about whom to fight, it is necessary to know the city's strength and the enemy's so that if the city is stronger one may recommend going to war, but if weaker, being cautious.
- G. You are right.
- S. First then, tell us the naval and military strength of our city, and then that of her enemies.
- G. No, of course I can't tell you it out of my head.
- S. Well, if you have made notes, fetch them, for I would greatly like to hear this.
- G. But, I tell you, I haven't made any notes either.
- S. Then  we  will  postpone  offering  advice  about  war  too  for  the present. 58

The  dialogue  highlights  the  reliance  on  the  numbers  of  troops  and resources  to  quantify  success  in  a  possible  conflict  and  shape  foreign policy accordingly. 59 The detail that such information would have also been  available  in  written  form  ( γέγραπται )  suggests  that  a  seasoned

- 58  Xen. Mem . 3.6.8-9 Οὐκοῦν , ἔφη , τόν γε βουλευσόμενον , πρὸς οὕστινας δεῖ πολεμεῖν , τήν τε τῆς πόλεως δύναμιν καὶ τὴν τῶν ἐναντίων εἰδέναι δεῖ , ἵνα ἐὰν μὲν ἡ τῆς πόλεως κρείττων ᾖ , συμβουλεύῃ ἐπιχειρεῖν τῷ πολέμῳ , ἐὰν δὲ ἡ τῶν ἐναντίων , εὐλαβεῖσθαι πείθῃ . | Ὀρθῶς λέγεις , ἔφη . | Πρῶτον μὲν τοίνυν , ἔφη , λέξον ἡμῖν τῆς πόλεως τήν τε πεζικὴν καὶ τὴν ναυτικὴν δύναμιν , εἶτα τὴν τῶν ἐναντίων . | Ἀλλὰ μὰ τὸν Δί᾿ , ἔφη , οὐκ ἂν ἔχοιμί σοι οὕτω γε ἀπὸ στόματος εἰπεῖν . | Ἀλλ᾿ εἰ γέγραπταί σοι , ἔνεγκε , ἔφη· πάνυ γὰρ ἡδέως ἂν τοῦτο ἀκούσαιμι .  | Ἀλλὰ μὰ τὸν Δί᾿ , ἔφη , οὐδὲ γέγραπταί μοί πω .  | Οὐκοῦν , ἔφη , καὶ περὶ πολέμου συμβουλεύειν τήν γε πρώτην ἐπισχήσομεν . See also the extensive education young Alexander received from some of the finest tutors that his father could hire to get him ready to rule. Plut. Alex. 5.
- 59  We may add as a further example Perikles' detailed account of Attic geography and Athenian naval forces to convince his fellow Athenians to persevere in their conflict with the Spartans and their allies: 'to these Perikles added other arguments, such as he was accustomed to do, as proof of their superiority in war.' Thuc. 2.13.7-9 ἔλεγε δὲ καὶ ἄλλα οἷάπερ εἰώθει Περικλῆς ἐς ἀπόδειξιν τοῦ περιέσεσθαι τῷ πολέμῳ .

commander like Xenophon would have made balance sheets comparing the two forces in an attempt to determine what military action to take. And in fact, he gives us such a balance sheet in his Anabasis , in a speech to  his  fellow  Greeks  that  includes  an  overview  of  the  strengths  and weaknesses of the two opposing forces:

If  anyone  of  you  is  despondent  because  we  are  without  horsemen while the enemy have plenty at hand, let him reflect that your ten thousand horsemen are nothing more than ten thousand men; […] moreover, we are on a far surer foundation than your horsemen: they are hanging on their horses' backs, afraid not only of us, but also of falling off. 60

Xenophon goes on to tell  his  men  that  they  should  not  worry  either about the terrain or about the lack of guides. His address is obviously rhetorical  to  the  point  of  absurdity  insofar  as  having  fewer  men  in  a foreign country is touted as a benefit; logic is turned on its head. But it reveals two important points pertaining to ancient probability. On the one  hand,  Xenophon  needed  to  address  his  men's  fears  because according to their own calculations their inferior numbers did decrease their  chances  of  returning  home  to  Greece.  On  the  other  hand, Xenophon's  men  also  noted  that  not  all  troops  were  alike,  and  that horsemen had different uses and benefits in particular circumstances, thus demonstrating combinatorial thinking where not only the number but also the type of troops are used to calculate odds of victory. 61

- 60  Xen. Anab. 3.2.17-19 εἰ δέ τις ὑμῶν ἀθυμεῖ ὅτι ἡμῖν μὲν οὐκ εἰσὶν ἱππεῖς , τοῖς δὲ πολεμίοις πολλοὶ πάρεισιν , ἐνθυμήθητε ὅτι οἱ μύριοι ἱππεῖς οὐδὲν ἄλλο ἢ μύριοί εἰσιν ἄνθρωποι· […] οὐκοῦν τῶν γε ἱππέων πολὺ ἡμεῖς ἐπ᾿ ἀσφαλεστέρου ὀχήματός ἐσμεν· οἱ μὲν γὰρ ἐφ᾿ ἵππων κρέμανται φοβούμενοι οὐχ ἡμᾶς μόνον ἀλλὰ καὶ τὸ καταπεσεῖν .
- 61  See also Polybios' verdict on the battle of Cannae (3.117.4-6) where about seventy thousand Romans died: 'both on this occasion and on former ones their numerous cavalry had contributed to the victory of the Carthaginians, and it demonstrated to posterity that in times of war it is better to give battle with half as many infantry as the enemy and an overwhelming force of cavalry than to be in all respects his equal.' τὴν μεγίστην χρείαν παρεσχημένου τοῖς Καρχηδονίοις εἰς τὸ νικᾶν καὶ τότε καὶ πρὸ τοῦ τοῦ τῶν ἱππέων ὄχλου . καὶ δῆλον ἐγένετο τοῖς ἐπιγινομένοις ὅτι κρεῖττόν ἐστι

To be sure, sheer numbers were certainly not the only criterion for deciding  a  battle.  One's  talents  as  a  general,  the  soldiers'  experience, their mental state, the character of the battleground, etc., all could prove decisive. But for Aristotle numbers were nevertheless a good indicator that allowed one to think probabilistically about the future and improve their  odds  of  success.  One  long  passage  in  the Art  of  Rhetoric is  worth quoting:

[A leader] should  know  all  the  expenses  of  the  state,  that  if superfluous, it may be removed, or if too great, may be curtailed […] of  these  matters  it  is  not  possible  to  acquire  a  general  view  from individual experience alone, but in view of advising about them it is further necessary to be well informed about what has been discovered among  others.  In  regard  to  war  and  peace,  the  rhetor  should  be acquainted with the power of the state, how great it is already and how great it may possibly become, of what kind it is already and what additions may possibly be made to it; […] These matters he should be acquainted with, not only as far as his own state is concerned but also in reference to neighboring states, and particularly those with whom there is a likelihood of war, so toward the stronger a pacific attitude may be maintained, and in regard to the weaker, the decision as to making war on them may be left to his own state. Again, he should know whether their forces are like or unlike his own, for herein also advantage or disadvantage may lie. 62

πρὸς τοὺς τῶν πολέμων καιροὺς ἡμίσεις ἔχειν πεζούς , ἱπποκρατεῖν δὲ τοῖς ὅλοις , μᾶλλον ἢ πάντα πάρισα τοῖς πολεμίοις ἔχοντα διακινδυνεύειν . Polybios echoes Xenophon by highlighting how different configurations of troops can generate different results that can be quantified loosely in terms of casualties.

- 62  Arist. Rhet. 1.4.8-9 (1359b-1360a) Ὥστε περὶ μὲν πόρων τὸν μέλλοντα συμβουλεύσειν δέοι ἂν τὰς προσόδους τῆς πόλεως εἰδέναι τίνες καὶ πόσαι , ὅπως εἴτε τις παραλείπεται προστεθῇ καὶ εἴ τις ἐλάττων αὐξηθῇ , ἔτι δὲ τὰς δαπάνας τῆς πόλεως ἁπάσας , ὅπως εἴ τις περίεργος ἀφαιρεθῇ καὶ εἴ τις μείζων ἐλάττων γένηται· […] ταῦτα δ᾿ οὐ μόνον ἐκ τῆς περὶ τὰ ἴδια ἐμπειρίας ἐνδέχεται συνορᾶν , ἀλλ᾿ ἀναγκαῖον καὶ τῶν παρὰ τοῖς ἄλλοις εὑρημένων ἱστορικὸν εἶναι πρὸς τὴν περὶ τούτων συμβουλήν . […] οὐ μόνον δὲ τῆς οἰκείας πόλεως ἀλλὰ καὶ τῶν ὁμόρων ταῦτα ἀναγκαῖον εἰδέναι , καὶ πρὸς οὓς ἐπίδοξον πολεμεῖν , ὅπως πρὸς μὲν τοὺς κρείττους εἰρηνεύηται ,  | πρὸς δὲ

While echoing Xenophon's emphasis on the size and character of an army and its resources, Aristotle is particularly concerned with how this information  could  be  manipulated  to  increase  the  calculable  odds  of success and help make decisions about the future. Notice, for instance, the correlation between addition and possible outcome as expressed by the use of the quantitative phrases πόσην ἐνδέχεται ὑπάρξαι (how great it may possibly become) and ἥτις ἐνδέχεται προσγενέσθαι (whatever it may be possible to add) relative not only to one's own power but also to that of their rivals. For Aristotle, even particular differences ( ὅμοιαι ἢ ἀνόμοιαι ) could be quantified, as shown through his use of the infinitives πλεονεκτεῖν (to claim more than one's share, to have an advantage, claim a  larger  share)  and ἐλαττοῦσθαι (make  smaller,  diminish,  reduce  in amount).  The  moral  meaning  of  the  verbs  expressing  greediness  and degradation  comes  from  their  more  technical  quantitative  meaning expressing addition and reduction. In this particular case the infinitives signal advantage and disadvantage insofar as they increase or diminish one's odds of success.

Both Xenophon and Aristotle, then, prescribe how leaders armed with detailed information could quantify the (un)certainty of war and peace and plan their future steps accordingly. A case in point is Demetrios the Besieger's decision,

though short of money, to double his army by new levies. And when some of his friends in surprise asked him, how he expected to pay them, when he found it difficult to support a smaller force; 'the more powerful we are', he replied, 'the weaker we shall find our enemies; and the more easily make ourselves masters of their country. From thence  tribute  and  free  gifts  will  come  in,  that  will  soon  fill  our coffers.' 63

τοὺς ἥττους ἐπ᾿ αὐτοῖς ᾖ τὸ πολεμεῖν . καὶ τὰς δυνάμεις , πότερον ὅμοιαι ἢ ἀνόμοιαι· ἔστι γὰρ καὶ ταύτῃ πλεονεκτεῖν ἢ ἐλαττοῦσθαι .

63  Polyaen. 4.7.1 s.v. 'Demetrius' Δημήτριος χρήματα οὐκ ἔχων διπλασίους συνέλεξε στρατιώτας· καὶ δὴ θαυμάζοντός τινος , πόθεν ἡ μισθοφορὰ τοσούτοις , ὅπου μηδὲ τοῖς ἐλάττοσιν ,  ' ὅτι ', ἔφη ,  ' βαρύτεροι ὄντες ἀσθενεστέρους τοὺς ἀντιπάλους ἕξομεν καὶ

The  passage  highlights the fascinating  discrepancy  in  perception between Demetrios and his friends. Whereas the latter are solely focused on their present circumstances, Demetrios looks to the future by using probabilistic thinking to determine the best course of action. In other words, his financial gambling is in fact an investment into making his chances of possible victory stronger ( βαρύτεροι ) relative to his soon-tobe weaker ( ἀσθενεστέρους ) enemies, by a factor of two, which would in turn  bring  him  a  significant  return  on  that  investment.  Demetrios' initiative,  like  our  earlier  examples,  therefore,  reveal  that  for  ancient decision-makers  statistical  thinking  became  what  psychologist  Gerd Gigerenzer  calls  'a  habit  of  mind.' 64 They  had  the  incentive  and  the inclination to convert various quantities into a single abstract value of uncertainty  to  make  informed  decisions  about  present  and  potential dangers.

## 5. QUALITATIVE PROBABILITY

Numbers are not the only means to express probability. Risk analysts Baruch  Fischhoff  and  John  Kadvany  (2011)  explain  why  estimative language plays a crucial role in communicating uncertainty in important socio-political  contexts:  because  analytical  judgements  are  inherently not certain, decision-makers use probabilistic language to reflect the estimates of the likelihood of developments or events. 65 Such language appears prominently in ancient philosophy and forensic oratory, though its use by ancient historians regarding practical decision-making has not received  extensive  attention.  In  fact,  historians  and  local  leaders  use terms  like εἰκός to  express  the  probability  of  an  outcome  during  the decision-making process. 66 In this regard, Thucydides's use of the term during the speech of the Korinthians on the eve of the Peloponnesian War is instructive: they explain to their Peloponnesian allies that 'For many reasons it is likely ( εἰκὸς ) for us to prevail: firstly, because we are superior in numbers and military experience, then because we follow all orders ... so if we win a single victory at sea, [the Athenians] are most likely ( κατὰ τὸ εἰκὸς ) defeated.' 67 Beyond its rhetorical flavor, Thucydides uses this passage to point out that likelihood of success is not merely a matter of guesswork but the product of calculation that took into account perceived experience, general inclination, circumstances and, of course, the sheer number of troops, ships and resources.

τῆς τούτων χώρας κρατήσομεν , καὶ φόρους οἴσουσιν ἄλλοι , [ ἄλλοι ] καὶ στεφάνους πέμψουσι τὸ πλῆθος τῶν στρατιωτῶν δεδιότες ἤδη .'

64  Gigerenzer 2002: 245.

65  Fischhoff & Kadvany 2011: 126-27.

Decision theorists interpret such estimations as expressions of 'qualitative probability', which represents 'a theory of probability based on qualitative  ordering  of  events  in  terms  of  their  likelihood  of  occurrence.' 68 In times of crisis, ancient leaders and communities ultimately had to choose between a set of difficult options, each with their own dangers and consequences, that could often be reduced to a binary response

- 66  For detailed discussions and examples of argumentation through likelihood ( εἰκός , εἰκότα )  in  forensic  oratory,  see  the  contributions  by  Michael  Gagarin  and  Craig Cooper in A Companion to Greek Oratory (Gagarin 2007: 27-36 and Cooper 203-19, respectively). Consider the famous hypothetical example of whether a weak man is (un)likely to be charged with assaulting a strong man (and vice-versa) (Arist. Rhet . 2.24.11). On the one hand, the weaker man would have smaller chances of success against a stronger man, which would make him wary of committing such a crime. On the other hand, given such general expectations of success, the stronger man would also be unlikely to assault a weaker man because everyone would think him to be the likely suspect; Gagarin calls it 'a reverse argument from likelihood.' Gagarin 2007: 32. For the first uses of the language of probability in Greek literature, starting with the Homeric Hymn to Hermes , see Kennedy 1995: 11-29. On the extensive use of εἰκός by Attic rhetoricians to suggest likelihood of guilt based on character rather than forensic evidence, see further Kennedy 1995: 64-80.
- 67  Thuc. 1.121.4 κατὰ πολλὰ δὲ ἡμᾶς εἰκὸς ἐπικρατῆσαι , πρῶτον μὲν πλήθει προύχοντας καὶ ἐμπειρίᾳ πολεμικῇ , ἔπειτα ὁμοίως πάντας ἐς τὰ παραγγελλόμενα ἰόντας […] μιᾷ τε νίκῃ ναυμαχίας κατὰ τὸ εἰκὸς ἁλίσκονται .
- 68  Narens 2007: 29. Note also Meusnier's (2008: 108) observations on the assumed discontinuity in recent scholarship between qualitative and quantitative approaches to probability, which leads to the perceived 'sudden' appearance of probability theory after its 'discovery' by Pascal, Fermat, and Huygens.

-  'yes'  or  'no',  action  or  inaction,  attack  or  defend,  etc.  Preferable courses of action could also be expressed through comparative adjectives. As with εἰκός , adjectives add 'gradations' of risk to these kinds of binary contrasts. Diodorus Siculus, for instance, portrays the convoluted history of the Successor Wars that erupted immediately after the death of  Alexander the Great, through a detailed description of the thought process  of  an  otherwise  unremarkable  commander  by  the  name  of Peukestes. This Peukestes was one of the many Macedonian leaders who had been prominent at Alexander's court, and who now sought to carve the dead King's empire among themselves. When he was asked by several others to send help against an increasingly belligerent and powerful Antigonos Monophthalmos, who wanted to take it all for himself, Diodorus tells us that:

At  first  [Peukestes]  paid  no  heed  to  them  […]  since  he  still  bore  a grudge for not receiving a generalship; but later, reasoning with himself,  he  conceded  that  should  Antigonos  be  victorious,  the  result would be that he himself would lose his satrapy and thus also risked ( κινδυνεῦσαι ) losing his life. Agonizing, therefore, about himself, and thinking that he would be more likely ( μᾶλλον ) to gain the command if he had as many soldiers as possible, he brought forth ten thousand archers, as they requested. 69

The passage's many verbs of pondering draw our attention on Peukestes' step-by-step thought process in a series of 'if… then…' clauses, as we are privy to how he determines what his options are, along with their probable consequences. In this context, the comparative μᾶλλον points  to Peukestes' deductive logic based on what scenario he deems more likely to occur.

- 69  Diod. Sic. 19.17.5-6 ὁ δὲ τὸ μὲν πρῶτον οὐ προσεῖχεν αὐτοῖς , μεμψιμοιρῶν ἐπὶ τῷ μὴ τετευχέναι τῆς στρατηγίας , ὕστερον δὲ δοὺς αὑτῷ λόγον συνεχώρησεν ὅτι κρατήσαντος Ἀντιγόνου συμβήσεται καὶ τὴν σατραπείαν αὐτὸν ἀποβαλεῖν καὶ περὶ τοῦ σώματος κινδυνεῦσαι . ἀγωνιῶν οὖν ὑπὲρ αὑτοῦ καὶ τῆς στρατηγίας μᾶλλον τεύξεσθαι νομίζων ὡς πλείστους ἔχων στρατιώτας προσήγαγεν , καθάπερ ἠξίουν , τοξότας μυρίους .

However, the full force of Diodorus' passage would be lost if we simply agreed that we are dealing with a 'rational' actor who eventually makes the objectively 'correct' choice. His account is not merely retrospective but captures the internal doubt and discomfort of Peukestes in having to make  a  difficult  choice.  We  see  the  commander  literally  agonizing ( ἀγωνιῶν ) over the preferable course of action. Having to help others at the expense of Antigonos, even though likely more beneficial, is not a particularly comforting thought considering the agonistic character of Macedonian politics. Peukestes settles on a solution by conjuring up different potential futures, mirroring Aristotelian decision-trees, each with their own series of steps and possible consequences that take into account military capabilities, geo-political circumstances, but also the personal character of his rivals; an otherwise notoriously difficult factor to quantify using formal statistical analysis.

Expressions  of  preference  might  strike  some  as  not  indicative  of 'proper' probabilistic thinking. But Spiegelhalter reminds us that whatever probabilistic model we adopt - be it classical or frequentist - it remains  true  that 'probabilities are constructed  based  on  existing knowledge, and are therefore contingent,' 70 an admittedly controversial statement that informs Bruno de Finetti's famous quote at the beginning of this paper, 'probability does not exist.' Spiegelhalter takes this statement to mean that 'probabilities are not states of the world […], but depend on the relationship between the 'object' of the probability assessment, and the 'subject' who is doing the assessing.' 71 The relational character of probability thus allows us some insight into how decision-makers can incorporate even subjective elements such as perceptions, experience, and emotions into their calculations by assigning to them qualitative values and priorities.

Indeed,  the  most  generative  strategy  that  ancient  decision-makers used to assess the future was the conceptual linking of the past with the present through perceived historical precedents. Particularly in warfare, a record of past encounters was touted as indicative of likely outcomes. 72

70  Spiegelhalter 2011: 20.

71  Spiegelhalter 2011: 20-21.

72  In the Latin context, we have examples from Caesar and Tacitus that express probability through precedent. Specifically, in Caes. BCiv. 3.73 Caesar gives a speech before

Returning to The Art of Rhetoric , Aristotle further qualifies the reliability of numbers to calculate odds of success by relying on the historical dimension  to  add  perspective  to  the  assigned  value  of  state  power.  He states succinctly that 'with reference to these matters he must also have examined the results not only of the wars waged by his own state, but also of those waged by others; for similar results naturally arise from similar causes.' 73 Aristotle understood that quantitative and qualitative probabilities  can  be  brought  together  by  decision-makers  to  generate helpful statistics informing the correlation between past outcomes and present circumstances. Polybios attributes such a statistical mindset to the Roman general's assessment of the Macedonian enemy before the battle of Kynoskephalai in 196 BCE, which would humble the Macedonian kingdom and establish a Roman presence in subsequent Greek affairs. Facing the professional army of Philip V, the Roman general Titus Quinctius Flamininus delivered a short speech to his troops in which he asked rhetorically:

Are these not the same Macedonians whom, when they held that desperately difficult position in Epirus, you compelled by your valor to throw away their shields and flee, never stopping until they got home to Macedonia? What reason, then, have you got to be timid now when you are about to battle the same men on equal terms? Why not foresee the past instead of dreading an opposite outcome, and dare? So, my

his troops after the battle of Dyrrhachium, urging them 'not to be discouraged, or give way to consternation, upon what had lately happened, but oppose their many successful engagements to one slight and inconsiderable check' ( ne ea quae accidissent graviter ferrent, neve his rebus terrerentur, multisque secundis proeliis unum adversum et id mediocre opponerent ). He was careful to point out that their single loss was due to their small numbers, as well as unprecedented circumstances and - alas - unfavorable fortune, which was bound to turn in their favor. Similarly, in Tac. Ann. 1.61-62. It is also worth noting Caecina's expertise; he was on his fortieth campaign. His experience of success and peril had made him fearless: Tac. Ann . 1.64.6 quadragesimum id stipendium Caecina parendi aut imperitandi habebat, secundarum ambiguarumque rerum sciens eoque interritus .

- 73  Arist. Rhet. 1.4.9  (1360a) ἀναγκαῖον δὲ καὶ πρὸς ταῦτα μὴ μόνον τοὺς οἰκείους πολέμους τεθεωρηκέναι ἀλλὰ καὶ τοὺς τῶν ἄλλων , πῶς ἀποβαίνουσιν· ἀπὸ γὰρ τῶν ὁμοίων τὰ ὅμοια γίγνεσθαι πέφυκεν .

men, encouraging each other, dash on to the fray and put forth all your strength. For with the gods willing, I feel sure that this battle will end like the earlier dangers. 74

Polybios uses the speech to bring attention to the morale boost that past victories gave the troops, a practice that many generals routinely used. But Polybios also touches on the deliberative process that informed the soldiers' confidence. In other words, precedent did not cause an automatic reaction, but was rationally consulted. Flamininus stimulates his men's observational skills to get them to realize that they are fighting the same enemy under comparable conditions, appealing to the image of the enemy's previous cowardice to drive home his point. He then encourages them to maintain their determination - based on past outcomes, there is no real reason for them to expect a different result. The interplay between past and future is elegantly highlighted by the call 'to foresee the past.' In using this phrase, Polybios stakes a claim that the Romans' stance was not merely a matter of courage and honor, but was the result of an informed decision based on previous encounters with similar outcomes, which in turn helped Flamininus calculate the projected risk of the battle.

Xenophon makes a similar pitch to his Greek companions at the outset of the Anabasis. He claims that he entertained - the gods willing many and beautiful hopes of salvation, 75 not  only  because  they  themselves were righteous pious men fighting against perjurers, bound to incur the wrath of gods. He also relied on the record of the Greeks in their past encounters with the Persians, starting from the Persian Wars and

- 74  Polyb. 18.23.3-6 οὐχ οὗτοι Μακεδόνες εἰσίν , οὓς ὑμεῖς προκατέχοντας τὰς ἀπηλπισμένας ἐν Ἠπείρῳ δυσχωρίας ἐκβιασάμενοι ταῖς ἑαυτῶν ἀρεταῖς φεύγειν ἠναγκάσατε ῥίψαντας τὰ ὅπλα , τέως εἰς Μακεδονίαν ἀνεκομίσθησαν ; πῶς οὖν ὑμᾶς εὐλαβεῖσθαι καθήκει , μέλλοντας ἐξ ἴσου ποιεῖσθαι τὸν κίνδυνον πρὸς τοὺς αὐτούς ; τί δὲ προορᾶσθαι τῶν προγεγονότων , ἀλλ᾿ οὐ τἀναντία δι᾿ ἐκεῖνα καὶ νῦν θαρρεῖν ; διόπερ , ὦ ἄνδρες , παρακαλέσαντες σφᾶς αὐτοὺς ὁρμᾶσθε πρὸς τὸν κίνδυνον ἐρρωμένως· θεῶν γὰρ βουλομένων ταχέως πέπεισμαι ταὐτὸ τέλος ἀποβήσεσθαι τῆς παρούσης μάχης τοῖς προγεγονόσι κινδύνοις .
- 75  Xen. Anab. 3.2.8 σὺν τοῖς θεοῖς πολλαὶ ἡμῖν καὶ καλαὶ ἐλπίδες εἰσὶ σωτηρίας . Notice again the emphasis on 'beautiful hopes' to suggest good chance of success. (See Ch. 1).

on to their most recent encounters as part of their support in Cyrus' bid to the throne of Persia, when 'you stood in formation against the descendants of those [ancient Persians], who far outnumbered you, and were victorious with [the aid of] the gods […] proving to be brave men.' 76 The repeated reference to τοῖς θεοῖς is noteworthy because it also echoes Flamininus' appeal to θεῶν βουλομένων to acknowledge the contingencies  of  war  and  the  inherent  dangers  that  lie  therein. 77 But  more  importantly, Xenophon's mention of a tradition of victory against the Persians served not only to spur morale, but also to suggest that past encounters were instructive on how to deal with the same enemy:

It is now more appropriate to be more daring to go against the enemy, for in the past you were ignorant about them, considering their host numberless, and nevertheless you dared to go against them with ancestral  resolution.  For  now,  when  you  have  already  had  proof  that they are unwilling to receive your charge even though they are many times more numerous, what reason is there for you to fear them? 78

Xenophon's speech is a masterstroke in mass persuasion that discloses the great lengths to which commanders would go to equate present conditions to successful past enterprises, particularly in unfavorable situations. Yet despite the rhetorical character, its probabilistic logic persists because, as in Polybios' example, it is grounded on a cumulative gathering of information. Starting from a point in time when the Greeks were ἄπειροι (inexperienced, unused to, unacquainted with) vis-à-vis the enemy's  military  capabilities,  their  experiences  gradually  increase  their knowledge of their enemy, as highlighted by the passage's many comparative adjectives. In such circumstances, a string of past successes is suggestive, at least in theory, of future victories. 79

76  Xen. Anab. 3.2.15 καὶ τότε μὲν δὴ περὶ τῆς Κύρου βασιλείας ἄνδρες ἦτε ἀγαθοί .

77  I will explore this topic in a forthcoming article on Polybios' conceptualization of contingency planning.

78  Xen. Anab. 3.2.16 νῦν δ᾿ ὁπότε περὶ τῆς ὑμετέρας σωτηρίας ὁ ἀγών ἐστι πολὺ δήπου ὑμᾶς προσήκει καὶ ἀμείνονας καὶ προθυμοτέρους εἶναι . ἀλλὰ μὴν καὶ θαρραλεωτέρους νῦν πρέπει εἶναι πρὸς τοὺς πολεμίους . τότε μὲν γὰρ ἄπειροι ὄντες αὐτῶν , τό τε πλῆθος ἄμετρον ὁρῶντες , ὅμως ἐτολμήσατε σὺν τῷ πατρίῳ φρονήματι ἰέναι εἰς αὐτούς· νῦν δὲ ὁπότε καὶ πεῖραν ἤδη ἔχετε αὐτῶν ὅτι οὐ θέλουσι καὶ πολλαπλάσιοι ὄντες δέχεσθαι ὑμᾶς , τί ἔτι ὑμῖν προσήκει τούτους φοβεῖσθαι ;

History, as such, was not merely didactic in a moral sense, but was thought to contain practical knowledge that could be consulted to determine one's odds of success, where numbers and attitudes were placed in a  historical  context  to  be  consulted  during  ostensibly  similar  circumstances. This tried-and-tested method eventually developed into a standardized form of education that culminated in the production of technical manuals. Xenophon's On Horsemanship , Aeneas Tacticus' On the defense of Fortified Positions , Onasander's On Strategy , and Polybios' now-lost On Tactics , are replete with precedents in various situations that were meant to inform a decision-maker's choices in matters of war and local administration.  Beside  oracular  consultation  and  divination,  then,  ancient thinkers also prescribed a probabilistic system of knowledge that rendered the future calculable and thus more imaginable.

## 6. PRECEDENT VERSUS ADAPTATION

Similarity is nevertheless not sameness, especially when statistics are involved. In fact, the logic that past successes necessarily translate into further victories is a probabilistic mistake, the so-called 'hot hand fallacy.' Especially in warfare, new encounters are independent events, and their odds of success will not depend strictly on the past; new conditions, information, and many other factors, can influence the outcome. Ancient historians were well aware of this logical fallacy and sought to render it intelligible for their contemporaries. Xenophon, for instance, expressed his support for the Common Peace of 371 BCE by equating the irredentism of Athens and Sparta to the compulsive behavior of gamblers and athletes:

- 79  A similar calculation was made by the Melians after having resisted the Athenians earlier in the war (Thuc. 3.91). But in the Melian dialogue Thucydides chooses not to emphasize this point and instead make an argument about the pitfalls of relying on hollow hope instead of rational calculation (Thuc. 5.116).

I for my part do not commend those men who, when they have become  competitors  in  the  games  and  have  already  been  victorious many times and enjoy fame, love winning so much that they do not stop until they are defeated and cease their training. Nor, on the other hand, do I commend those dicers who, if they win one success, throw double  stakes  ( περὶ διπλασίων κυβεύουσιν ),  for  I  see  that  most ( πλείους ) of these people become utterly impoverished ( ἀπόρους ). 80

This bullish attitude is well-known among psychologists and game theorists. 81 They agree with Xenophon that, while a lucky few might succeed, the great majority of those who adopt it are statistically bound to fail and become ἀπόρους ,  as  Xenophon's  use  of  the  comparative  adjective πλείους suggests. One must instead hedge their bets and be aware of circumstances and trends, and not 'engage in a contest of such a sort that we either win all or lose all'; 82 blind faith in past outcomes is not enough.

Xenophon is thus drawing attention to the essential skills of adaptation  and  improvement  that  decision-makers  must  possess.  Otherwise, they will suffer the fate of the Spartans at the hands of Kallias son of Hipponikos just outside of Korinth in 390 BCE during the so-called Korinthian War. Xenophon recounts how, upon splitting up their forces and returning to Lechaion, '[the Spartans] were by no means unaware that there were many peltasts and many hoplites in Korinth, but on account of  their  previous  successes  they  contemptuously thought that no one would attack them.' Their enemies, on the other hand, 'when they saw that [the Spartans] were few in number, but also unaccompanied by either peltasts or cavalry, thought that it was safe to attack them with their force of peltasts.' 83 Xenophon contrasts Kallias' observation ( καθορῶν -τες ) followed by his estimation ( ἐνόμισαν ) of the changed circumstances with  the  Spartans'  contemptuous  heedlessness  ( καταφρόνησις )  to  underline the importance of continually being mindful of changes and constantly striving to improve one's condition, especially in high stakes matters.

80  Xen. Hell . 6.3.16 ἀλλὰ μὴν οὐδ᾿ ἐκείνους ἔγωγε ἐπαινῶ οἵτινες ἀγωνισταὶ γενόμενοι καὶ νενικηκότες ἤδη πολλάκις καὶ δόξαν ἔχοντες οὕτω φιλονικοῦσιν ὥστε οὐ πρότερον παύονται , πρὶν ἂν ἡττηθέντες τὴν ἄσκησιν καταλύσωσιν , οὐδέ γε τῶν κυβευτῶν οἵτινες αὖ ἐὰν ἕν τι ἐπιτύχωσι , περὶ διπλασίων κυβεύουσιν· ὁρῶ γὰρ καὶ τῶν τοιούτων τοὺς πλείους ἀπόρους παντάπασι γιγνομένους .

81  Konnikova 2020a; 2020b.

82  Xen. Hell. 6.3.17 ἃ χρὴ καὶ ἡμᾶς ὁρῶντας εἰς μὲν τοιοῦτον ἀγῶνα μηδέποτε καταστῆναι , ὥστ᾿ ἢ πάντα λαβεῖν ἢ πάντ᾿ ἀποβαλεῖν , ἕως δὲ καὶ ἐρρώμεθα καὶ εὐτυχοῦμεν .

The example further indicates that in such real-time scenarios, like politics  or  sports,  rivals  learn  from  each  other  with  every  encounter, making the next clash all the more interesting because its outcome is not only determined by past results, but also by the changes that each side adopts in trying to predict the possible actions of the adversary; a really good team, for instance, is able to predict the opposition's predictions, as it were. The rise and rule of Rome offers a fascinating historical case study because Greek historians tend to explain it precisely as the result of the Romans' ability to learn from past failures and improve going forward.  Their  talent  is  apparent  when  they  manage  to  overcome  their more established Carthaginian rivals in the naval arena during the First Punic War. The war was rooted in the growing influence of the two Republics in the Western Mediterranean which made an eventual confrontation between the two powers virtually unavoidable. Polybios is particularly interested in this conflict and starts his Histories with it, because he interprets it as the first clear proof of the Romans' future greatness. He repeatedly mentions the Romans' traditional naval (in)experience, but adds that 'When they once conceived of the project, they took it in hand so boldly, that before gaining any experience in the matter they at once engaged the Carthaginians, whose hegemony of the sea had been undisputed for generations.' 84 The Romans, according to Polybios, knew that the key to eventual victory against the Carthaginians - or against

- 83  Xen. Hell. 4.5.12-13 καὶ ὅτι μὲν πολλοὶ ἦσαν ἐν τῇ Κορίνθῳ καὶ πελτασταὶ καὶ ὁπλῖται οὐδὲν ἠγνόουν· κατεφρόνουν δὲ διὰ τὰς ἔμπροσθεν τύχας μηδένα ἂν ἐπιχειρῆσαι σφίσιν . οἱ δ᾿ ἐκ τῶν Κορινθίων ἄστεως , Καλλίας τε ὁ Ἱππονίκου , τῶν Ἀθηναίων ὁπλιτῶν στρατηγῶν , καὶ Ἰφικράτης , τῶν πελταστῶν ἄρχων , καθορῶντες αὐτοὺς καὶ οὐ πολλοὺς ὄντας καὶ ἐρήμους καὶ πελταστῶν καὶ ἱππέων , ἐνόμισαν ἀσφαλὲς εἶναι ἐπιθέσθαι αὐτοῖς τῷ πελταστικῷ .
- 84  Polyb.  1.20.12 τότε δὴ πρῶτον ἐν νῷ λαμβάνοντες οὕτως τολμηρῶς ἐνεχείρησαν ὥστε πρὶν ἢ πειραθῆναι τοῦ πράγματος , εὐθὺς ἐπιβαλέσθαι Καρχηδονίοις ναυμαχεῖν τοῖς ἐκ προγόνων ἔχουσι τν κατὰ θάλατταν ἡγεμονίαν ἀδήριτον .

any enemy, in fact - was beating them at their own game, which meant being prepared to accept many losses for the sake of improvement. Looking at the record between the two fleets, people would understandably be tempted to bet on the Carthaginians, based on past results. But those gamblers would be sorely disappointed as the Romans were eventually able to routinely defeat their Punic adversary.

This tension between precedent and improvement also governs Diodorus Siculus' account of the interaction between the Carthaginian and Roman envoys on the eve of the First Punic War. On their part, the Carthaginians appeal to precedent to suggest that they are bound to win any future encounter, '[as] they wondered how the Romans dared to cross into Sicily while the Carthaginians were the masters of the sea, for it was obvious to all that, should they not protect their friendship, they would no longer dare to even wash their hands in the sea.' 85 By referring to their own record of success as φανερὸν πᾶσιν , the Carthaginians warned the Romans against trying to threaten their naval prowess. The Romans, by contrast, ostensibly emphasized the importance of accumulated experience to improve where they had failed in the past. Thus, while not denying the Carthaginians' present naval power, they issued a warning of their own, that the Carthaginians' prominence would ultimately prove their own undoing: 'for the Romans have always turned out to be pupils stronger than their teachers.' 86

These passages reveal an ancient understanding of the probabilistic feature regarding incremental success currently known among economists as a 'power law,' representing a relationship between two quantities, like the chances of victory going into battle, where changes in one quantity lead to a proportional relative change in another. Whereas in a game of dice where statistical data is collected from dice throws with the same aleatory chance, this incremental model suggests that one result say one battle between the Romans and the Carthaginians - will then pro- portionally impact the chances of the Romans during their next encounter, and so on. Sergio Da Silva, Raul Matsushita, and Eliza Silveira (2013) have looked at sports and war and have found that in both circumstances when antagonists compete, 'there emerges stasis, as each adaptation by one in countered by an adaptation by the other. The co-evolution between the antagonistic sides eventually reaches equilibrium and a fairly regular power law takes place.' 87 Eventually, since perfect counter-adaptation is unfeasible, one side is bound to gain the upper hand, which in turn  furthers  its  chances  of  success  with  successive  repetitions;  provided, of course, that it remains focused on adapting to circumstances and learning how to do things better. 88 We find the same phenomenon in the case of Roman success, who became increasingly more difficult to defeat in any single subsequent encounter because they kept learning, adapting, and improving. By the Third Punic War, when Carthage was razed  to  the  ground,  the  Romans  only  needed  marginal  refinement against their massively disadvantaged enemy.

85  Diod. Sic. 23.2.1 οἱ Φοίνικες θαυμάζειν ἔφασαν πῶς διαβαίνειν τολμῶσιν εἰς Σικελίαν Ῥωμαῖοι θαλαττοκρατούντων Καρχηδονίων· φανερὸν γὰρ εἶναι πᾶσιν ὅτι μὴ τηροῦντες τὴν φιλίαν οὐδὲ νίψασθαι τὰς χεῖρας ἐκ τῆς θαλάσσης τολμήσουσιν .

86  Diod.  Sic.  23.2.1  ' μαθητὰς γὰρ τοὺς Ῥωμαίους ἀεὶ ὄντας γίνεσθαι κρείττους τῶν διδασκάλων .'

Importantly, the Romans were not exceptional in this regard. Historians use the same logic of adaptation and refinement to explain the rise and fall of other erstwhile powers like Athens, Sparta, and Thebes. Xenophon explains Athens' long-term naval superiority through the voice of Prokles the Phliasian who observes that

you  already  possess  many  triremes  and  it  is  your  naval  tradition ( ναυτικὸν )  to  continually  add  to  them.  You  likewise  possess  as peculiarly your own all the arts and crafts which have to do with ships. Again, you are far superior to other men in experience of nautical affairs, for most of you get your livelihood from the sea. [… As a result,] you have engaged in very many and very great combats by sea, you have met with an exceedingly small number of misfortunes and have achieved an exceedingly large number of successes. Therefore, it is likely that the allies would like best to share in such danger if they were under your leadership. 89

87  Da Silva, Matsushita & Silveira 2013: 5382-83.

88  Da Silva, Matsushita & Silveira 2013: 5384-85.

89  Xen. Hell .  7.1.4-5 ἔτι δὲ τριήρεις κέκτησθε πολλάς , καὶ πάτριον ὑμῖν ἐστι ναυτικὸν ἐπικτᾶσθαι . ἀλλὰ μὴν τάς γε τέχνας τὰς περὶ ταῦτα πάσας οἰκείας ἔχετε . καὶ μὴν

The technical language used in the passage ναυτικὸν , τέχνας , ἐμπειριᾳ ... περὶ τὰ ναυτικά - suggests that Athens' consistency is not merely the result  of  singular  power,  but  also  of  military  and  logistical  skills diligently refined over time, as implied by the terms πάτρικον and βίος . Experience and skill thus work together to create the likelihood ( εἰκός ) that more allies will 'share in the danger' ( κινδύνου μετέχειν ),  which will in turn further facilitate Athenian success. By contrast, Agesilaos' rigid Spartans ostensibly lost their supremacy at the hands of Epaminondas' Thebans, who proved much more malleable to learn by trial and error, adapt, improve, and finally surpass their enemy. 90 In these examples we find echoes of Thucydides' dictum on the importance of adaptation: 'necessity states that, just as with a skill ( τέχνης ), improvements always prevail; and though unchanging customs may be best for undisturbed communities, constant necessities of action ( ἀναγκαζομένοις )  must be accompanied by the constant improvement of methods

ἐμπειρίᾳ γε πολὺ προέχετε τῶν ἄλλων περὶ τὰ ναυτικά· ὁ γὰρ βίος τοῖς πλείστοις ὑμῶν ἀπὸ τῆς θαλάττης· […] πλείστους γὰρ καὶ μεγίστους ἀγῶνας ἠγωνισμένοι κατὰ θάλατταν ἐλάχιστα μὲν ἀποτετυχήκατε , πλεῖστα δὲ κατωρθώκατε . εἰκὸς οὖν καὶ τοὺς συμμάχους μεθ᾿ ὑμῶν ἂν ἥδιστα τούτου τοῦ κινδύνου μετέχειν .

- 90  Plutarch has Antalkidas bitterly reproach Agesilaos for 'having taught those who were neither willing, nor knowledgeable about how to fight.' His first defeat was mockingly called 'a fine tuition fee that you claim from the Thebans for teaching them how to fight when they did not wish it, and did not even know how.' Plut. Ages. 26.2: ἦ καλὰ τὰ διδασκάλια παρὰ Θηβαίων ἀπολαμβάνεις , μὴ βουλομένους μηδὲ ἐπισταμένους μάχεσθαι διδάξας .  Plutarch  explains  that  having  to  regularly  fight against the Lakedaimonians ultimately forced the Thebans to become more warlike, 'such that they were trained ( ἐγγυμνασαμένους )  through the many campaigns of the Lakedaimonians against them.' The use of the verb ἐγγυμνάζειν highlights  a Theban mindfulness of past failures and, at the same time, a constant effort to improve one's chances of success. Purportedly, Agesilaos had contravened an ancient Lykourgan rhetra that specifically prohibited the Spartans to make frequent campaigns against the same enemy, in order that the enemy 'might not learn how to make war' (Plut. Ages . 26.3).

( ἐπιτεχνήσεως ).' 91 And since the world is forever bound to change, one must always remain vigilant, whether enjoying the height of glory or bearing the burden of defeat, because eventually one will be faced with opportunities as well as challenges to one's condition.

## CONCLUSION

This article proposes an alternative approach to the problem of probability in antiquity. By adopting a 'frequentist' model based on historical knowledge, we can trace a probabilistic mindset of decision-makers who developed  conceptual  tools  to  calculate  the  likelihoods  of  occurrence and odds of success in economic, social, and military initiatives. In turn, our discussion on ancient probability will further allow us to explore new avenues for research beyond the realm of ancient science about the formulation of risk in antiquity, and how ancient decision-making bodies understood and undertook contingency planning - both topics of future research. Finally, ancient probability invites us to reconsider the notion that the ancient Greeks were fundamentally 'past-oriented', and instead consider  a  speculative  attitude  towards  a  future  that  could  be  scrutinized, and even foreseen. Metaphorically speaking, in the valley of ancient history the future was not a sudden and mysterious shout, but an echo carried by the winds of the present hitting the mountains of the past.  The  Greeks  understood  that  only  by  knowing  the  environment could one hope to estimate the echo's path and the distances it traveled.

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