Defining power

During the course of a recent online discussion, David Friedman raised the question of how to define “patriarchy” in particular and “power” more generally.

I gave the following answer which I’m sharing with you in order to elicit broader commentary (ideally from people who actually know something about social choice theory):

Conceptually, defining “power” should be straightforward.

Borrowing from standard social choice terminology, under any Social Welfare Function, which maps from the set of all preference profiles (list of policy preference rankings, one for every member of the society) to a unique social preference ranking, if my preferences correlate more with the social preferences than yours (where correlation is defined in an appropriate way), I am more powerful.

In a dictatorship, the correlation is 1 if I’m the dictator.

In a democracy, the correlation is high if I’m the median voter, low if I’m a member of the political fringe.

Patriarchy is then a system in which men’s policy preferences are more highly correlated with the social preferences than women’s.

David raised the following problem with my definition:

Suppose one percent of the population prefer outcome A to outcome B, ninety-nine percent the other way around. The social preference function, in situations where it has to choose between the two, chooses A two percent of the time.

The group of people who prefer B have more power than the group who prefer A, but does it make sense to say that an individual member of the group has more power? Might it make more sense to use a definition in which the question is not whether the social choice function correlates with my preferences but whether a change in my preferences produces a change in the social choice function in the same direction?

I think that’s a very good point. So here is my updated definition of power:

If changes in A’s preference ranking are more highly correlated with changes in the social preference ranking than changes in B’s preference ranking are, A is more powerful than B.

Is this how people in social choice have always defined power? If not, is there a deep problem with this definition which didn’t occur to me?

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