linear vs. combinatoric social science

In mathematics, there’s a very rough distinction between “linear” things and “combinatoric” things. We are all familiar with linear science, but combinatorics is more subtle. Combinatorics simply means the math you need in order to count different combinations of things. For example, you may ask, “if I have ten red balls and twenty green balls, and I randomly draw three balls, how many different combinations of red and green do I get?” That’s a combinatoric question – counting discrete things.

Social science has lots of tools that exploit linear models: utility maximization, regression analysis, scale construction, etc. But we don’t have a lot of tools that address the combinatoric side of social life. To see what I mean, consider the issue of policy formation – why does government make some policies and not others?

• The linear answer (taken from the Median Voter Theorem in economics): Politicians offer policies designed to attract the median voter. Thus, the utility of a policy is approximated by how popular it is.
• The combinatoric answer (taken from Agendas, Alternatives, and Public Policies): Nature produces a stream of political issues and actors. Think of nature as drawing them from a big box of issues and people. If nature happens to simultaneously choose an issue and actor that “match,” then a policy gets made.

These are not inconsistent views, but they require very different toolkits. The first is about studying distributions of voters. The second is an arrival process. Metaphorically, the first model is a world of smoothness with thresholds. The second is chunky. Over the last hundred years or so of quantitative social science, we have lots of tools for smooth things. We have a few tools for chunky discrete things, like network analysis, but not enough. Ambitious quantitative social science PhD students should carefully think about that last sentence.

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