And in fact, giving Trump half a vote may be quite a "rational" thing to do. Let me explain.
First the cost to vote is non-trivial. A well-off person will often use a simple mail procedure or spend a LITTLE bit of time and gasoline voting -- call it $20 for the inconvenience.
A poorer person might lose $20 out of pocket -- or out $10 plus a 2-hour wait since D-leaning precincts in swing states often have deliberately degraded equipment -- so $20 even ignoring any inconvenience.
We'll suppose that you value your and your family's lives at $b if Trump wins (b = 200,000), and at $b + $a (suppose a = $20 million) if Trump loses.. Call the election a coin-toss; your expectation is $b + 10.00002 million if you don't vote (we add in the c = $20 savings from not voting), and voting increases your weighted arithmetic mean expectation whenever
(0.5+p)*20,000,000 + (0.5-p)*0 > 10000020
p > 1 in 1,000,000
if you do vote, where p is the probability your vote elects Biden.
This is a plausible estimate of your vote's chance to decide (if in a swing state) and for that state to swing the election. We've evaluated the cost to you of Trump victory as $20,000,000. Is that cost too high?
(Note that we are maximizing the weighted
arithmetic mean here. From the teachings of D. Bernoulli and Kelly we may prefer to maximize the
geometric mean. For that analysis we need to state $b, our residual "bankroll" if Trump wins. Suppose b = $200,000
I derive that voting is indicated whenever
p > .5 * ( ln(b+c) + ln(a+b+c) - ln(b) - ln(a-b)) / (ln(a+b) - ln(b))
which becomes
p > 1 in 91,400
if my arithmetic is correct. If b = 20,000, then p > 1 in 13,800. Kelly teaches that long-shots aren't worth so much. But if b = 20,000,000, p > 1 in 924,000, in the expected ballpark.
Summary: With parameters like ($20 M, $0.2 M, $20) -- though arguably extreme farfetched parameters -- you should waste your time and vote IF you're in a swing state but only if the election is CERTAIN to be extremely close, far beyond any reasonable a priori guess.
I think the above may be correct MATHEMATICALLY.
BUT we also derived mathematically that you may as well pee in your friend's swimming pool, or perhaps hope someone else will save the drowning child. The RESPONSIBLE act is to vote, and to vote sensibly.
The reason to vote therefore is the GOLDEN RULE: If many Biden voters do this Kelly calculation and fail to vote, while no MAGA-nuts so abstain, then you give victory to Trump.
To check the numbers, we do a thought experiment. Suppose some Omnimax AI determines that your vote IS the deciding vote, and offers you $25 million to vote for Trump. You lose the $20 million you get for saving democracy (and also earn the wrath and scorn of anyone who works out you've defected). But you get 25 million cold cash to play with! What do you say?
If the truth be told, I think many Americans would sell their vote for less than $25M, even for a mere million dollars, even if they knew theirs was the deciding vote. But suppose you get the whole $25 million. How do you rationalize shitting in the country's swimming pool? (I dunno. But it doesn't seem implausible that one family might decide to retire to some idyllic paradise, pamper themselves at a bigly rate and view their situation as a strange dream or nightmare.)
So what?
).
