Epigraphy, Servitudes, and Game Theory
...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....
---Peter Gould
In the study of Roman agricultural patterns it is important to have a conceptual framework in which to place the fragmentary information and evidence that is available from epigraphy, Roman law, and landscape archaeology. For the past few months I have been experimenting with Game Theoretical Models and the concept of Nash Equilibrium trying to see what type of land use models would arise.
The basis of game theory was first laid down in the late 1940's by the mathematician John von Neumann and the economist Oskar Mogenstern in their now classic book
the Theory of Games and Economic Behavior. In the book von Neumann gives the proof of the Minimax Theorem, which is central to game theoretic reasoning and that he first approached in 1928. In the 1944 book, von Neumann placed the theorem within the context of linear inequalities and the theory of convexity, which was later updated with more formal proofs of equilibrium states by John Forbes Nash.My current work on modeling land use and some of the environmental decisions made by Roman farmers takes its real start however, from a conversation that I had with Waldo Tobler, Emeritus Professor of Geography at California, Santa Barbara, about 8 years ago. I had just read Peter Gould’s paper on African farmers in General Systems Theory, a paper that would later lead me to his seminal work, Man against the Environment. I knew that Tobler was close to Gould and that he was also playing around with some game theory during these years, and so I asked Tobler about the paper. What was most impressive to me in all this was not really Gould’s mathematics, but rather his vision of what game theory might be able to do in geographical sciences, that even simple matrix games had a spatial component that few geographers had 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. 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. More recently however, some scholars like Dennis Kehoe, Cynthia Jordan Bannon and D. W. Rathbone, using legal inscriptions and the everyday account books of farms that survive 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, and summarizes the economic theory in his historical examinations and reconstructions. His is the sort of book that makes you anxious when you read it, as it gives you a good idea of how much you do not know and how long it takes to make any real progress in this area.
My own models are simply extensions of this kind of work. One group of Gould’s papers, from which my research certainly takes its inspiration, was written by him 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", were among the first attempts to use the concepts of game theory and Nash equilibrium to look into agricultural land use. These papers, and a few others, were also discussed in an early review article on these methods written by David Harvey, "Theoretical Concepts and the Analysis of Agricultural Land-Use Patterns in Geography." It is in fact from Harvey’s book, Models in Geography that my concept of geographic model derives.
Harvey asserts, in his review article on agricultural land use, 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 geography 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 me Bunge’s book is the most important work of geography in the 20th century and I still mine it for inspiration.
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.In his early research, Gould studied a group of African farmers around Kilimanjaro using decision theory to analyze 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.
Importantly, Gould realized that the game theory of the 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 payers 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. Gould uses the simple graphical solutions to the matrix games he creates which I found so attractive early on in Harold Kuhn’s lectures. For more on Kuhn and John Nash watch the video of a recent seminar they gave together at Princeton, here.
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 meaning 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. (For more on the interpretation of game theoretical results see Ariel Rubinstein's seminal paper Comments on the Interpretation of Game Theory or the Philosophy of Game Theory by Grune-Yanoff.
The current models I am working with are of course much more complex than anything Gould could have considered, as he 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 imagined, at least we hope they do.
The power of models in historical geography is 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 Roman farmers made or how they planted, rather they show me what possibilities there were and how to rank them. Most importantly however, they greatly inform my thinking about the Roman economy in its most empirical form, and since I do not have the disciplinary constraints on my ideas that an economic 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 simulations have the potential to shed light on the decision alternatives that face farmers and estate owners acting within primitive or developing economies. They give us a glimpse into how historically farmers interacted with their environment on a mainly cognitive level, allowing us to consider the choices they made and how their decisions affected the landscape around them. This to me, and to other geographers before me, like Gould and Harvey, is certainly a central geographical question.
For those interested I am using a software package that can calculate the Nash equilibria for games with large numbers of players, or in this case environmental variables called GAMBIT.
http://www.gambit-project.org/doc/index.html
It is an open source program and you can have a great deal of fun experimenting with variables and how they change the equilibrium outcome.
If we graph the error in positional fixation geographically using the difference in actual (modern) versus measured position we get a figure of the type shown below.
In the case of the Random walk, the error pattern that emerges is the result of the accumulation of errors that are independent of each other, so-called independent increments. The idea of independent increments is applicable in the case of voyages where dead-reckoning was employed because the error contributes cumulatively to the positional uncertainty and the errors are not systematic but rather random with many causes.
We can therefore express the error accumulation by the above equation. Each time a positional measurement is made it increases the overall error by a small increment. Assuming a Gaussian distribution of the error leads to the summation,
The variance of this summation is the quantity that we are looking to model, as it will allow us to correlate the root mean square error for positional measurements taken from the log books, and that which would have been incorporated into the charts themselves. If we solve for the variance we can see that it is increasing with the length of the voyage, as we would of course expect from serially correlated errors.
My research will consist in looking through the vast and very understudied collections of inscriptions from museums in Libya, Tunisia, Algeria, and other collections, along with writing geographical commentaries on some of the more famous inscriptions like Henchir Mettich, (pictured above in a photo I took in the storeroom of the Bardo in Tunis), Lamasba (Ain Merawa) and Aga Bey Koyu from the Usak Museum, along with many more.
