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:


VOTER B
P( Vote)
P(Not Vote)
VOTER A
P( Vote)
0.5
0.2
P(Not Vote)
0.05
0.25



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.

CITED:

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

Friday, July 27, 2012

Objectively Subjective

Something that has always bugged me is human’s ability to translate subjective, mental information about probabilities, happiness, guilt etc. into objective, quantifiable numbers. For me the problem really became pronounced when reading a paper asking students to assess from 0-100% how likely they felt their answer to a simulated SAT question was correct. Say that you are given 5 choices: A, B, C, D, and E. Ignore for a moment the inherent difficulty in forecasting using data, and instead focus on the process one would go through to make the assessment of how certain they are. First, you would have to think how well do I know this subject area? If your answer is ‘not too well’, then you have to translate that ‘not too well’ into some range. Let’s say I have a 30% chance I know the correct answer with certainty. This number is compared to the 20% chance that if you guess, you got it right. However, now you start going through the answers themselves and determine how ‘reasonable’ they seem. The mental machinations may exclude one obviously incorrect choice, but now we’re stuck with what do we mean by ‘reasonable’ in the context of some finite number. Let’s say I’m 100% certain D is wrong is wrong and 85% sure E is wrong. Knowing this means we might as well only select from A, B and C. With this restricted choice set, my probability of being right, incorporating my prior belief state, is about 35.29%, now isn’t that a nice number? However, we are ignoring one key problem, my initial assumption that I was 30% certain I knew the right answer. How am I to know that because I got cut off earlier in the day by some bozo, I am now just a little more pessimistic in my outlook? Because of me being in this “hot state”, I shave off 10% from my initial assumption. 
            
           This question gets even more interesting when looking at how people make absolute comparisons. George Miller, who recently passed away, was a pioneer in the field of short-term memory, writing a now rather famous paper entitled, ‘The Magical Number SevenPlus or Minus Two: Some Limits on our Capacity for Processing Information’.  One experiment of particular interest to this discussion deals with peoples’ ability to discern differences in tones, Prof. Miller sums it up nicely:

“When only two or three tones were used the listeners never confused them. With four different tones confusions were quite rare, but with five or more tones confusions were frequent. With fourteen different tones the listeners made many mistakes. These data are plotted in Fig. 1. Along the bottom is the amount of input information in bits per stimulus. As the number of alternative tones was increased from 2 to 14, the input information increased from 1 to 3.8 bits. On the ordinate is plotted the amount of transmitted information. The amount of transmitted information behaves in much the way we would expect a communication channel to behave; the transmitted information increases linearly up to about 2 bits and then bends off toward an asymptote at about 2.5 bits. This value, 2.5 bits, therefore, is what we are calling the channel capacity of the listener for absolute judgments of pitch. 
 So now we have the number 2.5 bits. What does it mean? First, note that 2.5 bits corresponds to about six equally likely alternatives. The result means that we cannot pick more than six different pitches that the listener will never confuse. Or, stated slightly differently, no matter how many alternative tones we ask him to judge, the best we can expect him to do is to assign them to about six different classes without error. Or, again, if we know that there were N alternative stimuli, then his judgment enables us to narrow down the particular stimulus to one out of N /6.”
The takeaway from all his results is that humans have an innate capacity to make an absolute judgment among 7 different items on a uni-dimensional scale. For example, if someone were given 14 shades of green and asked which ones are different, humans would usually say that 7 of them are the same and 7 are different. Now one must remember that these are questions about single dimensions of OBJECTIVELY knowable items, like color, sound, taste etc. The world out there is filled with the unknown. When pollsters and academics ask questions to subjects relating to “enthusiasm to vote”, “dislike with the president’s economic policy” or “probabilities that your answers are right”, participants are doing their best to bring all these factors together and spit out a number. What I’m saying is that the results that these processes glean may be telling us little about people’s true tastes and instead depend heavily on how many choices participants are given, as well as, other factors related to framing of the questions asked.