Consequentialism and voting

In a large election, an individual vote is almost certain to make no difference to the outcome. Given that voting is inconvenient and time-consuming, this raises the question whether rational citizens should bother to vote.

It obviously depends on the citizen's values. For a completely selfish person, the answer may well be 'no'. Different election outcomes typically don't matter too much for an ordinary citizen's selfish interests; and a miniscule chance of a medium-sized gain does not offset the cost in time and inconvenience.

But most people aren't completely selfish. (And if the few completely selfish voters stay at home, that's no reason for concern.)

What if a potential voter cares about the outcome for others? To simplify, what if she is a pure act-utilitarian who wants to maximize the total expected happiness (or whatever) in the electorate? It is often said that this would still not make voting rational, so that we need to postulate some intrinsic value to voting, or a non-consequentialist obligation to vote.

Along the same lines, in his 1980 paper "Rule utilitarianism, rights, obligations and the theory of rational behavior", John Harsanyi uses the voting case to argue that rule utilitarianism leads to better outcomes than act utilitarianism: rule utilitarians would vote, act utilitarians would abstain.

But let's think this through. Here is one of Harsanyi's examples.

EXAMPLE 1. 1000 voters have to decide the fate of a socially very desirable policy measure M. All of them favor the measure. Yet it will pass only if all 1000 voters actually come to the polls and vote for it. But voting entails some minor costs in terms of time and inconvenience. The voters cannot communicate and cannot find out how many other voters actually voted or will vote.

Under these assumptions, if the voters are act utilitarians then each voter will vote only if he is reasonably sure that all other 999 voters will vote. Therefore, if even one voter doubts that all other voters will vote then he will stay home and the measure will fail. Thus, defeat of the measure will be a fairly likely outcome.

Is this correct? Let's figure out the decision matrix for an arbitrary member of the group.

We'll assume that everyone loses 1 util by voting. If everyone votes, this means that the group has lost 1000 utils in total. To get an interesting social dilemma (or an argument for rule utilitarianism), we want the state in which everyone votes to be better than the state in which everyone stays at home. So the net utility of the "very desirable measure" M must exceed 1000 utils. Let's say it is 2000 utils. The decision matrix for an arbitrary act-utilitarian voter now looks like this.

... 998 others vote999 others vote
don't vote...-998-999

Here, voting has highest expected utility iff the probability of 999 others voting is at least 1/2000. Our voter does not have to be "reasonably sure", as Harsanyi claims, that all the others will vote. Only if she is very confident that some of the others will stay at home is it rational for her to abstain.

Admittedly, in real life it may be reasonable to assign a probability of less than 1/2000 to the assumption that everyone else in a large group will do their share. But in real life we also rarely need absolutely everyone to do their share in order to reach a desirable outcome. Moreover, if we may be confident that at least one person will not show up, this is typically because it is reasonable to expect that at least one person isn't motivated or has forgotten or is unable to come. But if the chance of such a disturbance is in fact greater than 1/2, then groups of (act-utilitarian) non-voters perform better, in the long run, than groups of (rule-utiliarian) voters who often waste almost everyone's efforts.

So Harsanyi's example doesn't work. Nor do his other examples.

I think this illustrates a general fact: people aren't very good at calculating expected utilities -- not even experts in decision theory. We often use heuristics, such as only looking at the most probable state. Of course the (by far) most probable state is that your vote will make no difference. Similarly: reducing your carbon-offprint won't affect global warming, donating to cancer research won't affect whether new cures will be found, going vegetarian will not prevent the rise of antibiotic resistant bacteria. But for rational decision-makers, this is irrelevant. What matters is the expectation of the difference. It's worth sitting down and doing the math.


# on 03 April 2014, 04:16

Hi Wo, great post. I am not sure that this affects your argument, but it's worth asking what would happen if (a) everyone did sit down and do the math; (b) all agents are alike and rational (c) this was common knowledge. Write V for the proposition that you vote and (~V) for the proposition that you don't; and write p for the common (and commonly known) subjective probability that an arbitrary one of the voters votes. Then there are two obvious Nash equilibria: p = 0 and p = 1 (nobody votes or everybody does). But are there any others?

Write N for the number of voters and K for the ratio of the relative benefit of voting when N-1 others vote to the relative disbenefit of voting when < N-1 others vote (= the inconvenience of voting). (So in your example, N = 1000 and K = 1999). Then in a non-extreme equilibrium with u your subjective utility function, u (V) = u (~V) (otherwise you would definitely vote or definitely not vote, so p = 1 or p = 0). So u (V) - u (~V) = 0

So if we write Q = p^(N-1), we have:

u(V) - u(~V) = 0 = KQ + (-1)(1-Q). So Q = 1/(1 + K). So p = (1 + K)^(-1/(N-1)) =def. p*

If now we imagine repeated such votes, so that voters calculate p on the basis of the observed relative frequency of votes in previous rounds, then one equilibrium is where a proportion p*, strictly between 0 and 1, of people vote every round. If we hold n fixed, then as K gets bigger and bigger p* falls: for a really important measure that requires unanimity a *smaller* proportion of people will vote for it than if it is only moderately important, in non-extreme equilibria.

In any case, the fact is that if we as act-utilitarians all sit down and do the maths, rather than just be rule-utilitarians, one possible result is that some proportion p*, where 0 < p* < 1, of people will vote for it. This outcome is Pareto inferior to either of the other possible outcomes. So maybe this is an objection to you after all. (I haven't read the Harsanyi though, so maybe this isn;t any help to him either.)

# on 03 April 2014, 18:25

Hi Arif, very interesting!

I don't quite understand your simplification of the decision problem -- what exactly is K? --, but I think you're right that there is an equilibrium with mixed strategies. (I looked at the case with 3 voters; here the mixed equilibrium is at p(Vote) = m^{-1/2}, where m is the utility of the very desirable measure.)

By your line of reasoning, it could also happen that everyone has observed that nobody goes to vote and consequently keeps abstaining in the next election. Such a society of act utilitarians would also be worse off than a society of rule utilitarians who always vote. Adding mixed options merely adds a further, even worse, equilibrium profile.

I'm always unsure about how to handle such cases with multiple equilibria. I'm tempted to postulate a deliberation dynamics that requires agents to go for the best equilibrium. If there's common knowledge that everyone is an ideal act utilitarian, this would mean that everyone must vote. Then your scenarios in which we have observed that others don't vote could not arise.

# on 04 April 2014, 19:29

Thanks wo.

I meant K to be this ratio: (the utility difference that voting makes in the case that N-1 others vote) / (the utility difference that it makes in case < N-1 others vote). I ignored the fact the numerator of this fraction itself depends on N. But this doesn't make a difference to the overall point, which is that as N increases the mixed equilibrium is more enthusiastic and as K (or your m) increases it is less enthusiastic.

On the stipulation that everyone is an ideal act utilitarian: if that just means that they are perfectly rational, concerned with the welfare of others etc. then that doesn't imply that everyone must vote. If it means that there is some additional incentive for each person to vote (as in Australian elections) then that changes the payoffs, so now it is a different decision situation. If it means that by chance and in fact everyone chooses to vote the first time, despite there being no rational compulsion to do so, then fine - but then maybe Harsanyi could say that the act-utilitarian needs to invoke this additional bit of luck to get a happy result that rule-utilitarians could get without it.

# on 06 April 2014, 13:00

Right. If we assume that it is rationally permissible to assign negligible credence to everyone else aiming at the best equilibrium (i.e. voting), then there is no guarantee that a group of ideal act-utilitarians will achieve the optimal outcome. As I said, I'm tempted to make such a probability assignment rationally impermissible, but admittedly that's not part of standard decision theory.

The general question is whether there's a code of rules for individuals which, if everyone follows it, maximizes the expected overall utility in the group (in some to be specified sense of "expected"). It's easy to come up with cases where rule-utilitarianism leads to sub-optimal outcomes.

# on 05 May 2014, 16:52

Ho Wo (add they'd say in battle school),

I'm uncertain about your generalisation to the carbon reducing case. After all, in those cases the disutility is often not negligible, it's ongoing, and most importantly, if one wanted to impact global warming there's almost certainly more effective things one could do. (Take up an extra job with the time saved, and use the money to buy rainforest, or whatever.) Do you still think the maths actually will point where you want it to? (I'm not in a position to check right at the moment.)

# on 05 May 2014, 19:08

Hi Ole,

I thought offsetting should count as a way of reducing one's carbon footprint. But I agree with the general point: it's not obvious what the maths will say in this case, especially since we have to balance climate impact against other moral and non-moral causes. On the other hand, some reductions in carbon footprint come at no significant costs at all (eat less red meat, use efficient appliances, put your computer to sleep overnight, insulate your house, etc.), so here we don't even have to start calculating.

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