## what are the major contributions of mathematical sociology?

Here’s a question: what are the main contributions of mathematical sociology? Although I’ve published a little bit on the topic, I don’t know things deeply enough to provide examples. So I am asking our friends at Permutations (and our readers) to provide examples. In answering this question, I stipulate the following rules:

- Statistics does not count. That’s not theory building. It’s data analysis.
- Descriptive math does not count. For example, the definition of centrality doesn’t count. You aren’t deducing things as much as computing things, which is important, but not the point of this post.
- Let’s leave out simulations. I want theorems.

In other words, start with some assumptions that define a model and then use mathematics to deduce some-new non-obvious conclusion.

To give you a sense of what I mean, let me list some core contributions of mathematical economics. These are all (a) considered central to economics, (b) are examples of deductive math, not stats/descriptions/simulations, and (c) actually require non-trivial mathematics. Each of these is considered a landmark contribution of social science.

- Debreu’s general equilibrium theorem.
- Existence of at least one Nash equilibrium in finite games.
- The solution of Black-Scholes equations using the heat equation.
- Arrow’s impossibility theorem.

What’s the equivalent in sociology? If there are none, why not? Don’t we have models that you can prove things about?

How about Granovetter’s “Threshold Models of Collective Behavior” in AJS (1978), where he demonstrated that collective behavior is not necessarily the simple result of consensus or an average of preferences — would that count?

I wonder why exclude simulations?

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NatalieNovember 5, 2009 at 4:13 am

Natalie:

1. Simulations are important (I’ve published my own), but I’m interested in theorem proving. Simulations are estimates of model behavior. Important, and good enough in many cases, but not really what I am talking about.

2. I don’t remember GRanovetter (78) well – are there theorems or simulations? I might admit that as a good case.

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fabiorojasNovember 5, 2009 at 4:28 am

To be honest, I don’t know enough about “theorems” to know what counts as one. However, Granovetter (1978) presents an analytical model and shows the distribution of results given certain ranges of inputs. Fancy greek letters and graphs of curves included. I think he says that simulations confirm the analytical model, but that’s not presented.

I’m not sure I understand about what makes simulations different — simulations provide estimates of model behavior, by using random distributions as inputs, rather than arbitrary discrete values. For social processes, wouldn’t that be more rigorous and robust, rather than less? (assuming the distributions chosen are theoretically driven) Maybe this is out of my depth…

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NatalieNovember 5, 2009 at 4:41 am

How about Breiger’s 1974, “The Duality of Persons and Groups”?

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braydenNovember 5, 2009 at 4:49 am

I can understand why you’d be personally interested in theorems, but if you’re talking about the “major contributions of mathematical sociology,” it doesn’t make sense to exclude simulations. Just look at the membership and section award recipients of the mathematical sociology section.

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Thorstein VeblenNovember 5, 2009 at 4:55 am

Of course, the mathematical social science on thresholds was widespread in economics before 1978. However, from the economics perch, the area of mathematical sociology that seems most prominent is work on agent-based modeling, which is certainly related to Granovetter’s paper.

Also – man, that’s a high bar you’ve set! Debreu and Arrow, it can very easily be argued, wrote the two seminal economics papers of the 20th century! This is like asking for economists who have looked at sociological subjects and expecting contributions on the level of the absolute titans of sociology proper.

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kevincureNovember 5, 2009 at 5:07 am

Natalie:

A theorem is a mathematical statement that says “if you assume X, then logically Y must follow.” Mathematicians would not consider a simulation result a theorem because you are looking, as you correctly note, at random inputs. It is logically possible that you missed some input that falsifies your theorem. Thus, logically proved theorems are considered much stronger than simulations. However, mathematicians find simulations useful because they suggest what the average looks like in a model and that is often good enough. But in the end, it’s no substitute for proof.

Regarding Granovetter (1978), I looked it up and there is no proof anything in it. It’s important work, but it’s conceptual. You don’t see a statements like “under the following conditions, I can prove the following…”

Brayden: Looking at Brieger (1974), I wouldn’t count it. There is actually nothing that’s proved. It’s a hugely important sociological point, but it amounts to a simple matrix transformation that’s tautological I.e., if you have an NxM person-group matrix, you can make the NxN person matrix or the MxM group matrix. A crucial observation, but certainly not a theorem in the normal way that the word is used. UPDATE: Rereading it, ther is a short 1-line proof, but still, it’s more of a linear algebra observation than what most mathematicians would call proof of a new result.

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fabiorojasNovember 5, 2009 at 5:11 am

Thorstein: (a) Ahem, I am one of the math soc award winners – and I did a simulation to get that award!

(b) I am excluding simulations from this discussion because they are not mathematical in the sense that mathematicians use the term. Math is proof. That’s the standard. Period.

Now, I am not denying the extreme usefulness of simulations, but it’s not a deduction, it’s an estimate. Apples and organges.

(c) Why should I care about proof? True, in the social sciences, approximate knowledge is what we can live with, but there is also value in establishing the logical foundations of the discipline. Math is one (but not the only) way to do that. It can also be useful – mathematical proofs can yeild new and interesting results.

Kevin: High bar? Damn right.

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fabiorojasNovember 5, 2009 at 5:18 am

Fabio, Great question. I’m curious to see what others say. I like your four from economists though was a little surprised by #3. I noticed that nothing from decision theory made it, like an expected utility representation.

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Mike M.November 5, 2009 at 5:20 am

Formal approaches are being advanced (and called for) in organizational and strategy research, a recent AMR special issue features some of this work, see http://journals.aomonline.org/InPress/main.asp?action=issue&p_id=4&p_short=AMR&issue_id=63&v=34&n=2&d=April%202009

Org theory and strategy, though, are probably net importers of mathematical intuition from economics (game theory, Bertrand/Cournot, Debreu — naturally much of the econ math in turn can be linked to physics, as discussed by Mirowski), though unique insights are also emerging in the process (as an example, Rich Makadok has done some interesting formalization on the capability development-factor market issue).

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tfNovember 5, 2009 at 6:40 am

I think that theorems may not be the only valuable contributions mathematical models can have (and I’m not talking simulations here). Building math models involves basically 3 steps. First there is what I call “the abstraction step” in which you map some sociological phenomena into formal entities. Second, there is the “derivation step” – here you play with your formal entities and derive things within the formal domain. This can be theorems, new concepts that are based on the more basic ones and data (e.g. simulations). Third, you take your formal “conclusions” and map them back to the phenomena domain – that is interpreting them. Either one of the three steps can be non-trivial and a source for contribution in my view. Most importantly I think that building models can help clarify concepts and assumptions so that more precise statements (e.g. hypotheses) can be developed. This is a contribution in itself in my opinion.

This can be extremely useful in integrating or reconciling seemingly conflicting theoretical approaches. For example, in one of my graduate seminars I wrote a paper that reconciles conflicting views about turning points, showing they are actually compatible if we define their basic terms and assumptions carefully. Interestingly, in my model I proved an interesting theorem which says that the faster a change in society is, the more likely it is that people agree about its cause. But I doubt this theorem has any importance.

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amitNovember 5, 2009 at 7:17 am

I agree with previous posts that your focus is quite narrow. Simulation studies have provided major insights, and certainly do require mathematics and formal modeling. Moreover, by excluding descriptive math, you exclude most of SNA, which is arguably one of the most important contributions of sociology to social science… Nevertheless, even within this narrow focus, it’s still an interesting question. One of my favorite contributions would be the model of Raub & Weesie (1990; AJS), proving that network closure can promote cooperation.

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RenseNovember 5, 2009 at 9:15 am

Fabio: You asked for a theorem, and I point to a paper with a theorem. Besides providing a theorem, Breiger (1974) is also considered a classic contemporary sociological theory piece. Just sayin’.

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braydenNovember 5, 2009 at 2:19 pm

There’s plenty of theorem proving in mathematical demography. Try the classic works on mortality selection, such as Vaupel, Manton and Stallard (1979) and Vaupel and Yashin (1985), which show how population-level aggregate rates of anything that can only be done once (dying, divorce of a particular marriage) can look completely the opposite of any of the underlying individual-level dynamics.

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ElizabethNovember 5, 2009 at 3:19 pm

How about Peter Blau’s 1977 book “Inequality and Heterogeneity”? I don’t recall that he used equations per se but his theorems and assertions were based in mathematical logic.

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mattbrashearsNovember 5, 2009 at 3:33 pm

[…] case you haven’t noticed… Fabio over on orgtheory is asking what Mathematical Sociology has done for the discipline lately: Here’s a question: what are the […]

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In case you haven’t noticed… « PermutationsNovember 5, 2009 at 3:36 pm

The first thing that comes to mind (from someone who is NOT a sociologist) is Social Balance Theory.

I think it qualifies in that it is a mathematical generalization of Heider’s theory of balance and it generates formal predictions about social structure that are empirically testable.

It may not qualify in that it technically comes from the social psychology literature, but I am trying to be inclusive.

http://en.wikipedia.org/wiki/Social_balance_theory

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Drew ConwayNovember 5, 2009 at 3:42 pm

There are theorem’s out there. An older one is in “A Stochastic Model of Social Mobility” by Robert McGinnis, American Sociological Review, Vol. 33, No. 5 (Oct., 1968), pp. 712-722, published at a time when Math Soc was sorting out what it was. Lots of mobility processes, in this pre-log-linear and pre-survival-analysis era were written down as Markov models. This literature, however, didn’t offer many proofs, just references to known features of Markov models, proved elsewhere.

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Steve MorganNovember 5, 2009 at 4:42 pm

I have a remarkable proof that fits Fabio’s criteria perfectly, but unfortunately this comment box is too small to contain it.

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KieranNovember 5, 2009 at 4:52 pm

Aside from the mathematical sociology mentioned by Steve Morgan, and given that Drew Conway allowed a non-sociologist to intrude, what about the logic of collective action … The Free Rider Problem … The Mobilizer’s Dilemma …?

And I do believe that Dr. Rojas’ bar is awfully high. That said, this might be enough to increase the membership of ASA’s mathetical sociology and Rationality and Society sections.

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Brian PittNovember 5, 2009 at 5:19 pm

Isn’t the insistence on “theorems” for sociological analysis pretty much the same as Parsons’ attempt to create an abstract theory which could be applied to all societies? Theorems are highly abstract and operate independently of historical context. Sociological analysis cannot be explanatory independently of that context. Economics can get away with theorems (such as the equilibrium theorem) because its analysis is done at a very abstract level (the individual, rational actor).

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GuillermoNovember 5, 2009 at 5:29 pm

Besides, if by “theorem” what you mean is something like the following: “any spherically symmetric solution of the vacuum field equations must be stationary and asymptotically flat”.

I believe sociology is light years away from that.

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GuillermoNovember 5, 2009 at 5:32 pm

Despite being a contributor to http://permut.wordpress.com/ I’m going to go ahead and bite the bullet. There are no theorems proven by sociologists as important as the ones that you cite in economics.

Possible partial explanations:

1. The content sociologists are interested in is harder to express in mathematical theorems. I think its conceivable that there is something to this but this doesn’t mean that the marginal utility of theorem proving is lower in sociology.

2. Sociology has under-valued theorem proving. I think this is true, but it’d be hard to prove.

3. The people drawn to sociology are, on average, less mathematically oriented. This is undeniably true, but one could argue it is merely an effect of explanations 1 and/or 2.

Like other commenters, I strongly dispute the suggestion that theorems proved is the only measure of the contributions of mathematical sociology.

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Michael BishopNovember 5, 2009 at 5:40 pm

How about Harrison White on vacancy chains?

From the perspective of somebody trained in physics, vacancy chains are equivalent to models of virtual particle propagation.

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Michael F. MartinNovember 5, 2009 at 5:50 pm

Also, Simmel on triads?

Seems to have inspired some interesting new work:

http://arxiv.org/abs/0906.2893

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Michael F. MartinNovember 5, 2009 at 5:52 pm

Not to hijack the discussion, but if there is consensus among us that mathematical sociology does not, at present, approach the level of “sociology theorems”, as suggested by Fabio, then what are the current aims of this incipient field? My somewhat educated guess is that the goal is to translate sociological theories to mathematical language in order to make it easier to perform simulations (i. e., number 3 in Fabio’s list). Am I correct?

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GuillermoNovember 5, 2009 at 6:56 pm

“My somewhat educated guess is that the goal… ”

Why does a science, if socilology is actually a science, have to have a goal, except for knowledge? You’re saying that mathematical sociology has no worth except to create simulations easier?

So is Mathematical Sociology an application? An application you run when you need a new game, perhaps?

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larrydunbarNovember 5, 2009 at 8:48 pm

Of course we shouldn’t forget Asmov’s psychohistory…:)

Sorry. Fascinating topic but I’m obviously in over my head. Pardon the digression.

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Fred H SchlegelNovember 5, 2009 at 10:10 pm

“You’re saying that mathematical sociology has no worth except to create simulations easier?”

No, not at all. I was not making any value judgments, nor do I think its potential is reduced to that of an “application”.

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GuillermoNovember 5, 2009 at 10:31 pm

Guillermo, I don’t think we should give up on theorems. But I agree that simulations are worthwhile. I agree that methodological advances are important – and will further our theories. Another important thing for mathematically minded sociologists to do is to learn about models used in other fields that might be applied to sociological phenomena.

Finally, and I can’t back this up with very solid evidence, but I feel like having studied game theory and evolution using mathematical models, helps me think about my empirical research on popularity and norms. I haven’t proved any theorems related to it, but I still feel like experience with math helps me structure my thinking about my work.

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Michael BishopNovember 5, 2009 at 11:23 pm

“I haven’t proved any theorems related to it, but I still feel like experience with math helps me structure my thinking about my work.”

Very true.

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GuillermoNovember 5, 2009 at 11:39 pm

Like Steve Morgan, I think the theorems are out there. Though for various reasons– good and bad– not as important in sociology. I’d also second Matt Brashears’s nomination of Blau’s structural theory as introducing an important set of theorems (though I think Blau and Schwartz is where the theorems are; and pehraps Blau and Schoenherr for the work on org structure; oh, and there’s Mayhew’s structural sociology which is all about theorems). As far as my favorite theorems, I’d point to Feld’s series of papers (often with Grofman) on the ‘class size paradox’ and especially my favorite, the 1991 AJS paper Why Your Friends Have More Friends Than You Do.

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ezrazuckermanNovember 6, 2009 at 12:50 am

I found Michael Chwe’s “Structure and Strategy in Collective Action” (AJS Vol 105 No 1 1999) very enlightening. Not sure if it qualifies as mathematical sociology though.

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AlexNovember 6, 2009 at 4:06 pm

I’m a bit surprised that this esteemed line of commenters has failed to mention what to me is the most obvious example: Simon’s (1952) formalization of Homans’ theory of sentiment and interaction in small groups.

Yes, Herbert Simon is not a “sociologist” but the paper was published in ASR. In any case, a “real” sociologist has also dealt with and further developed this model, in a section of the book that he wrote by himself called “Four Social Structure Theorems” (Fararo 1989: 120-129). Of course, (non-trivial) mathematical derivations from “first principles” can also be found in Fararo and Skvoretz 1987, where they attempt to formalize and integrate Blau’s structural theory and Granovetter’s “Strengh of Weak Ties” argument (you can also find “real” math in other Fararo and Skvoretz papers). As an AZ grad who took Networks with M. McPherson, I would also have to say that most of Bruce Mayhew’s work is “mathematical” in the restrictive sense that Fabio defines the term above. One important non-trivial proposition that Mayhew derived is that “oligarchy” can emerge simply as a function of group size without presuming inherent individual differences, etc.

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OmarNovember 6, 2009 at 6:07 pm

“In any case, a “real” sociologist has also dealt with and further developed this model, in a section of the book that he wrote by himself called “Four Social Structure Theorems” (Fararo 1989: 120-129).”

Yes, Thomas Fararo is one of the most important examples of “mathematical” sociological theory. I did not intend to be derisive of mathematical sociology in my previous comments. I was just pointing out that at the moment it is still incipient. If, at some point in the future, it achieves a way of being a more precise depiction of reality than current theoretical language, so be it.

And I would definitely classify Herbert Simon as a “real” sociologist.

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GuillermoNovember 6, 2009 at 7:13 pm

fabio,

does Gould AJS 2002 meet your criteria?

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gabrielrossmanNovember 6, 2009 at 8:27 pm

Some other examples, which just came to mind:

Jasso’s justice function is derived mathematically. In fact, she is probably the best example of a “mathematical purist” in the terms laid out by Fabio above.

Cartwright and Harary (1956) proof that if a graph is balanced (all triples multiply to a positive sign) then it partitions into two disjoint cliques with all positive relations within and all negative relations between is probably considered the most important example of applied math in sociology. Yes, the article was published in Psychological Review, but the main impact was on social network analysis. For instance, James Davis’ Human Relations paper on balance has a ton of theorems and proofs.

Roger Gould’s (2002) last paper on status hierarchies is full of proofs in the Appendix.

Heise’s (1977) affect control theory is “mathematical” in that the relevant predictions are derived from differential equations with a closed form solution.

Breiger (1974) may not count, but Breiger and Pattison’s work on “Role Algebras” does. Enough theorems and proofs to satisfy even the narrowest definition of “mathematical.

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OmarNovember 6, 2009 at 8:42 pm

The reason a theorem may not be easily found in mathematical sociology is that Sociology, in general, is Generational, while economics is not. A generation is one of many dimensions in the Grand Dimension. It is hard for math to cross dimension, especially when these dimensions are not understood or even talked about, except in economical terms.

Economics is of one dimension.

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larrydunbarNovember 6, 2009 at 9:54 pm

Fabio: I hope you will publish some synthesis report for that post and the ongoing discussion; I do not have the time to read it all but would be interested to see what comes out of it!LikeLike

Fr.November 7, 2009 at 4:10 pm

I am sure there are plenty of Sociology Theorems that are published in Economics journals are read as Economics. So the question is which theorems are actually Sociology theorems.

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BonaraNovember 7, 2009 at 7:52 pm

I don’t think I understand what make Social Network analysis metrics etc be “descriptive.” They are deduced from definitions of nodes and ties. Like a couple of other posters, I consider social network analysis to be sociology’s most important recent contribution. But if we are interested in the non-Network contributions of Math Soc, then the biggest utility of them (in my mind, at least) is that a formal definition of theories adds substantial clarity and allows contradictory or tautological theories to be eliminated (e.g., see Samuel Lucas’s article in latest Rationality & Society. Sociology could use more articles like that).

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Andrei BoutylineNovember 7, 2009 at 11:54 pm

I thought this thread would get a few math soc types laying it out for me. But it seems this has hit a nerve… definitely worth a sequel!

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fabiorojasNovember 8, 2009 at 3:53 am

“But if we are interested in the non-Network contributions of Math Soc, then the biggest utility of them (in my mind, at least) is that a formal definition of theories adds substantial clarity and allows contradictory or tautological theories to be eliminated…”

That’s exactly what was in my mind when I wrote my previous comments.

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GuillermoNovember 8, 2009 at 4:49 am

Thinking more about this overnight…

There is one application of mathematics which is not statistics, descriptive mathematics or strictly speaking material for simulations, but has been found to have an application to sociology: game theory. The works of Boudon and Swedberg are seminal in this respect.

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GuillermoNovember 8, 2009 at 2:43 pm

Boudon and Swedberg are seminal for the influence of game theory in sociology? Please explain.

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Steve MorganNovember 9, 2009 at 4:02 pm

Concurrency?

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jimiadamsNovember 9, 2009 at 10:56 pm

“Boudon and Swedberg are seminal for the influence of game theory in sociology? Please explain.”

Well, I can’t use this space as a mini-blog post, but I can give a very quick bibliography. Boudon writes about the use of game theory in sociology in his 1981 book “The logic of social action: an introduction to sociological analysis”.

Swedberg, for his part, wrote a 2001 article in Theory and Society, titled: “Sociology and game theory: Contemporary and historical perspectives”.

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GuillermoNovember 10, 2009 at 5:32 pm

Actually, Swedberg mentions Boudon as excellent, not as influential. He would be right to do so if he were talking about France, but I am not sure Boudon was ever influential in the US.

My own list of influential authors in sociology viz. game theory would be Coleman, and Swedberg/Hedstrom for mechanisms.

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Fr.November 10, 2009 at 5:51 pm

“Actually, Swedberg mentions Boudon as excellent, not as influential. He would be right to do so if he were talking about France, but I am not sure Boudon was ever influential in the US.”

Well, I wasn’t adopting the US-centric perspective. Boudon is definitely considered one of the forerunners if you take world sociology as a whole.

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GuillermoNovember 10, 2009 at 6:05 pm

Guillermo: Thanks. I asked because I have been Richard’s colleague at Cornell since he came here in 2002 or so, and I have never once heard him advocate for game theory approaches in sociology. Perhaps he is beaten to punch by some of my other colleagues! So, I should read his paper, which I never have.

As for Boudon, I associate him with various forms of formal argumentation, but perhaps because I work mostly on inequality-related questions, I know his Markov-inspired mobility work more. I did once have a look at the 1981 book, and so I should track it down again.

Finally, related to my first point, I would put Doug Heckathorn at the top of any list.

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Steve MorganNovember 10, 2009 at 6:27 pm

“As for Boudon, I associate him with various forms of formal argumentation, but perhaps because I work mostly on inequality-related questions, I know his Markov-inspired mobility work more.”

Besides his work on education (which uses path analysis), Boudon did game-theoretical work to explain the fall of the Soviet Union. I have the reference at home, so I’ll provide it a bit later.

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GuillermoNovember 10, 2009 at 6:33 pm