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Wolfgang Balzer, Karl R. Brendel and Solveig Hofmann (2001)

Bad Arguments in the Comparison of Game Theory and Simulation in Social Studies

Journal of Artificial Societies and Social Simulation vol. 4, no. 2,
<http://jasss.soc.surrey.ac.uk/4/2/1.html>

To cite articles published in the Journal of Artificial Societies and Social Simulation, please reference the above information and include paragraph numbers if necessary

Received: 17-Jul-00      Accepted: 01-Feb-01      Published: 31-Mar-01


* Abstract

The aim of this note is to clarify and to correct some arguments which are used in the debate about the comparison of discrete social simulation with other methodologies used in the study of social phenomena, notably those of game theory. Though part of what will be said also applies to non-discrete simulation, the arguments are investigated only as far as the discrete case is concerned. The main claims against each of both scientific approaches are considered in particular, i.e. "impossibility" of game theory and "unsoundness" of simulation studies. Regarding the latter, arguments are presented that items occurring in simulation studies correspond to the formal constituents of a scientific theory, and thus a comparison of both approaches on the same level is justified. The question whether a superiority of one of the two approaches can be stated is illuminated in the light of four dimensions: empirical adequacy, theoretical fruitfulness, social relevance, and simplicity. This leads to the conclusion that both claims are unjustified and should be avoided in the debate about the role and merits of social simulation.

Keywords:
Social simulation, game theory, discrete event simulation, model theory, confirmation, impossibility theorem

* Background

1.1
In order to understand the current debate on the comparison of simulation and other, different methods it may be helpful to recall the background. Discrete event simulation is a rather recent approach. Though the first programs using such techniques for the study of social events[1] occurred three decades ago, it seems fair to say that this approach received real impetus only in the last decade. This is documented by a great number of simulation studies of social phenomena over a wide range of rather different applications, ranging from anthropology, through economics to sociology.[2]

1.2
One feature accompanying that development was that researchers who started work with discrete simulation often did not use the traditionally established formal methods and models. The shift towards discrete simulation led to lesser use of other types of simulation, and to a withdrawal of intellectual labour from established mathematical fields, most notably from game theory. Game theory had established itself as the dominant approach in the mathematical modelling of social phenomena, and it had even succeeded to 'take over' parts of the previously dominant economic paradigm given by general equilibrium theory. Now a new, rival paradigm had occurred on the scene: discrete simulation. The fact that an increasing number of interesting studies were made in the new field, withdrawing manpower and interest from game theory, certainly is reason enough for a strong debate about the evaluation of the new approach and an evaluation of both against each other - which one is better and/or more promising - and some struggle for the interest of young researchers who can choose between the alternatives.

1.3
In order to avoid too strong a commitment to either of the two sides of this debate, which now in fact is taking place, let me point out the following. First, such debates are a common phenomenon in the history and sociology of science. Under the influence of the work of Feyerabend (1962) and Thomas Kuhn (1962) they have become widely recognized under the label of incommensurability. Second, and contrary to the image that has built up in the public debate, incommensurability and the kind of debates for priority we have before us, do not imply irrationality of science or the relativization of scientific truth. It has been argued and shown by examples that in the domain of physics, from which the original examples of incommensurability were taken, incommensurability can go, and usually does go, together with some kind of comparability. Often a relation of approximative reduction can be established between the two rival approaches.[3] Third, it is a bit one-sided to present the two approaches as an alternative. Another less aggressive vocabulary would describe the approaches as complementary to each other. Game theorists have begun to use simulation to supplement their paper-and- pencil studies, and some simulators are backing their programs by game theoretic models. However, in the light of the struggle for scarce resources (manpower and funds) this harmonious description cannot completely cover up a basic antagonism at a social level. Finally, it should be pointed out that neither of the approaches can, nor will, 'swallow' the other. Therefore, if complete victory in the battle for priority would mean to acquire command over the rival approach, victory is almost impossible.

1.4
As 'usual' in such debates, the arguments used by the opponents are not always on the highest scientific level. 'Usually', there is no knock-down argument for showing the priority of one side over the other, so a major goal of contributions to the debate is to convince the reader or listener, in a way that may well go beyond austere scientific argument and include rhetorics and other means. In the debate among game theorists and simulators there are two such arguments which deserve closer scrutiny.

* 'Impossibilities' for Game Theory

2.1
A seductive way of arguing against the rival approach is in terms of impossibility claims. If such a claim can be justified this means that the opponent program can not possibly achieve certain kinds of results and this means a hard blow. Two kinds of impossibility claims are put forward by simulators in the current debate.

2.2
First, simulators claim that the systems and phenomena they are after are too complex to be treated only in terms of mathematical or game theoretic models.[4] More sharply: it is impossible to treat such complex models adequately only in terms of game theory. Two meanings of this claim should be distinguished. First, it may be meant as an informal, practical claim: "no researcher with medium capabilities will be able presently to provide adequate, purely game theoretic models for complex social phenomena". Though this might be further detailed, it is clear that such a practical claim has political and rhetorical overtones. Such a claim may have a decisive influence on a young researcher's decision of which way to go, but it is too sweepy for a comparison of the two approaches in the spirit of scientific argument.

2.3
On second reading, the claim may be understood as a hard fact that can be scientifically justified. There are hard facts of such a kind in science: think of the impossibility of analytically solving the n-body problem for n > 3 in the natural sciences, or Arrow's impossibility theorem in the social sciences. For the present discussion two points are relevant. The first is that such impossibility results are very rare, and usually quite difficult to prove. Second, and more importantly, these impossibility results are proved (or justified) by the practitioners of the very theory which they limit. In the case at hand this would mean that an impossibility theorem saying that game theory cannot be applied to certain cases must have been proved by game theorist. To the best of my knowledge, there are no such proofs, and we conclude that in its hard meaning the impossibility claim is unfounded.

2.4
A second, similar impossibility claim made by simulators is that the phenomena they study by simulation are chaotic or non-linear and therefore are not amenable to mathematical modelling like differential equations or 'linear methods' of whatever kind. As in the first case, we have two different claims. The practical one, which is basically political and not very relevant for the present discussion, and a systematic one which turns out to be unfounded. There are no general impossibility theorems in mathematics showing that certain 'complex' systems cannot be modelled by differential equations (whether linear or not). Impossibility theorems are only available for concrete cases, stating that a certain concrete equation has no solution that satisfies some special requirements. Such theorems, however, are far from justifying a general impossibility claim of the form considered here.

* 'Unsoundness' of Simulation

3.1
In the other direction a negative claim is implicitly made by game theorists when they require that simulation studies should be backed by the knowledge of an underlying theory, which usually would be game theory.[5] Such claims are intriguing textbook instances of rhetorics. They seduce the reader into accepting the presupposition on which they rest. The presupposition is that the simulation studies of which the requirement is made are not backed by the knowledge of an underlying theory. This presupposition is called up in the reader because if the presupposition was not satisfied the statement would be rather trivial, saying something like "a study which is backed by the knowledge of an underlying theory should be backed by the knowledge of an underlying theory". A further implication which seems to be intended in such arguments is that what is not backed by the knowledge of an underlying theory is not properly scientific. In view of this, I will call the claim implicit in the presupposition an unsoundness claim. The presupposition amounts to claiming that the attacked approach is not fully scientifically respectable - it lacks knowledge of an underlying theory - and is unsound in this sense.

3.2
Rhetorics aside, the question is whether the presupposition that certain simulation studies are in this sense unsound holds water. This is by no means an easy question, and its answer requires some preparation.

3.3
Let me begin by recalling some elementary notions, which often seem to be used in different meanings in such discussions. In science we are faced with basic notions such as laws, theories and models. Laws are statements that summarise relations among observed variables, objects, or events. Theories represent hypotheses about the structures or processes in an environment. They differ from laws in making reference to unobservable objects or mechanisms. Models are descriptions of the environmental conditions, both overt and hidden, for an experimental or observational setting. Thus, a model is required to indicate the manner in which a law or theory applies to a particular situation.

3.4
In the non-statement view of scientific theories, a hypothesis H is syntactically a universal (or still more complex) sentence that has the property of a cluster law: it contains two or more relational terms, which cannot be conjunctively separated.[6] An initial condition I for H is an atomic sentence or a conjunction of such sentences formulated in the vocabulary of H.[7] A model of H, on the other hand, in the understanding shared at least by all formally educated scientists, is a (set theoretic) structure in which the terms of H can be, and are, interpreted, and in which H comes out as true under such interpretation. One hypothesis H in this way corresponds to a whole (proper) class of models M(H) (of H). Similarly, we may identify an initial condition I for H with the class M(I) of all models of I.[8] These model classes provide a semantics for the sentences. On the semantic level usually no distinction is made between models corresponding to a hypothesis and models that would correspond to an initial condition. Usually, both these features are considered in one single model. This can be done because among all models of a hypothesis H there are those which in addition are models of the initial condition I, i.e. elements of M(H) M(I). In other words, one single model can satisfy both the hypothesis and the initial condition.

3.5
Going through the literature we find that all notions of a scientific theory which have been discussed are such that a scientific theory should consist at least of a hypothesis H and some set of data from which initial conditions Ix for the different models x of H can be picked. Equivalently, a theory must consist at least of two classes of models (representing a hypothesis and data).[9] A scientific theory contains further items, most notably some specification of intended systems, i.e. systems (whether real or abstract) to whose study H should contribute and from which the data have been, or can be, obtained. In addition to this condition on the formal parts of a theory, there is an essential pragmatic requirement that each scientific theory has to satisfy. It is required that there should be a group of practitioners of the theory, of people who intend to, and try to, apply the theory. Without the latter condition we would have to consider a host of more or less bizarre 'theories' entertained by single persons. This condition has to be read in a diachronic way. At least after some initial phase in which its fate is still under doubt a proper scientific theory must be acknowledged by a group of practitioners. The doubts of potential adherents may be overcome either by empirical or theoretical successes or by justified claims about the future of such successes, or by both. A new theory thus may have in the beginning of 'its life' a hard time in which its status as a scientific theory remains open. This observation can be corroborated by case studies from the history of science.

3.6
In the light of these notions, we may ask whether simulation studies satisfy the requirements for scientific theories. If not, then game theorists' presupposition of unsoundness would be satisfied. Simulation studies would not be, or correspond to, scientific theories, and would indeed need sound theoretical backing (in terms of, say, game theory).

3.7
A simulation study basically consists of several runs of a given program. At the level of program code, we should distinguish proper rules and facts. Facts are atomic sentences, whereas proper rules are represented by formulas that are more complex. This distinction corresponds to that between data and hypothesis in a theory. Computer scientists often ignore this distinction, and just speak of 'data bases' including proper rules. However, as the distinction between hypotheses and initial condition(s) is a purely syntactical and precise one, it may be applied in the realm of programs without any ambiguity, even if computer applicants themselves do not (yet) care about it. We thus can speak of proper rules in a program, i.e. rules which are more than a mere conjunction of atomic sentences.

3.8
A computer program is a system of rules that cause a computer to change its initial state and enter a successor state. The successor state becomes the new initial state and is changed according to the rules, and so on. A sequence of such state changes beginning at a primary state and ending in a final state which can not be changed by the defined rules any more is called a program run. The memory contents, or the data a user has entered at the beginning of a program run is called input. In contrast, the memory contents or the data read by a user at the end of a program run is called output. Input and output are in this view no rules and therefore not part of the program itself. The rules are always flexible enough to be applied to a class of inputs. The same rule system can generate many (in principle indefinitely many) different program runs, depending on the input. A program can therefore be seen as a rule system in which every element of a class of inputs can be transformed in a correlating output. In case the rules are formulated in a standardised format they can be transferred to other computer systems and can be used in different application domains. To come back to simulation, a run in a computer simulation corresponds to a program run described above.[10] Some runs may be repetitions beginning with the same initial data in order to study the probabilistic behaviour of the program (if it contains probabilistic elements), in other runs the initial data - including parameters - may be varied in order to study the range of applicability and the robustness of the program. It is difficult to draw a sharp boundary between these two types of 'repetitions'.

3.9
There is an obvious correspondence between the items occurring in a simulation study and the formal constituents of a theory. Seen from the simulation side, the proper program rules correspond to a hypothesis, each run corresponds to a model, and the initial data in one run correspond to a set of initial data which are satisfied in a model. Seen from the theory the hypothesis corresponds to the proper program rules as well as to all possible runs (with all possible initial conditions); the data of the theory correspond to a class of initial data which may be used in simulation runs, a model corresponds to a simulation run, and a model satisfying initial condition(s) corresponds to a simulation run beginning with just these initial condition(s).

3.10
By means of these correspondences, we may transfer the requirements for scientific theories to the domain of simulation as follows. The counterpart of a scientific theory in the domain of computer simulation consists of:
  1. a set of proper program rules (a 'proper program') which corresponds to the theory's hypothesis, and
  2. a set of sets of initial data which correspond to the theory's data.[11]

3.11
Data files often give the initial data on the computer side. When the program is started the data from one data file is read and used as input. A program plus initial data may be called a counterpart of a scientific theory, and thus a 'scientific theory' in a derived sense, if it satisfies the other pragmatic requirements noted above. On this account a simulation program gives rise to the formal items which a 'sound' scientific theory needs. It has a constituent corresponding to a hypothesis or a class of models: the set of proper program rules, and it has a constituent corresponding to the theory's data: the files containing the initial data.

3.12
On the formal side, therefore, simulation studies satisfy the requirements for scientific soundness. Unsoundness claims, if correct, must be justified by reference to the pragmatic components introduced above. This leads to the questions whether there is a set of intended systems and a group of practitioners in a simulation study.

3.13
Intended systems need not be concrete, real systems; they may be abstract, conceptual systems as well. We cannot insist that intended systems should be concrete, real systems because many theories in social science do not have such intended systems either. Think of the theory of international trade for 2 countries, 2 goods and 2 factors, or of the spatial theory of voting for cases with n parties where n is sufficiently large. But if in the abstract case conceptual systems are admitted as intended systems, then most simulation studies do have sets of intended systems which they intend to study by means of simulation. So the ultimate assessment of simulation studies for scientific soundness reduces to the existence of a group of practitioners.

3.14
For many simulation studies, there is no such group. There is just one person who has written the program, produced or collected the data, and run the simulations. However, one-person groups do not count as groups here. So should we say in these cases that such one-person simulation studies do not have all the necessary features to pass for a sound, scientific activity?

3.15
Again, the answer is by no means easy. The situation is complicated by the diachronic nature of the assessment. As already mentioned, a scientific theory in the beginning of its 'life' may be in the same situation as a one-man simulation study, i.e. lacking a group of practitioners. There are cases of theories that for some initial period were acknowledged and practised by just one person, but after some time were acknowledged by others and thereby became full-fledged scientific theories. In the light of this, the criterion of having a group of practitioners should be applied only with hindsight, after a period long enough to allow for other persons to show their interest in the new theory or the new simulation study. If the criterion of practitioners is applied in this way, then some simulation studies in fact satisfy all the requirements for scientific soundness. There is a number of such studies, which not only were acknowledged by other researchers but also were repeated by other research groups. Their members thereby became practitioners.

3.16
To summarise this discussion, the only remaining feature in the assessment of scientific soundness of simulation studies is applicable only with hindsight. It does not apply immediately when a new study is published. From this, we conclude that the presupposition of game theorists' demand for theoretical backing of simulation studies made within a narrow time horizon, and the implicit claim that these studies are scientifically unsound, can be rejected. It can be rejected because simulation studies have all the features necessary for sound scientific activity. Put briefly, every simulation study has the same status as a scientific theory in its state of being proposed for the first time. To refer to some 'underlying theory' therefore is beside the point. The simulation study itself in the sense just clarified is such an underlying theory. In the analogous case of a new theory (instead of a simulation study) a similar demand would amount to requiring that the new theory is justified by reference to some other, 'underlying' theory. Such a demand we have not found anywhere in the literature.

* Global Comparison?

4.1
There is another sense in which we could understand the unsoundness claim discussed in the previous section, namely when we take the referent of 'underlying theory' to be not a single theory but rather a whole domain of theories which can be regarded as a unit in terms of their 'family resemblance'. Clearly, game theory is such a unit. So the unsoundness claim for simulation studies might be read as expressing that game theory as a whole is superior to the simulation approach as a whole, and that therefore single simulation studies should be backed game theoretically.

4.2
But how to justify the superiority of game theory, how to justify, in general, the superiority of one scientific approach over another, when both have overlapping intended applications? Unfortunately, philosophy of science at present does not offer much help for settling that question. Besides the purely practical criterion according to which the approach is the better one which turns out to be more successful, the potentially relevant general notions of confirmation, explanatory coherence, simplicity and the like are not in a state that would allow their application to the present case.

4.3
In this situation, we would like to address three aspects that seem to be particularly relevant to the present debate. The question under discussion in the following is to assess priority claims among two competing scientific 'approaches'. The meaning of 'approaches' here needs clarification. The two approaches under discussion are game theory and discrete event, social simulation. Clearly, by 'approach' we cannot mean 'theory' because the meaning of the latter term was fixed above in a much more local way. 'Game theory' as well as the simulation approach comprises many theories (in the above sense)[12] which are similar to each other in certain ways. In the absence of an established term, we will use 'approach' to refer to the two families of theories. There are at least four dimensions along which we could check for priority: empirical adequacy, theoretical fruitfulness, social relevance, and simplicity.

  1. If one of the approaches is empirically more adequate in the sense of being better confirmed than the other this would yield a strong argument for its superiority. In the case before us, one might say that many simulation studies are done with artificial data, data that are automatically created. Such studies are not referring to a real system and therefore cannot be said to be empirically confirmed. This is certainly true, but does not yield a strong argument for game theory's being empirically better confirmed. In game theory, there is a similar tendency to avoid direct contact with data and empirical systems. Most game theoretical work is done in a mathematical way, without reference to concrete data and systems.

    Yet there are game theoretic models from which empirical implications can be obtained on a qualitative, macroscopic level. The spatial theory of voting, for instance, has models yielding realistic percentages of equilibrium numbers of voters who abstain, numbers that can be compared with empirical data. Applications of the 'folk theorems' on repeated games yield predictions about stable outcomes based on data about the initial distribution of strategies in the population. This would be an argument in favour of game theory if simulation studies were not able to provide similar predictions. Nevertheless, simulation studies can be used in the same way. There are simulations about spatial movements of actors in the spirit of Schelling's original model that can be fitted with actual data.[13] The percentages of cooperation in PD simulations including whole neighbourhoods can be compared with experimental data from laboratory experiments,[14] and distributions of producers/predators/protectors resulting from 3P simulations can be compared with actual such distributions in existing societies.[15]

    It is difficult to find an asymmetry here. Even if a greater number of studies with empirical implications could be found in the game theoretic approach, one still might counterbalance this by pointing to its history being 3 or 4 decades longer than that of the discrete simulation approach.

  2. Concerning theoretical fruitfulness, there is no doubt that the game theoretic approach was and is immensely fruitful. The differentiation of the original model into a large number of different special models has not lost impetus, and many researchers construct their new models on a game theoretic basis. But a similar level of activity is observed in the social simulation area. The mere numbers of course would favour game theory, but this would ignore that discrete social simulation is much younger than game theory. We would expect bibliometric investigations to show that the rate of increase in activity now is greater in the simulation approach than in game theory.

  3. The social relevance of either approach has two aspects. First, it can be discussed in terms of empirical predictions, and their use for society. As stated above, these predictions are of the same qualitative, macroscopic kind in both approaches. So whatever their social utility, they do not yield a real difference between the approaches. Secondly, social relevance can be discussed in terms of the power of an approach to form simple 'regulative' ideas that penetrate into everyday life, and there are used as a more or less explicit basis for practical reasoning and decision. This power rests on 'interesting' theorems or phenomena that are first proved or made visible with the help of a particular model from either approach. In game theory, the discovery of PD situations can count as an interesting phenomenon, which is the way to becoming a firm part of people's present day intellectual outfit. After that, the discovery that pure defection can be avoided in certain iterated settings might acquire equal, public attention. In discrete social simulation, no such spectacularly 'interesting' discoveries and findings are available. Some first indicators, that such discoveries could be possible, in my view are results like that of Axelrod's 'rise and fall of empires' Axelrod (1997, p.124) or of Hegselmann's clustering in solidarity networks Hegselmann (1994), or Fischer and Suleiman's recurrent patterns of political coalitions in an 'Israel like' environment Fischer and Suleiman (1999). Along this line, game theory presently has the edge on simulation.

  4. Last but not least, a traditional dimension of comparison of theories or 'approaches' is that of simplicity. Though there is no commonly accepted notion of simplicity of a theory, or a commonly accepted notion of complexity, some remarks can be made.

    Obviously, it makes little sense to say that one of the two approaches is simpler than the other tout court. In order to compare for simplicity we have to refer to a common application. The question is whether in view of some particular, or all, applications that can be, or have been, treated both in game theoretic terms and by means of simulation, there is a difference in simplicity. A priori, the area of overlap of possible applications should be very big, due to the unrestricted nature of game theoretic models which, in principle, can be applied to almost everything. In practice, however, it is difficult to find examples. One example are the PD tournaments which after Axelrod (1984) were studied more carefully both with paper and pencil and with simulations. Without closer inspection it is difficult to claim that one of the two approaches is simpler in this area would require closer inspection, but this is difficult because of the lack of acknowledged criteria.

    Without closer inspection, two things can be said. First, simulation is simpler when the application aims at exploring a space of possibilities to find new, interesting ones. Conversely, game theoretic study is simpler when the goal lies in the consolidation and integration of many possibilities in one homogenous picture. These points can be perceived in the PD studies. New, stable strategy combinations were found in the simulation results,[16] but integrating all of them into a neat model required the cumulated game theoretical knowledge available in the form of its theorems.

    There is one more aspect to simplicity that, though hardly relevant at the group level, can be very important at the level of a single individual researcher. In order to write a discrete, social simulation program the essential task is to formulate some rules for the individual behaviour of the simulated agents in a computer language. Often these rules can be integrated in simulation tools or shells. The simulation then can be run by taking a ready made simulation environment and just plugging in the rules one has formulated. In this way, one can in fact generate very complicated and complex systems by very simple rules.[17] A game theoretic study of the same rules would involve the build-up of a full game theoretical model. Assuming that the researcher is a social scientist who initially does not know game theory nor any programming language we can ask which way is easier and in that sense simpler: learning game theory or learning a programming language? Though this also is an involved matter, it seems to me that learning game theory so far as to be able to construct the particular model one is after is harder than learning a programming language to the extent that one can formulate the rules of behaviour and run a program. This difference will become more salient with the development of further tools for computer simulation, which has already begun.

* Conclusion

5.1
After having looked at three kinds of polemic arguments which are put forward in the debate between game theorists and discrete, social simulators, none of the arguments is found convincing. Claims of priority resting on such arguments therefore should be considered with suspicion.

5.2
Game theory may be said to have a slight advantage both in the number of established models and in models that gave rise to simple, socially relevant 'images' or ideas. However, this advantage must be corrected in the light of its higher age, and in the light of the development of new tools for simulation.

5.3
We conjecture that the priority debate will end as it often did in similar cases. Once the new approach (here: simulation) has been firmly established institutionally, there follows a period in which both parties accept the respective other approach as legitimate, and the strive for scientific priority is held on a very low level, at least in direct confrontations.


* Notes

1 One of the first such programs, which uses the model of cellular automata, is Schelling (1971).

2 A typical example is Gilbert and Doran (1994).

3 See, for instance, Scheibe (1997, 1999).

4 For example: "Complex adaptive systems are so intricate that there is little hope of a coherent theory without the controlled experiments that a massively parallel computer makes possible", Holland (1992, p. 27/28). It should be mentioned that in the same paper the author also stresses the need of theory.

5 For example Binmore (1998) ends his review of Axelrod (1997) saying that Axelrod's "conjectures can only be evaluated in a scientific manner by running properly controlled robustness tests that have been designed using a knowledge of the underlying theory".

6 A cluster law H with just two terms can be conjunctively separated if it is equivalent to a sentence A and B where A and B each contain only one of the two terms, see Balzer et al. (1987).

7 In the following, we will use the terms 'data' and 'observation sentences' interchangeably. An initial condition is just a (sub)set of the set of data or observation sentences.

8 Whereby we can assume that models of the latter kind have the same type as the models of H.

9 As just explained, one can as well consider the intersection of two such classes, and have a theory consisting of at least one class of models in which also certain observation sentences are true. The observation sentences may be 'distributed' over different models.

10 To get an idea of such a rule based simulation system, see Balzer (1999).

11 The theory's data are 'distributed' over different models.

12 'Intermediate' levels are discussed in philosophy of science under the labels of 'theory net' and 'theory evolution', see Balzer et al. (1987).

13 See e.g. Nowak and Vallacher (1994).

14 See Liebrand and Messick (1996).

15 See Albert and Balzer (2000).

16 Though the folk theorems assure that these combinations are stable, this does not mean that we 'knew' these combinations just because we knew the theorems. These theorems, see e.g. Osborne and Rubinstein (1994), cover a huge set of possibilities. To see the point, consider an analogous situation in quantum physics. We would not say we know the energy operator just because we know the standard theorems about (unbounded) operator algebras.

17 In SMASS, for instance, which uses Prolog, the formulation of a rule can be a very simple matter of a couple of hours, since Prolog syntax is almost like that of natural language, see Balzer (1999).


* References

ALBERT, A. and W.Balzer (2000) Towards Computational Institutional Analysis: Discrete Simulation of a 3P Model, in G.Ballot and G.Weisbuch (eds.), Application of Simulation to Social Sciences, Hermes Science Publishing Ltd, Oxford, 195-208.

AXELROD, R. (1984) The Evolution of Cooperation, New York: Basic Books.

AXELROD, R. (1997) The Complexity of Cooperation, Princeton: UP.

BALZER, W, .C.U.Moulines, J.D.Sneed (1987) An Architectonic for Science, Dordrecht: Reidel.

BALZER, W. (1999) SMASS: A Sequential Multi-Agent System for Social Simulation, in R.Suleiman et al. (eds), 2000: Tools and Techniques for Social Science Simulation, Heidelberg: Physica-Verlag, p. 65-82

BALZER W. and V.Dreier (1999) The Structure of the Spatial Theory of Elections, Brit.J.Phil.Sci. 50, 613-638.

BINMORE, K. (1998) Review of 'The Complexity of Cooperation: Agent-Based Models of Competition and Collaboration' by Axelrod, The Journal of Artificial Societies and Social Simulation, 1 (1), January < http://jasss.soc.surrey.ac.uk/1/1/review1.html>

FEYERABEND, P.K. (1962) Explanation, Reduction and Empiricism, in: H. Feigl and G. Maxwell (eds.), Scientific Explanation, Space and Time, University of Minnesota Press, Minneapolis.

FISCHER I, .and R.Suleiman, (1999) Representation Methods and the Emergence of Inter-Group Cooperation, in R.Suleiman et al. (eds.), 2000: Tools and Techniques for Social Science Simulation, Heidelberg: Physica-Verlag, p. 218 - 239

GILBERT N. and J. Doran (eds.) (1994) Simulating Societies, London: UCL Press.

HEGSELMANN R. (1994) Zur Selbstorganisation von Solidarnetzwerken unter Ungleichen, in K.Homann (ed.), Wirtschaftsethische Perspektiven I, Berlin: Duncker & Humboldt, 105-129.

HOLLAND, J.H. (1992) Complex Adaptive Systems, Daedalus 121, 17-30.

KUHN ,T. (1962) The Structure of Scientific Revolution. University of Chicago Press.

LIEBRAND, W.B.G. and D.M.Messick (1996) Computer Simulations of Sustainable Cooperation in Social Dilemmas, in R.Hegselmann et al. (eds.), Modelling and Simulation in the Social Sciences from the Philosophy of Science Point of View, Dordrecht: Kluwer, 235-247.

OSBORNE, M. J. and A. Rubinstein (1994) A Course in Game Theory. Cambridge: MIT.

NOWAK and R.R.Vallacher, A. (1994) Dynamical social psychology, NY: Guilford Press.

SCHELLING, T.C. (1971) Dynamic Models of Segregation, Journal of Mathematical Sociology 1, 143-186.

SCHEIBE, E. (1997) Reduktion physikalischer Theorien, Vol.1, Berlin: Springer.

SCHEIBE, E. (1999) Reduktion physikalischer Theorien, Vol.2, Berlin: Springer.
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