Free will and mathematical models

Where does free will fit into mathematical models? This question has come up quite a bit in my graduate networks seminar, where my students and I have been reading a lot of work by Bruce Mayhew. The reason for the question isn’t difficult to find: Mayhew, more than most scholars, argues very strongly against the invocation of individual will or decision making as an explanatory variable. Indeed, he sometimes mocks such accounts as boiling down to, “People do things because they want to,” which is a rather unsatisfying explanation for things. Yet, at the same time, this seems to lead unavoidably to the conclusion that people don’t do things because they want to, but because they are impelled to by forces beyond their control. And who wants to live life believing that?

Fortunately, this particular dilemma is fairly easily resolved. And that’s at least in part because pretty much all of us already agree with Mayhew- we just don’t realize it yet.

Okay, so that was a deliberately provocative bit of language meant to get you to click the “more” button, but I still think it’s essentially true. Many of us- and by that I include myself, my students, and you- are social scientists. That means that we are trying to find scientific explanations for social phenomena. In other words, we’re seeking explanations that are generalizable and apply widely over many different individuals, many different groups, and so on. If that’s the case then we already agree with Mayhew that human behavior is at least partly determined- it’s determined by the factors that we include in our theories and models. If, in contrast, human behavior weren’t determined- if we really did just do whatever we wanted, whenever we wanted to, there wouldn’t be a social science because there wouldn’t be enough predictability to human behavior to make it possible. Of course, you might argue that we do things because we want to, but the reasons that we want certain things are the proper domain for sociology. Fine, I’m willing to entertain that notion, but if there are external forces or factors that can cause enough of us to want the exact same things routinely enough to give rise to regular social structures then, really, individual desire is just a mediating variable between the external factor and the behavioral outcome. It’s a distinction without a meaningful difference.

So, then, where does free will appear in mathematical models? Interestingly enough, I do think it’s there. The trick, however, is to ask how we would recognize free will in data if we were to observe it. If we see behavior or outcomes that are reliably associated with some covariate we think, “Ah-HA! The outcome is being produced by the covariate!” Or something to that effect, anyway, allowing for the fact that correlation is not causation. As a result, an individual’s free choice only matters, and only becomes observable, to the extent that it is unconnected to covariates. When behavior is reliably associated with a covariate it isn’t free so much as driven (potentially) by that covariate. In other words, free will appears in our models in the error term.

And this is, I think, the ultimate solution to my students’ consternation at Bruce Mayhew. He isn’t claiming that people don’t have the ability to make choices- I actually don’t think he was terribly interested in a question that philosophical. Instead, he’s making the very simple argument that explaining something by simply invoking individual choice is unsatisfying and even unscientific. As social scientists it isn’t our job to decide whether or not people can choose some things freely. Rather, our job is to explain how everything else- those things that people can’t or won’t choose freely- works. Likewise, mathematical models of human behavior might seem to lack heart and soul to some people, and this reaction is understandable, but that’s exactly what makes them useful.


4 Responses to Free will and mathematical models

  1. Michael Bishop says:

    Nice post. I think I come to the same conclusions from a somewhat different route.

    It may or may not be relevant to sociological practice but I think that our common usage of the term free will isn’t really coherent. It requires revision if we want to understand it scientifically. When I mentioned this in the comments at I was asked for references and I suggested Dennett. A brief summary of his position is available in this wikipedia entry on one of his books:

    Here is another interesting essay of his:

    You can find links to some Yudkowsky essays on the topic here:

  2. Rense Corten says:

    Nice post. The same arguments apply to things like love, by the way… I once freaked out a bunch of undergrads by claiming that “love is in the residual” while discussing a paper on marriage markets :).
    But what about Rational Choice models? They are typically mathematical, AND obviously rely on the notion of individual choice. You might say that the sociological value of such models is in the constraints influencing people’s choices, but still individual preferences (“will”) play a crucial role.

  3. Michael Bishop says:

    Rense’s comment brought me back to this post. You know, I think concerns about “free will” leads to a bias against quantitative work. Of course, in some people there is a bias towards difficult math for aesthetic and/or signaling reasons.

  4. mattbrashears says:

    You raise a good question, Rense, but I think you basically provide the answer at the same time. Rational choice models are elegant because they assume a common basis for making decisions- rationality as defined according to some external and ostensibly objective standard. If everyone makes decisions according to this same standard, then what determines behavior is the set of circumstances one is embedded within. Granted, those circumstances may involve anticipation of another’s behavior (e.g. in a prisoner’s dilemma game) but if all actors are, indeed, rational then the nature of the game is what matters, not the actors.

    Things don’t get better, really, if we shift to bounded rationality, as these models essentially argue that behavior would be rational if only actors had enough knowledge/capacity, and deviations from rationality reflect these constraints on operations. Clearly actors are not unlimited, but that only gets us so far.

    In effect, the only place where choice enters into rational choice models is in the name.

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