Tuesday, February 21, 2017

Total and average epistemic and pragmatic utilities

The demiurge flipped a fair coin. If it landed heads, he created 100 people, of whom 10 had a birthmark on their back. If it landed tails, he created 10 people, of whom 9 had a birthmark on their back. You’re one of the created people and the demiurge has just apprised you of the above facts.

What should your credence be that you have a birthmark on your back?

This seems a plausible answer:

  • Answer A: (1/2)(10/100)+(1/2)(9/10)=1/2

Let’s think a bit about Brier scores, considered as measures of epistemic disutility. If everybody goes for Answer A, then the expected total epistemic disutility will be:

  • TD(A) = (1/2)(100)(1/2)2 + (1/2)(10)(1/2)2 = 13.75

That’s not the best one we can do. It turns out that the strategy that minimizes the expected total epistemic disutility will be:

  • Answer B: 19/110

which yields the expected total disutility:

  • TD(B) = 7.9.

The same 19/110 answer will be optimal with any other proper scoring rule. Moreover, what holds for proper scoring rules also holds for betting scenarios, and so the strategy of going for 19/110, if universally adopted, will make for better total utility in betting scenarios. In other words, we have both an epistemic utility and a pragmatic utility argument for the strategy of adopting 19/110.

On the other hand, the 1/2 answer will optimize the expected average epistemic and pragmatic utilities in the population. But do we want to do that? After all, we know from Parfit that optimizing average pragmatic utilities can be a very bad idea (as it leads to killing off of those who are below average in happiness).

Yet the 1/2 answer has an intuitive pull.


William said...
This comment has been removed by the author.
IanS said...

You could ask the simpler question: What should your credence be in ‘coin landed heads?’. The birthmarks make the arithmetic more complex, but do they add any extra conceptual issues? With the simpler question, the setup would be (one version of) Bostrom’s Incubator.

I’m a bit worried about adding scores over different people. Isn’t the point of scoring rules that I want my beliefs to be close to the truth. By contrast, adding betting winnings could make sense – you could imagine that all the winnings/losses are paid to some external worthy cause.

Alexander R Pruss said...

Yeah, you're right that the birthmarks are an unnecessary complication. The same issues arise for the coin throw.

William said...

If you go Monte Carlo here, it looks like the 0.5 is correct:
# R program: test birthmark on coin flip, either 9 of 10 or 10 of 100, single pick
trialtrials <- 100
trials <- 10000
results <- rep(0, trialtrials)

for(k in 1:trialtrials) {
marks <- 0
for(i in 1:trials) {
t100 <-sample(rep(1:100))
t10 <- sample(rep(1:10))
t2 = sample(1:2)
if(t2[1] == 1) {
if(t10[4] != 5) {
# Birthmark
marks <- marks + 1
} else {
if(t100[3] < 11) {
# Birthmark
marks <- marks + 1
results[k] <- marks
This gives a scatter plot around 5000/10000.

So the problem is then the very severe penalty, when you do this guess on everyone, for guessing you are in the situation with only 10 people. The scoring rule (after it's applied to everyone created) causes a much bigger loss for guessing the 10-man situation when it's the 100-man than for guessing the 100-man situation when it's the 10-man one, and so the odds are adjusted to avoid that bigger loss.

If you only pick one person (you) and then start the whole process over, it's 50/50.