* a note on the futility of voting
Topics: Democracy
06 Nov 2000
From: Ervan Darnell
If I don't send this today, it's pointless. So, I'll leave it a little
unfinished:
Consider the simple case for whether you should bother to vote. Assume
your vote is an independent event from how other people vote. That is, you
are not beholden to some authority telling you how to vote as a bloc. If
so, your vote is completely irrelevant and you have a negative expected
value for even bothering to bat an eyelash while thinking about it.
Your vote matters only in the case of tie, or in the case of bringing the
tally close enough that the elector really gets to decide, or there is
another election called, something to that effect. Whatever you take the
trigger to be, the probability is essentially the same as for being the tie
breaker.
Let
n -- number of people voting. I'll take this to be 40M for a presidential
race.
and 200K for a county election in a large urban area
p -- probability that a given person prefers a given candidate, i.e. the model
is that everyone votes randomly, but with a certain distribution.
Q(n) -- probability of your vote mattering for a given n
The chance of a tie is Q(n) = (n Choose n/2) * p^(n/2) * (1-p)^(n/2)
This is a big number, too large (or small depending on evaluation order)
for x86 floating point, so it's necessary to apply Stirling's formula (n! ~
= e^-n* n^n * sqrt(2*pi*n)) and take the log, which yields:
log(Q(n)) = (n + .5) * log(2) - .5 * log(pi) - .5 * log(n) + (n / 2) *
log(p) + (n / 2) * log(1 - p)
If the election happens to be absolutely and utterly even, i.e. p=.5, the
chance of a tie is 10E-4.3, i.e. about 1 in 10,000, amazingly high. But
p=.5 exactly is unrealistic. Here is the chance of your vote mattering for
different values of p:
p log10(Q(40M)) log10(Q(200K))
---------------------------------------------------------------------
.5 -3.9 -2.7
.501 -39 -2.9
.51 -3478 -20
With a bias of only 0.1% and if the outcome of the election is worth $10T
to you, an amount much greater than the annual GDP, your expected value is
still 10E-24 cents, which I believe is pronounced one septillionth of a
cent, and is approximately equal to the amount of your income Gore wants to
let you keep.
"But what if everyone felt that way?" In that case, n would be much lower
and the analysis would change.
"But what if only Republicans (xor Democrats) felt that way?" That's a
better question. It's the same effect as voting as a bloc. If you a
priori publicly bind yourself to someone else's decision for how you will
vote, your vote is paradoxically more valuable. At this level, it's rather
similar to a prisoner's dilemma [1,2] and your vote becomes more valuable
by virtue of having agreed to "cooperate".
As a practical matter, a large percentage of eligible voters do vote
because the frictional cost is low enough to not worry about it. However,
when it comes to the quality of voting I think the above analysis does
apply. That is, people will not develop the information necessary to make
an informed decision, and rationally so, because their interest in the
outcome is far less than the energy required to affect it even when the
consequences can be profound. This is democracy: turning control of your
life over to someone else, who shouldn't and doesn't care.
------------------
[1] http://iserver.saddleback.cc.ca.us/div/la/neh/prisoner.htm, a simple
explanation
[2] http://www.constitution.org/prisdilm.htm, some interesting links, only
a few of which I have explored.
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