Friday, August 23, 2013

Through Pollock's Eyes:

Reflections on the Fractal Nature of Geographic Curves, Abstract Cartographic Spaces and the Last Pool of Darkness

The idea that one gets a better and better approximation of the length of a shoreline by measuring it in finer and finer detail is false; the series of approximations does not converge to an answer, it just gets bigger and bigger, to infinity.... ....................................................................---Tim Robinson, The Connemara Fractal

Many of my best mathematical reflections seem to come to me while wandering around in museums, especially the Museum of Modern Art in New York City. The abstraction of shapes and colors found in the paintings of many of the 20th century's greatest masters, all of which line the walls there, remind me of spatial and geographical forms that have been bled of reference, seemingly residual landscapes and spaces devoid of human action; the proverbial blank on the map or a vast bringing together of spatial silences. Recently, while standing in front of one of Jackson Pollock's great drip paintings, the connection that I have often felt between mathematical and geographic spaces and the planar surfaces created by the abstract expressionists suddenly crystallized in my mind as geometry. Fractal Geometry. I am of course not the first to think of this connection. J.R. Mureika for example, studied perceptual color space, and related it to the fractal forms of Pollack's paintings ("Fractal dimensions in perceptual color space: A comparison study using Jackson Pollack's art," Chaos 15 (2005))
Both geographic spaces and most of Pollack's drip paintings are fractal in nature. In contrast to lines on a map, which are one-dimensional, and are pure generalizations of reality, fractals consist of patterns that recur on finer and finer scales. Because of this 'scaling', fractals can build up natural shapes of immense complexity like coastlines, boundaries and even full landscapes. The Pollock drip paintings are similar and when looked at in finer and finer scales, such as those shown in the detail images below, are much more like geographical curves than cartographic maps actually are. His paintings, strangely enough, unlike maps, can be seen as scaled portions of the whole at larger and larger scales. Several scholars have looked at the fractal dimension of Pollock's paintings using one of the easiest, at least from a mathematical perspective, methods to determine dimension. A method known as box-counting. Mathematically, box-counting is relatively easy to calculate, and to program and its origins go back to at least the 1930s when it was known as Kolmogorov entropy after its Russian inventor. The dimensions calculated for Pollack's paintings show them to be somewhere in the range of 1.3 to 1.7, and unlike those of typical cartographic expressions, are actually closer to that of real world geographic curves.

Real geographical curves are so complex in detail that their lengths are often infinite, or to put it in a more accurate sense, indefinable. Many of these curves, such as those representing a coastline, are statistically 'self-similar', which means that each portion can be considered as a reduced-scale model of the whole, much like we see in a Pollock drip painting. [1] . This peculiar feature of self-similarity, which is an artifact of scaling (something Galileo would have loved), can be described mathematically as a type of dimension, which unlike normal curves, is fractional.
This fractional dimensional property of coastlines was first posited in a formal way by Benoit Mandelbrot in his classic article in Science from 1967, entitled, "How Long is the Coast of Britain ? Statistical Self-Similarity and Fractional Dimension."

Self Similarity of the von Koch Curve. It looks the same no matter what scale we look at it in.

In that article [2] Mandelbrot studied the form of geographic curves, and simpler examples like the von Koch curve shown above, for which the concept of 'length' has no apparent meaning. The von Koch curve is built up by an algorithmic procedure that at every stage in the operation the middle third of each interval or line segment is replaced by the other two sides of an equilateral triangle. These types of curves, to use Mandelbrot's language, can be considered as "superpositions of features of widely scattered characteristic sizes." As he puts it, "as even finer and finer features are taken into account, the total measured length increases, and there is usually no clear cut gap or crossover, between the realm of geography and details with which geography need not be concerned". [3]

Mandelbrot set of a quadratic function in the complex plane. Self-Similarity at every scale

One of the best reflections on this problem of crossover and scaling in cartographic thought comes from the writer and mapmaker Tim Robinson's essay, 'The Connemara Fractal' published in his Setting Foot on the Shores of Connemara and other essays. Robinson, who is one of the great writers on cartography and what for him is the solitary process of map making, writes in that he is "not very interested in maps from a technical point of he will move on to the more interesting questions of what it is like to make a map...insofar as I can untangle my memories of the process." The process of map making that Robinson is speaking is "a long walk," and "an intricate, knotted itinerary that visits every place within its territory." The idea of a long and very detailed walk that Robinson invokes was suggested to him by the extraordinary form of the southern coast of Connemara. Robinson tells us that, "it looks so complicated as to be unmappable; it is a challenge to be unraveled." ...a very Mandelbrotian series of images indeed.

Robinson crosses over to details that according to Mandelbrot "geography need not be concerned", when he starts looking at actual distances along the twisting and knotted coast of Connemara. What he finds are not mappable distances, but rather impossibly infinite curves. Robinson writes that the "distance from Ros a' Mhil to Roundstone is only about 20 miles, but the coastline in between is at least 250 miles long, even as estimated on a small-scale map." He continues, "when I wrote that I was ignorant of the work of Benoit Mandelbrot, who had proved that an outline as complex as a coastline does not have a definable length." Robinson first learned of Mandelbrot's work through a newspaper article sent to him by a reader of his essays on Connemara and he calls Mandelbrot's work "a disturbing doctrine---disturbing to one who fondly imagined he had walked a coastline with due attention to its quiddity....for [Mandelbrots paper] was an annihilating critique of my essay's imagery."

In most respects Mandelbrot's paper was not really a critique of Robinson's essay, but rather a profound theoretical statement that opens up a clearing for us to conceive of a very different, and inherently less positivist conceptual foundation for the cartographic spaces we create. For as Robinson says, the process of map making, like that of discourse must stop somewhere. To be true of the world of fractals is to be infinitely and indefinitely seeking a precise measurement of length or to possess the unexplainable talent of a Jackson Pollock. But to find our way in the real and empirical world it appears that you and I will have to be satisfied with our approximations and our finitude.

[1] For more on Pollack's paintings as fractals see, "The Visual Complexity of Pollack's Dripped Fractals by R.P. Taylor, at

[2] Benoit Mandelbrot, "How long is the coast of Britain? Science 156 (1967) 636-638.

[3] For more on the concept of fractal dimension and measurement see M.C. Shellberg and Harold Moellering's classic paper, "Measuring the Fractal Dimension of Empirical Cartographic Curves."

Tim Robinson is one of the most interesting cartographer's working today. His books on the Aran Islands and Connemara are must reading for anyone interested in the practice of cartography and its relationship to landscape. His short essay called, Interim Reports From Folding Landscapes, imagines the process of cartography in a way that few, if any, other cartographers have ever been able to describe in detail.

Robinson writes that, "A map is a sustained attempt upon an unattainable goal, the complete comprehension by an individual of a tract of space that will be individualized into a place by that attempt."

In Connemara, the last pool of darkness, Robinson reflects on the time Ludwig Wittgenstein spent in the remote parts of Ireland writing and thinking. Wittgenstein, as is well known loved these remote and barren places. These rocky shores seem to have enabled him, in a way not possible in places like Cambridge, to reflect on mathematics, logic and his sins. Landscapes like those of Iceland, western Ireland and the fjords of Norway were the places he worked best and so it seems with Robinson who like Wittgenstein (or not so like him) was a Cambridge trained mathematician.

Thursday, August 01, 2013

Modeling Roman Land-use, Centuriation and Environment: Cartography, Geography, and Game Theory

The foundations of the science of land measurement lies in practical experience, since the truth about sites or area cannot be expressed without lines that can be geometrically measured.

De arte mensoria

My interest in models started more than 30 years ago when I was first a student of engineering, where models are your bread and butter and of course mathematics is your language of expression. Modeling of this kind comes quite naturally to me because ever since I was a child I have only thought in mathematical terms, and for me everything reduces to analysis. I suppose this why I am not an historian or a traditional geographer, although it certainly does not explain why I left engineering. In any case it is the spatial analysis potential of Geographic Information Systems for historical and archaeological studies that most excites me today. 

At Tipasa, Algeria, 2010

As far as the Roman models are concerned the basics go back to when I was a senior in college and I was privileged to have taken a graduate course in game theory and simulation.

It was there that I first wrote a paper on the iterated Prisoner’s Dilemma, which in my mind is an inherently spatial problem. I experimented with this well before the modern age of easy mathematical programming and software like Mathematica.  Back then, looking at even the simplest matrix games was a complicated programming problem that involved juggling hundreds of IBM punch cards. The series of lectures given by Harold Kuhn at Princeton, on the foundations of game theory in the 1980’s, also made a deep impression on me[i]. The graphical techniques for solving matrix games that he laid out in the lectures I found intrinsically beautiful in their simplicity[ii].

Searching for Food and Ruins on Sicily
After that class, I was hooked on Game Theory and for a number of years studied in the economics department of Columbia, not as an enrolled student, but rather as someone who just sat in on the graduate classes on game and decision theory. The professors at that time, in the late 1980s and early 1990s were quite easy going, and as long as one made contributions to the classes, sitting in was not a problem. I remember struggling at first with the presentation and mathematics in Drew Fudenberg’s classic text[iii]. Later however, I became immersed and indeed somewhat obsessed by some of the measure theoretic problems. It was then that I worked my way through many of the foundational texts on game theory, like The Theory of Games and Economic Behavior, by Von Neumann and Morgenstern, Smith’s Evolution and the Theory of Games, and John Nash’s important equilibrium papers[iv]. To get an historical sense of the theory’s development, I delved deeply into the early papers of Emile Borel[v], Denes König,[vi] and Ernst Zermelo. Zermelo’s proof that chess is determinate, which he accomplished using a form of backward induction amazed me in its depth[vii]. One of the great stories in the history of game theory, which I always loved, is about Guillaume Guilband, the French economist, who discovered that the first statement of the solution of a matrix game was actually written down in 1713. This happened by chance, when he purchased the treatise on probability, Essay d’Analyse sur les Jeux de Hazard, written by Pierre Redmond de Montmort, from one of the bookseller’s stalls that line the Seine in Paris[viii]. Reading these early books and papers, mostly in the middle of the night, sitting in 24 hour dinners in New York City, was for me a great time, and one of the most intellectually rewarding of my life. I was also spending a great deal of time at the Marshall Chess Club and so I thought, wrongly, that perhaps Zermelo would improve my chess game.

My current work on modeling land use and some of the environmental decisions made by Roman surveyors and farmers takes its real start however, from conversations that I had over the years with Waldo Tobler, Emeritus Professor of Geography at California, Santa Barbara. Tobler and I discussed many subjects, such as his bi-dimensional regression method[ix], a form of which I used in my Cartographica paper on Waldseemuller,[x] and more complex topics like the equivalence of Fred Bookstein’s thin-plate splines[xi] to Tobler’s second order tensor fields. I used Booksteins’ splines in a slightly modified form in my medieval chart models[xii].  This whole family of geo-rectification and image registration methods has modeling applications in historical geography and cartography that have yet to be fully exploited.

A few years ago I read Peter Gould’s paper on African farmers in General Systems Theory[xiii], a paper that would later lead me to his seminal work, Man against the Environment. I knew from our discussions that Tobler was close to Gould and that he was also playing around with some game theory during these years, and so naturally I asked Tobler about the paper.  Surprisingly, Tobler still had discussion notes on Game Theory, which he kindly sent to me along with copies of his own notebooks on the subject.  What was most impressive to me in all this material was not really Gould’s mathematics, but rather his vision of what game theory might be able to do in geographical sciences. Gould’s agricultural applications showed that even simple matrix games had a spatial component that few geographers had ever thought to utilize.

One of the things that Gould wrote and that struck me as profound was that , “we have all too often lacked, or failed to consider, conceptual frameworks of theory in which to examine man's relationship to his environment, the manner in which he weighs the alternatives presented, and the rationality of his choices once they have been made.” The rationality part instantly jumped out at me. As you may or may not know, the idea of rationality is an area of hot debate when it comes to questions of the Roman economy and law. There are many scholars, especially after Finley’s seminal book called The Ancient Economy, who believe that to consider Roman farmers and landowners as ‘rational’, in the sense of their maximizing the yield from their farms and thinking about market forces, is to project too much of a modern conception of a market economy onto the past. The book was important, as it was one of the first to draw from other disciplines besides ancient history, and to call for the mathematical modeling of ancient economies. Finley wrote that models were, “the [only] way to advance our understanding of the ancient economy” and against “the continual evocation of individual facts” that are accumulated by historians and classicists.

More recently, some scholars like Dennis Kehoe[xiv], Cynthia Jordan Bannon[xv] and D. W. Rathbone[xvi], using legal inscriptions, and the everyday account books of farms that survive only as papyrus fragments, have started to use economic models and things like the theory of the commons to talk about Roman markets and agricultural estate management. Each of them in their own way incorporates many of the terms and categories of game and decision theory in their analysis. Perhaps the best book that accepts and summarizes the presence of ‘rational’ actors in the Roman economy is a book by Paul Erdkamp, entitled, The Grain Market in the Roman Empire: a social, political and economic study. Erdkamp puts forward many different models in the book, while he also summarizes the relevant economic theory in his historical examinations and reconstructions. His is the sort of book that makes you anxious when you read it, as it reminds you of how much you do not know and how long it takes to make any real progress in this area.
For me the most direct influence on my thinking about a Roman market economy would be the papers of Peter Temin[xvii] and David Mattingly[xviii]. Temin is very interested in analytical models of the Roman economy and for him they serve two purposes. First, they provide “a simplified description of events that can be repeated and discussed.” Second they allow economists and economic historians “to test counter-factual propositions.” Models, if they are good, allow you to ask what would have happened if the institutions and economic variables were different. Using the categories of K. Polanyi[xix] Temin argues that the “economy of the early Roman Empire was a market economy.” His analysis shows that although there was not a single empire wide market for all goods, local markets were connected around the Mediterranean, and that the economic system of Rome can be looked at as an enormous conglomeration of interdependent small micro-markets.

Mattingly says much the same.  After conducting a survey of rock-cut olive oil presses in Roman North Africa, he concluded that oil production happened on a far larger scale than would be needed for the local population. Using this kind of archaeological evidence, and other hints found in the landscape itself, he determined that farmers and estate owners must have been heavily invested in an export market.  Mattingly’s study of the regional differences in Roman Oleoculture based on the spacing of trees is another area in which economic or game theoretic modeling might be useful[xx]. In any case, it now seems to be generally accepted among  most scholars who study the ancient economy that the large market in food, wine, olive oil, lumber, and bricks created by the growth of Rome to a population of about one-million people went some way to stimulating the formation of a diffuse market economy.

Scholars like Mattingly, teach you how important it is to get out into the field, whether it is to count trees or to just get a feel for the topography. It makes you realize that historical geography is not something you only do while sitting at a desk. This is one thing that younger geographers who use GIS from an early time in their education seem to forget and it is important to remind them of. Modeling is about the real world, whether it be historical or contemporary.  A bonus to all this is that since most of my fieldwork revolves around North Africa and Southern France I get to spend a great deal of time in the Mediterranean sun.

The effect of fieldwork on my research and on my life cannot be overestimated and it is only when I am traveling ‘off the map’ that I begin to think of myself as a geographer. I shall never forget walking through the ruins of places like Tipasa or coming across megaliths on the Island of Bornholm.  When I am back at home or at the Library, I often cannot help but reflect on the words that Albert Camus wrote in his essay Return to Tipasa.

Albert Camus at the Ruins of Tipasa

Camus recalls those moments by the sea, strolling amongst those silent walls and fallen stones using words in a way that I could possibly hope to and his writing about this place pulls a sense of meaning and emotion straight from my heart. He says that in Tipasa, “under the December light, as happens but once or twice in lives which ever after they consider themselves favored to the full, I found exactly what I had come seeking.” Camus in one of the most moving passages that I have read in any book, remembers what it is like to stand there, “From the forum strewn with olives could be see the village down below,” Camus continues, saying that, “No sound came from it [the village]; wisps of smoke rose in the lipid air. The sea likewise was silent as if smothered under the unbroken shower of dazzling cold light. From the Chenoua a distant cock’s crow alone celebrated the day’s fragile glory. In the direction of the ruins, as far as the eye could see, there was nothing but pock-marked stones and wormwood, trees and perfect columns in the transparence of the crystalline air.” These words still make me recall why it is I like to travel so much.

The Columns of Tipasa with Chenoua in the Background
There are plenty of places like this left in the world and there is also plenty of field work one can do as an historical geographer, especially if one is considering things like Roman land usage, surveying and cartography. This is simply because there are just so many unsolved problems and so much unexplored territory. One project that I am working on at the current time is an attempt to reconstruct the field layout patterns as described in the Albertini Tablets[xxi]. These are Roman real-estate documents dating from the fifth century AD which concern the sale of uncultivated lands somewhere along the Tunisian and Algerian border. The concept of uncultivated lands, or subseciva, is a point of Roman law that many of the surviving inscriptions from North Africa are concerned with. I became very interested in them after a visit to Tunisia, where I photographed the Henchir Mettich inscription in the Bardo Museum[xxii] and was able to look at many of the inscriptions myself. Epigraphy can greatly inform any study or model of Roman landuse, especially in North Africa, since so many inscriptions survive; most having to do with landownership and agricultural law. It is unfortunate however that so many of the studies fail to recognize the spatial component that these inscriptions contain.

Henchir Mettich Inscrption
The idea that history has a spatial component is of course not a new one, but the tools we have are. Even Marc Bloch, in his classic book, French Rural History: an Essay on its Basic Characteristics, understood how important the spatial distribution and differences in things like field patterns and old walls are. Bloch looked to examine rural and agrarian history in their full complexity and for him that required getting into the field and walking the old paths and fields. It meant looking at old maps and trying to come to terms with the daily routine of farming and the smells of hay and manure. Through his tramping the countryside and talking to actual framers he developed a kind sensitivity to his surroundings that few historians locked in Paris libraries could have. Many modern geographers still lack this sensitivity and as William Bunge once said “’Amateur’ field geographers can speak with authority about the clarifying effects on the mind of direct physical danger in the real world and there exists a terrible antagonism between field geographers and armchair academics. Not only do those in their armchairs think and write junk, obfuscation, obscurantism, and endlessly convoluted self-referral to their literature in windowless libraries, they do not care about the human condition.[xxiii]” It is Bunge’s words that I remember when I am out in the field in places like Tunisia and Algeria.

My own models of Roman cartography and land-use are simply extensions of the studies of others using a more multi-disciplinary framework[xxiv]. One group of Gould’s papers, from which my research on Roman agriculture certainly takes its start, was written in the 1960's. His papers, "Wheat on Kilimanjaro: the perception of choice in game and learning model frameworks," and "Man against His Environment: a game theoretic framework[xxv]", were among the first attempts to use the concepts of game theory and equilibrium to look into agricultural land use. These papers, and a few others, were also discussed and summarized in an early review article by David Harvey called, "Theoretical Concepts and the Analysis of Agricultural Land-Use Patterns in Geography[xxvi]."  It is in fact from Harvey’s paper and his later book, Explanation in Geography that my concept of geographic model derives[xxvii].

Harvey asserts, in his review article, that at the time he was writing, many geographers tended to ignore theoretical breakthroughs from other disciplines, mainly on the "grounds that they proved too abstract to help in the search for unique causes of specific events." To counter this he quotes from William Bunge, whose book Theoretical Geography transformed the discipline and opened up an analytical window for the field, suggesting a more theoretical and inherently mathematical approach to the study of geographical and spatial distributions. To many geographers Bunge’s book is revolutionary.  K.R. Cox went as far to say that it is “perhaps the seminal text of the spatial-quantitative revolution. Certainly in terms of laying out the philosophical presuppositions of that movement it had no peer.[xxviii]” My attraction to Bunge is not just mathematical. He was an outsider and a radical voice, both in his geography and in his politics, which is something that I am very sympathetic to.  To me Bunge’s book is the most original work of theoretical geography in the 20th century and I still mine it for inspiration[xxix].

Most of Harvey's paper is dedicated to outlining the requirements for a set of theoretical and conceptual elements to constitute a model in geography. A model, according to Harvey, requires a set of relationships to be established that somehow link the input, status and output variables in a specific way. This linkage must quantify the model mathematically in order for it to be tested. For Harvey, the relationships of the variables in the model can be of three distinctive types:

1. Deterministic relationships which specify cause and effect sequences.
2. Probabilistic relationships which specify the likelihood of a particular cause leading to a particular effect.
3. Functional relationships which specify how two variables are related or correlated without necessarily having any causal connection at all.

For agricultural models Harvey makes a distinction between two types of frameworks, one in which the underlying structure is normative and therefore, describes what ought to be under certain assumptions. The second, is descriptive, and describes what it is that exists. These distinctions are extremely important when we try to interpret game theoretical models, especially in something as difficult to conceptualize as the Roman economy.

In his early research Gould, using a normative game theoretic model, studied a group of African farmers around Kilimanjaro and analyzed how they decided what to plant in varying environmental conditions. Gould understood the patterns of land-use and the choices made by farmers are the result of decisions made either individually or collectively and that it might be useful to try to model those decisions in a game theoretical framework. In Gould's models the environment is one player and the farmer is another. Each of the players is faced with a number of different strategies the solution of which is the game's equilibrium. Using simple matrix games he was able to construct cartographic representations of various equilibrium alternatives that could be compared to what was in the fields.

Gould certainly recognized the limited applicability of his models and that they were highly general. This is not meant as a criticism. Early on in the history of studying ‘primitive’ economies scholars knew that there was an inverse relationship between the general applicability of a model and its ability to accommodate local empirical reality. There are just too many variables for it to be otherwise. Max Weber, in his Economy and Society said as much when he wrote, as early as 1922, that, “The more sharply and precisely the ideal model has been constructed, thus the more abstract and unrealistic in this sense it is, the better it is able to perform its functions in formulating terminology, classifications and hypothesis.[xxx]” The best models therefore, work to highlight certain features of reality while suppressing others, thus allowing the simplification of empirical data.
Importantly, Gould realized that the game theory of his time was still algorithmically primitive and that his results determined neither how the farmers actually behaved nor how they should have behaved in an absolute sense, but rather how they should behave if they want to achieve particular results. In strategic games, such as the one Gould proposed in his papers, Nash equilibria are a set of actions amongst the players that lead to a steady state. It is a position in the game in which each player holds the correct expectation about the other player and behaves and acts rationally according to his choices. 

The concept of equilibrium is not so straightforward here as one might think, and it can be interpreted in several ways. For example, when we say that a physical system is in equilibrium we might mean that it is in a stable state, one in which all the causal forces internal to the system are in balance. This is the traditional economic meaning of equilibria. The variables are dynamic however, and the balance between them that makes up the equilibrium can be thought of as networks of mutually constraining relations. Equilibria can then be considered as endogenously stable states of the model. Some scholars however, interpret game theoretic equilibria as being explanatory of the process of strategic reasoning alone. For them a solution must be an outcome that a rational agent would predict using the mechanisms of rational computation alone. The interpretation of equilibrium states is still a matter of discussion in the literature of game theory and has interesting philosophical implications to how we view and interpret what the models tell us outside of their mathematical formalism[xxxi].  Several commentators have described game theoretic formalism as logic, in that it simply provides a structure that is not really empirical. In other words it simply provides a formal structure with which we can talk about strategies; syntax rather than semantics. I like this interpretation, mostly because one of the biggest influences on my more abstract thought comes from the work of Ludwig Wittgenstein whose Tractatus Logico-Philosophicus highlights the difference between form and meaning[xxxii].

The current models I am working with are of course much more complex than anything Gould or Harvey could have considered, as they lacked both the mathematics and the computing power. New techniques like quantal response functions, which allow us to look at probable actions, are much more powerful and yield much more interesting results. They were first introduced by McKelvey and Palfrey in the late 1990s and considered mathematically for the possibility that the players will make mistakes and therefore they give more realistic results than anything Gould could have imagined; at least we hope they do[xxxiii].

Specifically, the models I am working on concern the complicated problem of market integration in the growing of grain and olives in North Africa, from the height of the empire until the coming of the Vandals around 500 AD. Market integration in the sense I mean it here can be defined as the degree to which a market succeeds in adjusting to shocks in supply and in harvest yields. This problem in the grain and olive markets in North Africa is dominated by a small number of important facts, some geographical, and others having to do the population’s consumption patterns.

Because grain is only harvested once a year, but is consumed throughout, a price cycle is established such as the one shown in the graph above. Right after the harvest there is a great deal of grain available for purchase at a cheap price. As the year goes on however, the price goes up, as the supply of grain diminishes. The cycle here is not a normal supply and demand relationship because the demand for grain is inelastic and does not diminish as the price increases. This inelasticity in demand is brought about by the fact that the population has no real alternatives as far as food is concerned and must consume the grain from the harvest at least until some critical price is reached. This is further complicated because most of the sharecroppers and small local framers did not have the means to take advantage of the cheap prices by buying in bulk right after the harvest and therefore had to deal directly with the price fluctuations. We read in Philostratus and Columella,[xxxiv] for example, that when these critical prices are reached, or during difficult harvest years, many farmers would begin to consume fodder and animal feeds in order to ward off starvation. Because ancient yields of grain and other crops were low and much of the production was in the hands of small landholders, the market supply fluctuated more heavily than the harvests themselves would lead one to believe.
One of the most critical characteristics of pre-industrial food prices from ancient to early modern times was their inherent volatility. Even a minor disruption of supply could cause prices to multiply rapidly. The data on actual prices of grain during the Roman Empire is very limited but the strategies that where employed by farmers and grain dealers are few, and can therefore be modeled using game theoretical techniques.
Game theory, as has been mentioned before, deals with decision and strategy. One of the major decisions facing a farmer was whether or not to hold back on the sale of some grain to take advantage of the later price rises. Another strategy would be to hold grain over to the next year in order to guard against a crop failure. These strategies are complicated and not straightforward in the case of smallholders, who may have to pay their rent in kind, or in money, directly after the harvest, and may therefore be forced to dump their grain stocks on an already oversupplied and low price market.
Geographical and ecological factors are also critical[xxxv]. The vagaries of weather in North Africa can create large dynamic fluctuations in yields from year to year and season to season. My models deal with ‘dry farming’ in an area that only typically sees around 400 mm of rain annually and so environment is a big concern for all of these small land holders. These environmental fluctuations can be quite local and concern only microregions, thereby leading to different compensatory strategies[xxxvi] that are unique to the inelastic demand for food in Roman markets. The two most important of these from a modeling perspective are: first, transporting surpluses to regions experiencing shortage; second, storing surpluses until the next harvest year, a process known as carry-over. Carry-over is always a risky venture as the next harvest may yield higher surpluses and depress prices further, or the stored grain may spoil or be damaged by adverse weather. All of these factors play into the decisions made by farmers and affect their livelihoods in profound ways.
Over the last few years many other interesting models that are directly related to the environment and land use have developed[xxxvii]. I simply mention Herbert Gintis’ game theoretic foundation for the evolution of private property rights and territoriality as a good example[xxxviii]. Gintis attempts to unify the various strains of environmental, historical and legal issues that go into the possession of territory and land and in doing so provides a game theoretic framework to discuss the underlying philosophical issues associated with the concept of territory[xxxix]. Like so many game theoretical models, these more applied versions, simply give us a framework for thinking about intractable problems.

The most interesting development in the last few years and that I think will be very important to the study of the spatial factors in the Roman economy is called Agent-Based Computational Economics. This form of modeling studies the structure of economies form the ground up, as dynamic systems of interacting agents. Doing this creates interesting overlaps between network theory, which is so much in vogue now, and the paradigm of complex adaptive systems. In agent-based models, the "agents" are "computational objects modeled as interacting according to rules" over space and time, not real people. The rules are formulated to model behavior and social interactions based on incentives and information. Agent based models have applications across the Roman economy and are quite useful in looking at the spatial networks of trade, land use and environment at the various scales in which they develop.  Leigh Tesfatsion, from the University of Iowa, provides perhaps the best introduction to these models in her seminal "Agent-based Computational Economics: Modeling Economies as Complex Adaptive Systems.[xl]

Agent-based computational economics is similar to, and overlaps with game theory as an agent-based method for modeling social interactions. But there are some major differences from the standard methods of game theory which make agent-based methods more useful for looking at my primary interest, Roman land use. For example in these methods the events bein modeled are driven solely by initial conditions, whether or not equilibria exist or are computationally tractable. This means that the agents, whatever they may be, are historically locked into a set of actions, which have been chosen from a larger set of possible actions by their previous decisions. For small, local networks of interactions, like we see developing in the spatial constrained Roman economy, this gives us a more realistic feel for what the actor’s possibilities were and how local market dynamics influenced their decisions.

For me the power of models in historical geography derives from the fact that you can look at many different scenarios and compare them with the little actual historical data you have. I would never assert that what I am doing actually gives me any definitive answers on what decisions land use decision Roman farmers made or how they planted, rather they show me what possibilities there were and how to rank them. This goes for any of the mathematical models that I have attempted from Waldseemüller onwards.  Like Gould, who wrote that his papers were, “very much written in the spirit of speculation, to provide a framework for examining some traditional but still intractable geographical materials,” my investigations are qualitative experiments, helping to arrange questions about an historical and spatial pattern of land use. Most importantly however, these models greatly inform my thinking about the historical perception of space in its most empirical form, and since I do not have the disciplinary constraints on my ideas that an economic or ancient historian might, I can push the limits of the models for purely theoretical and curiosity reasons. 

My hope is that these methods will yield an 'experimental' historical geography; an acceptance of simulation as a method in historical studies. These game theory models really do have the potential to shed light on the decision alternatives that face farmers and estate owners acting within agrarian or developing economies. They give us a glimpse into historically how farmers might have interacted with their environment on a mainly cognitive level, allowing us to consider the choices they had and how their decisions affected, changed and formed the landscape around them.  To me, and to other geographers before me, like Gould and Harvey, this remains an important and central geographical question.

[i] Harold Kuhn, Lectures on the Theory of Games, (Princeton: Princeton University Press, 1952)
[ii] One of the best modern introductions to matrix games can be found in Julio Gonzalez-Diaz, An Introductory Course on the Mathematical Theory of Games, Graduate Texts in Mathematics, Vol. 115 (Providence: American Mathematical Society, 2010). In this work one can find mathematical introductions to the important algorithms for the solution of matrix games by M. Krein and D. Milman, “On the Extreme Points of Regularly Convex Sets,” Studia Mathematica 9 (1940): 133-138 and the seminal paper by L.S. Shapley and R. Snow, “Basic Solutions of Discrete Games,” Annals of Mathematical Studies 24 (1950): 27-35.
[iii] Drew Fudenberg, Game Theory (Cambridge MA.: MIT Press, 1991).
[iv] For reprints of the important papers by Nash and other important game theorists see Harold Kuhn, ed. Classics in Game Theory, (Princeton: Princeton University Press, 362).
[v] Emile Borel, Le Hasard (Paris: Alcan, 1914). For more on Borel and the early development of Game Theory see Robert Leonard, Von Neumann, Morgenstern and the Creation of Game Theory: from chess to social science, 1900-1960 (Cambridge: Cambridge University Press, 2010) 57-62
[vi] Denes Konig. “Uber eine Schlussweise aus dem Endlichen ins Unendliche,” Acta Litterarum ac Scientiarum III (1927) 121-130.
[vii] Ernst Zermelo, “Uber eine Anwendung der Mengenlehre auf die Theorie des Schachspiels”, Proc.Fifth Congress of Mathematicians, (Cambridge: Cambridge University Press, 1913) 501-507. There is a great deal of discussion about what it is that Zermelo actually proved. It is generally accepted that his 1913 paper, “On the Application of Set Theory to the Theory of the Game of Chess,” gives the first formal proof in the Theory of Games. What that proof shows is however a matter of some controversy. For more on this see Ulrich Schwalbe and Paul Walker, “Zermelo and the Early History of Game Theory,” Games and Economic Behavior 34 (2001) 123-137. .
[viii]J. Waldgrave, “Minimax Solution of a 2-person, zero-sum game, reported in a letter from P. Montmort to N. Bernouilli, transl. and with commentary by W. Kuhn in W.J. Baumol and S. Goldfeld (eds.), Precursors of Mathematical Economics  (London: London School of Economics, 1968) 3-9
[ix] Waldo Tobler, Bi-dimensional Regression, Geographical Analysis 26 (1994): 187-212.
[x] John Hessler, “Warping Waldseemüller: a Phenomenological and Computational Study of the 1507 World Map,” Cartographica 41 (2006): 101-114.
[xi] Fred L. Bookstein, “Principal Warps: Thin-plate splines and decomposition of deformations,” IEEE Transactions on Pattern Analysis and Machine Intelligence 11 (1989): 567-585.
[xii] For more on these models see the Washington Post article and John Hessler, “Bi-dimensional Regression Re-visited: studies in the geometry of the medieval Portolan chart,” and Random Walks Across the Atlantic: Stochastic Processes and the Geometry of the Early Renaissance Sailing Chart,” .
[xiii] Peter Gould, “Wheat on Kilimanjaro: The Perception of Choice within Game and Learning Model Frameworks, General Systems Theory 10 (1965): 157-166.
[xiv] Dennis P. Kehoe, The Economics of Agriculture on Roman Imperial Estates in North Africa, (Göttingen: Vandenhoeck & Ruprecht, 1988) and Law and the Rural Economy in the Roman Empire (Ann Arbor: University of Michigan Press, 2007)
[xv] Cynthia J. Bannon, Gardens and Neighbors: private water rights in Roman Italy (Ann Arbor: University of Michigan Press, 2009)
[xvi] Dominic Rathbone, Economic Rationalism and rural society in third century Egypt: the Heroninos archive and the Appianus estate, (New York: Cambridge University Press, 1991)
[xvii] Peter Temin, “A Market Economy in the Early Roman Empire,” Journal of Roman Studies 91 (2001) 169-181
[xviii] David J. Mattingly, “Olive Cultivation and the Albertini Tablets,” L’Africa-Romana 6 (1989): 403-415.
[xix] K. Polanyi, The Great Transformation (New York: Farrar & Rinehart, 1944)
[xx] David J. Mattingly, “Regional Variation in Roman Oleoculture: Some Problems of Comparability,” in Landuse in the Roman Empire, edited by Jesper Carlsen (Rome: L’Erma Di Bretschneider, 1994) 91-106
[xxi] Christian Courtois, Louis Leschi, Tablettes Albertini (Paris: Aux Editions, 1952). For the geographic context of the tablets see R. Bruce Hitcher, “Historical Text and Archaeological Context in Roman North Africa: The Albertini Tablets and the Kasserine Survey,” in Methods in the Mediterranean edited by David B. Small (New York: E.J. Brill, 1995)
[xxii] John Hessler, A Cartographic Commentary on the Henchir Mettich Inscription from Central Tunisia: Roman Surveying in the Medjerda Valley,
[xxiii] Bunge always thought geography was a field subject and that its roots lie in trying to say something of the human condition.  See his “Geography is a Field Subject”, Area 15 (1983) 208-210.
[xxiv] John Hessler, Fourier Finds Caesar: A Study in the Physical Evidence of Roman Agarian Law and Cartography Using Period Functions and Remote Sensing,
[xxv] Peter Gould, “Man against His Environment: A Game Theoretic Framework,” Annals of the Association of American Geographers 53 (1963):290-297.
[xxvi] David Harvey, “Theoretical Concepts and the Analysis of Agricultural Land-Use Patterns in Geography,” Annals of the Association of American Geographers 56 (1966): 361-374.
[xxvii] David Harvey, Explanation in Geography (London: Edward Arnold, 1969). There are other books that cover the same material and are especially useful for gaining historical perspective such as R.J. Chorley and P. Haggett, editors, Models in Geography (London: Methuen, 1967).
[xxviii] K.R. Cox, “Classics in human geography re-visited: Bunge, W., Theoretical Geography. Commentary 1, Progress in Human Geography 25 (2001) 71-73.
[xxix] William Bunge, Theoretical Geography (Lund: Royal University Department of Geography, 1966). For a modern appreciation of this text one should see Michael Goodchild’s, William Bunge’s Theoretical Geography at
[xxx] Max Weber, Economy and Society, (Berkeley: University of California Press, 1978) 21
[xxxi] Ariel Rubenstein, “Comments on the Interpretation of Game Theory,” Ecometrica 59 (1991): 909-924 and R. Aumann, “What is Game Theory Trying to Accomplish?,” in Frontiers in Economics, ed. by K.J. Arrow and S. Honkapohja. (Oxford: Blackwell Publishing, 1987)
[xxxii] John Hessler, Finding the Antipodes: Mathematical Constructivism and the Changing Logic of Cartographic Objects, 1960-1975 (2011)
[xxxiii] Richard McKelvey and Thomas Palfrey, “Quantal Response Equilibria for Normal Form Games,” Games and Economic Behavior 10 (1995): 6-38 and Quantal Response Functions for Extensive Form Games,: Experimental Economics 1 (1998): 9-41.
[xxxiv] Columella, On Agriculture, 2.9.14
[xxxv] For a good introduction to the interaction between human geography and the fluctuation of ecological factors see Karl S. Zimmerer, “Human Geography and the ‘New Ecology’: The Prospect and Promise of Integration,” Annals of the American Association of Geographers 84 (1994) 108-125
[xxxvi] For more of microregions and Mediterranean ecology see Peregrine Horden and Nicholas Prucell, The Corrupting Sea: a study in Mediterranean History (Oxford: Blackwell, 2000)
[xxxvii] Nick Hanley and Henk Folmer editors, Game Theory and the environment (Northhampton, MA: Edward Elgar, 1998)
[xxxviii] Herbert Gintis, “The Evolution of Private Property,” Journal of Economic Behavior and Organization 64 (2007) 1-16.
[xxxix] For more on this see my presentation at the 2010 Annual Conference of the Association of American Geographers, Economic Foundations of Roman Cartography: Bounded Rationality, Territory and Epigraphy, 100-300 AD.
[xl]Leigh Tesfatsion, "Agent-based Computational Economics: Modeling Economies as Complex Adaptive Systems," Information Sciences, 149(2003) : 262-268.