Friday, November 2, 2012

Political Independence

          I’m getting rather perturbed with this constant stream of naysayers, who claim that voting is unimportant. One argument trotted out on a consistent basis goes something like this. A liberal individual voting in a very conservative county, in a severely conservative state, has very little likelihood of swaying national, state-wide, and/or county-wide elections due to the sheer numbers game. One vote out of millions will do nothing to change an election. Many people counter with arguments about one’s civic duty to vote, or how people in other parts of the world are willing to die for the right to vote. All of these arguments are persuasive on moral and philosophical grounds, but if we are dealing with avid utilitarian’s, who are hell bent on claiming that the likelihood one’s vote matters is low, none of these arguments prove that the ends justify the means. What all these counter-arguments miss, however, is that there is a pernicious, downright improper assumption running through these anti-voting epithets. That problem is of course the assumption of independence. When I say independence, I’m talking about probabilistic independence, having no thoughts of anything more patriotic or interesting.

           Let’s take a step back and talk about this sort of independence and then it hopefully will become clear why these anti-voting authors’ assumption does not bode well for how the real world works. All independence says is given two events, A and B, knowing that A has already happened, the likelihood of B happening has not changed, and vice versa. Thus, two events being independent translate into two events that have no relationship to one another. Take this simple example. If I go home and decide to cook myself dinner, the likelihood that my next door neighbor, who I have never met, cooks dinner is unchanged. This is for both physical reasons, we don’t share a living space, and social reasons, we don’t have any means of communicating with one another. However, when I decide to cook dinner, my roommate’s likelihood of cooking dinner does change. Either for a physical reason, I’m taking up the stove so he can’t cook, or a social reason, he sees me cooking dinner and decides he can just take some of my food. So we would say that the likelihoods of me and my next door neighbor cooking dinner are independent, while the likelihoods of me and my roommate cooking dinner are dependent.

         Hopefully, one can see where I’m going with this. All those authors who decry about the likelihood one’s vote matters assume that a person’s choice to vote is independent of everyone else’s choice to vote. If everyone’s decision to vote is independent, then the likelihood that I vote having any sort of meaning on the election is rather small. However, if my decision to vote persuades another family member, or friend in my social network, to vote, and their decision also causes another to vote, then the cascading dependencies can really make the numbers a bit more favorable. Let’s go to a simple statistical example I’ve concocted. In the table below, we can see the joint distributions between two people, person A and person B voting, where they have dependent probabilities:

P( Vote)
P(Not Vote)
P( Vote)
P(Not Vote)

The table is very simple, so bear with me as I walk you through it. For example, sum across both rows and you get the marginal probabilities of A voting and not voting:

Marg. Prob A Votes = P(Voter A Votes | Voter B Votes) + P(Voter A Votes | Voter B Not Votes) = .7

Marg. Prob A Not Votes = P(Voter A Not Votes | Voter B Votes) + P(Voter A Not Votes | Voter B Not Votes) = .3
       You’ll notice if we sum these two numbers together we get 1, and that makes sense because Voter A can only vote or not vote, thus they will do either/or 100% of the time. We can now ask questions regarding conditional probabilities. Given that voter B has voted, what is the probability that voter A will also vote? To answer this, we restrict ourselves to only the first column and see that voter A votes with P(Voter A Votes) = .5 and voter A does not vote with P(Voter A Not Votes) = .05. However, the sum of these two are only .55 . So to find the probability that voter A votes when voter B has voted, we simply divide P(Voter A Votes) = .5 by our restricted sample space, the marginal probability of B voting, which is equal to .55 . So we have P(Voter A Votes| Voter  B Votes) =  .5/.55 = .91. We can now verify these two voters are dependent because the likelihood that A votes changes whenever B decides to vote or not vote. So doing the same process we would find P(Voter A Votes | Voter B  Not Votes) = .2/.45 = .44 . Now things are getting interesting, given that B votes the probability that A votes nearly doubles. This means that voter B’s decision to vote also increased the likelihood that voter A would vote by a factor of 2.

       Now let’s complicate things a little bit more. Let’s say that we have a pool of 100 voters, what is the likelihood that a single voter could sway the election? Well if we assume that every person’s vote is independent that is very easy, it’s merely 1 in 100, or .01. However, using our numbers from before let’s say voter B decides to vote before voter A. So voter B has a probability of .01 of swinging the election, but because her decision to vote has that added benefit of increasing the probability that A votes by .91 - .44 = .47, the likelihood that voter B’s vote is the decider is now .01 + .01*.47 = .0147.

      What this example is trying to illustrate is that if there are externalities to voting then the likelihood one’s vote matters can start to creep upwards. Given that our social networks are increasing faster and faster with the plethora of social media, these small spillovers start to add up. However, is a person’s choice to vote independent of everyone else’s choice to vote? Recent work by Betsy Sinclair at the University of Chicago, which looked at political canvassers in Los Angeles, seems to suggest that the independence assumption is incorrect. Networks influence political activity via social pressure to conform. Previous research had found that people in social networks where with high rates of political activity conform to social pressure and are more likely to acquiesce and vote. The important point in Sinclair’s work is that the messenger matters, if the politically active person trying to gin up votes is not in the same group as the person they are persuading then the effects on political behavior is small. However, if both people are members of similar groups, then the effects may be very large. For hermits that lack any social networks, the likelihood that there vote matters is infinitesimally small, however, for those people inculcated in vast networks of friends, families, and co-workers, their vote may make more of a difference than pundits make you believe.


Sinclair, B., McConnel, M., and Michelson, M. “Local Canvassing and Social Pressure: The Efficacy of Grassroots Voter Mobilization.” Forthcoming Political Communication, July 7, 2010.

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