.... continued from Pascal's Wager: A Counterobjection to Many Gods
The clue to my resolution about Pascal's Wager was that I was
distressed by the possibility of Pascal's Wager being valid. If it was
valid, and if I accepted it, my expected utility would be infinitely
positive, and you'd think that would be something to feel good about.
Yes, my utility for the near future would be bad, but I am normally
rational enough to endure some suffering and work in the interest of
future benefit.
What I came to believe was that expected utility is not in all cases a
good basis for deciding one's actions. It works in almost every
everyday decision, but not in contrived situations such as Pascal's
Wager. It is similar to how Newtonian physics works in everyday
situations, but in extremes such as speeds near the speed of light and
extremely small scales, it doesn't work.
In this case, I continue to weigh the impact of the choice on the very
likely case more heavily relative to expected utility than the impact
on the very unlikely case. It is well known that people sometimes act
contrary to expected utility, such as by being risk-averse, and
sometimes will change their minds if the discrepancy is pointed out to
them. However, after reflection, I still believe in my choice.
It wasn't easy to reject such a well-established principle, but the
basis on which I had always accepted expected utility is statistical.
It depends on infinity, and a different kind of infinity than is
involved in Pascal's Wager. Namely, expected value is the value you'd
expect for the average after infinite trials. It does not work very
well when there is a single trial.
The expected value for rolling a standard six-sided die is 3.5. If you
roll it once, you would hardly say you "expect" to get a 3.5. If you
roll it twice, then you can reasonably say you expect the average of
the rolls to be 3.5. That is the most likely value, and the next most
likely values are close to it. Because the distribution of a die roll
is uniform, it doesn't take many rolls before expected value becomes
really expected.
The distributions in Pascal's Wager are far from uniform, and there is
one trial. Either god exists or he doesn't. You don't get a different
lifetime or eternity in which to try again. The probabilities for some
finite utilities are large, and the probability for infinite utility
is small. If an agnostic accepts Pascal's Wager, they can hardly say
they "expect" infinite utility (heaven). They fully expect that their
consciousness will cease when they die and they will accrue no further
utility for themself.
There are other well-known cases in which expected utility breaks
down. For example, the St. Petersburg paradox due to Bernoulli. In
that hypothetical scenario, one is told about a game in which someone
will flip a coin until it comes up tails. The player begins with a
dollar, and each time the coin comes up heads, their money quadruples.
The dilemma is, what is the most one should one pay for the
opportunity to play this game? The expected value of the game is
infinite. If expected value is a good way to make choices, then one
should give up everything they have if necessary to play. Few people
would give up anything they weren't comfortable to part with.
Bernoulli's answer was that expected utility is not proportional to
expected value of money, but I suspect the paradox could be restated
to avoid that objection.
Again, it's not a very good bet only if only one trial is offered. If
you are offered the chance to play many times, then it becomes a
better bet. If you consider what is the most likely result of playing
the game once, it is that you win one dollar. If you play twice, 2
dollars and 5 dollars are equally likely. If you play three times, 6
dollars is more likely than 3 dollars, i.e. an average of 2 dollars is
more likely than an average of 1 dollar. So you might be willing to
pay more per game to play the game three times than one. The amount of
the most likely average (mode of the mean) of all the games approaches
infinity as the number of times the game is played approaches
infinity. If you are offered the chance to play the game a great many
times, you could still reasonably "expect" to make a profit over time
if you payed a large amount each time.
Yet expected utility works in everyday decisions, even some that are
one-time opportunities. Although they are one-time opportunities,
their consequences and probabilities are comparable to some other one-
time decisions. As such, you can talk about an expected utility
resulting from a set of different one-time decisions, and the expected
utility resulting from some policy for making decisions. It should
turn out that you can really expect the best utility from a policy of
maximizing expected utility - applied to most decisions.
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