We have recently been subjected to a barrage of messages from corporations, Bollywood stars and page three familiars exhorting us to go out and vote. The subtext is that if you do not vote, you do not "care". After the poor turnout in South Mumbai, there was much anguish and soul-searching by perpetually anguished, professional soul-searchers such as Barkha Dutt on exactly this matter.I wish to point out that it is perfectly rational for voters who "deeply care" (in a sense which I will make precise below) to abstain from voting. The argument is very simple and runs as follows. It is extremely improbable for a voter to be able to influence the outcome in a large election. A voter can be influential only if the other voters are exactly divided in their votes for the best candidate. One does not need a Ph.D in probability theory to realize that (i) this is an extremely unlikely event and (ii) this probability will decline rapidly as the number of voters increases. For instance if there are only two candidates and voter preferences over the candidates are equally likely, then the chances of being influential is 0.5 if there are three voters, about 0.03 if there are 1000 voters and very close to zero if there are 10,000 voters. Therefore, even though you care deeply about the outcome of the election, your expected payoff from voting is likely to be very small; if you have to offset these gains against the cost of voting (these costs are not necessarily monetary; they may represent the discomfort of standing in queues and so on) you may decide quite rationally not to vote even if these costs are very low (as they are in places like Delhi and Mumbai).
The argument above for not voting involves a curious inconsistency. Suppose all voters argued in the same way and concluded that they should not vote. Then every voter would be influential and would gain by voting! A formal game theoretic way of saying this is to say that for all voters not to vote, is not a Nash equilibrium of the game (Nash, here, is John Nash of ``A Beautiful Mind''). So what is the Nash equilibrium here? Suppose that there are N eligible voters (N large) with different voting costs denoted by c. Assume that the proportion of voters with voting costs less than c is given by F(c). Clearly F(c) increases as c increases. Assume that each voter benefits an amount α (let us not quibble at this moment about how these things are measured) if her preferred candidate wins. A "caring" voter has a large positive α and an apathetic one, presumably a small positive one. Let p(n) denote the probability of a voter influencing the outcome when n voters actually vote. It is clear that p(n) declines as n increases. Let c* be a solution to the equation p(NF(c*))α =c*. Some harmless assumptions regarding the functions p and F (continuity, etc) will guarantee the existence of a solution. The Nash equilibrium of the game is that voters with costs below c* will vote while those with costs above c*, will not. The point here is that the turnout on which voters' decisions to vote are based, is exactly the one generated by those decisions.
Is the discussion above "useful" in any sense? I think it is, if you are interested in motivating voters to vote. If your message is "Vote because you can choose a better Government", you are trying to get voters to increase their α. This is not likely to have a large effect because the p(n) term is already very close to zero. A better strategy is to emphasize that voting is duty just like paying taxes and not throwing garbage into your street. The effect of this is to add a positive constant K on the left hand side of the equilibrium equation. Voters get this benefit independently of the outcome of the vote - you can think of this as the "warm glow" you get when they put that ink on your index finger. It is quite easy to verify that if K is large enough, you get a corner solution where all voters irrespective of their voting costs, vote.
5 comments:
Interesting.
Where does the cost of *not voting* figure in this? I'm thinking about arguments like "The [party that shall not be named] is (almost) pure evil. It should not come to power. At Any Cost."
Quite possibly, this would be clubbed together with "benefits" of voting. But, isn't there some (behavioral economics / psychology) evidence that perceptions of "potential benefits" and "potential harm" have an asymmetrical influence on people's choices / motivations / propensity to act?
One simple question,
The power of a person's vote to affect the outcome of a poll is also dependent on the winning margin, right? That is to say, a vote may not make much difference in Amathi or Rae Bareli, but may create a much bigger difference in an poll where the margin of winning is, say: 100 or 500 votes. How do we account that in this calculation?
Rahul, I'd like to add my point-two-penny wisdom to this. Yes I agree that there are logical arguments for viting and abstaining as it is probably one of those undecidable logical questions. I have to point out one strategy that a muslim progressive group is following. Their party is contesting in four constituencies in TN, including Ramanathapuram. They did not join the coalition with DMK because they were not given enough seats.But in a bid to keep non-secular parties out at any cost, they are promoting and consolidating teh status of teh DMK member in Ramanathapuram. It is paradoxical and perhaps a joke to some that they are campaigning against their own candidate, but like I believe common logic may not answer such questions.
Also, I'd like to know why Barkha Dutt (read NDTV) is making such an ass of herself by going on with the hackeneyed "National" versus " Regional " party descriptions when everyone knows that Congress and BJP are regional parties themselves (only they are confined to the "National-language" belt!). The other aspect of this outdated thinking is that the south parties such as DMK PMK ADMK are referred to as non-ideology based , unlike the CPs, and hence they will move to the largest bidder. As arguments go - this is strange. It's another slavish feature of the english-speaking individuals that while they recognize Marx as an idealogue (and Martin Luther, Martin Luther King, and the Advani types) They have no thought of Periyar and Ambedkar, Muthulakshmi Reddy (just a sample, there are many more) and other important champions of dravidian liberation movements! It's quite ridiculous. I almost don't think there's any need to vote and maintain the democracy for the benefit of silly progress-faking journalists like the NDTV and co...
Thanks for that Rahul (aka OLO). There is one additional random variable that must be accounted for here, i.e., the voting decision of the voter. Since this is often quite random and not necessarily aligned with the interests of the individual or group, not voting and voting would have approximately the same effect. So the important thing is to have your name on the voters' list. This you understand well.
Thanks for the comments.
I agree with Abi that gains and losses may be treated asymmetrically by voters (and decision makers generally). A good example of such an approach is ``Prospect Theory'' of Kahnemann and Tversky (the former won a Nobel for this some years ago; the latter didn't because he had died earlier). However my point remains that however you reckon your gains and losses, once you mulitply it by the infinitesimal probability of the event occurring, the expected benefit is very small and could well be overturned by very small costs. A very crude analogy might be the following: the consequences of being struck by a meteorite will be a catastrophic but the extremely low probability of such events means that we are unlikely to spend money and effort taking safety precautions for them.
I think my argument holds even in the case that Arun discusses, that of ``close'' races. Suppose there are two candidates A and B and that each voter is an A or B voter with equal probability, drawn independently. By the Laws of Large Numbers etc, you must believe that this is an extremely close race. However the probability that the aggregate votes of all voters except a given voter, are exactly equal or differs by one, is infinitesimal. I think voters know this. Why? For instance, there is hardly any discussion in General Elections about what happens in case of a tie (in fact, I don't think most people know what is supposed to happen; I certainly don't). For elections with small numbers however, these things are important and widely known. For instance, in the Lok Sabha, the Speaker has a tie-breaking vote. Young Somnath Chatterjee was booted out of the CPM on this issue! As an aside, one of the most bizarre tie-breaking rules is used for meetings of the Board at University College, London. The stuffed, enbalmed body of the philosopher Jeremy Bentham is carried to these meetings; in case of a tie, his vote is recorded for the proposal which is the most radical departure from the status quo.
AmOk says that some voters vote without any reference to payoff ``calculations''. This might happen in two ways, I suppose. One is that their decision to vote or abstain is idiosyncratic. This is not a bad thing because excessive calculation might convince then that voting is not worthwhile. On the other hand, they may vote for a random candidate. If large numbers of voters do this, we might get weird outcomes (large numbers of votes for unknown candidates) which we don't really observe.
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