Please note the primary topic of this thread is the math. It doesn’t matter what you believe : given a situation with an infinite or very high potential reward, a significant cost for a chance to get that reward, and a low probility of experiencing an outcome where you receive that reward, how does a rational agent compute what action to take?
Religion is just being used as an example. Don’t try to apply your personal beliefs, realize this is just a math problem with labels on the variables from religious ideas. So it’s sidetracking the thread to claim practicing religion has no real cost or that all monotheistic beliefs have a high chance of heaven : you may believe this to be literally true but this isn’t what the math problem says.
But, “Don’t waste your time with things that have a low probability of happening” isn’t a mathematical principle. I’m not totally sure I understand the OP. but to the extent that I do understand it, you haven’t found any flaw in the math behind Pascal’s Wager.
I think it is indeed a flaw, whether or not “Allah” is a cromulent example of the flaw. There is just as much reason to believe that there is a god which will condemn you for believing in Jehovah as there is to believe in one who will reward only such belief. Whether or not that god is Allah is irrelevant to the math. So the average expected value of belief is zero, not infinity.
Ok, well yes. But we need more math. I gave a modern day Pascal’s wager :
You are a robot that has been tasked with making your owner money
The powerball is such that the expected value is positive and better than any other investments
You have a moderate amount of money, far smaller than the betting pool of the powerball
So clearly if you invest it all on the powerball, roughly 999 out of 1000 you lose all your owner’s money. But the return on the 1 in 1000 chance is better than any other outcome for your owner.
So how do you rationally choose what to do? What mathematical rule or principle do you use to choose where to invest?
Maybe you need to know (or assume) something about Master’s risk aversion? Then you can apply a quantitative theory like utility theory, prospect theory, portfolio theory, etc., to decide what to do. From your point of view as a robot, humans are all crazy anyway, so it is hard to judge if any particular approach be more “rational” than the others. For example, you think maybe your owner should instead invest the money in a nice, relaxing holiday so he doesn’t die of stress-related complications at age 35 and will therefore make more money over a longer expected lifespan, but nobody’s listening.
The expected utility, or value to you of a payoff is not (necessarily) equivalent to the expected dollar value. Winning two million dollars probably wouldn’t be twice as valuable to you as winning one million dollars; and certainly, winning an infinite amount of money wouldn’t be infinitely better than winning a million dollars.
Although, I’m not sure this objection would apply to the actual Pascal’s Wager. If Pascal is positing infinite happiness, isn’t he already speaking in terms of utility rather than monetary value?
He is. In Pascal’s wager, there is Christianity, offering infinite happiness (and there is no way to quantify the probability of this according to Pascal; you are supposed to assume it is at least philosophically possible), and lack of belief in God, offering some zero or finite utility. The problem is that he dismisses alternatives such as other gods out of hand, while admitting that since God is incomprehensible the uniqueness of Christianity can not be justified by reason or rational thinking.
Perhaps, if we follow his line of thought that reasoning is of no avail here, we can criticize him for then trying to present the situation in terms of mathematical game/decision theory.
The problem is more in set theory than probability. Pascal’s wager assumes the size of the set of possible religions offering infinite rewards and infinite punishment is 1. As has been noted above, it falls apart if the size of that set is > 1.
As for DrDeth’s comment, the size of the faith is clearly not an indicator of its potential truth, or early Christianity would not have qualified. I doubt anyone believes that the problem of God can be solved with a vote.
Doesn’t solve it. The flaw here I see now is the distribution. You don’t just care about getting the highest expected value - you want to choose a set of actions where the median outcome over the probability distribution is good, not the average from the tail where you win an unbounded reward. Not quite sure how to describe this.
For those who say you lose nothing if wrong, I think it depends on what level of infinite the reward is. If it is countably infinite, then rewards in this life have to count for something, so you by definition have something to lose. The only way in which this life doesn’t count for anything is if the heavenly reward is uncountably infinite, like each infinitesimal moment there is a better reward than an entire life on Earth. I personally reject this possibility because if I were altered enough upon reaching Heaven that I could experience such bliss, then I would no longer be me.
Any good investor doesn’t invest their entirety into anything, even if it is a good chance to payoff well.
A good algorithm would “invest” some amount of the available funds into the lottery, but not its entirety. It is not either or.
It is also gambling, and the first rule of gambling is to never wager more than you can afford to lose. If the robot wagers more than you can afford to lose, it is both a bad investor and a bad gambler.
The rational robot would purchase some small number of tickets, using only a small fraction of its resources. For instance, I purchase a couple of tickets sometimes when I notice that I have “pot odds.” I could buy more, and increase my chances of winning, but it quickly becomes diminishing returns.
Keep in mind that it is going to be pretty hard for the lottery to be the best investment you can find. I can find you junk bonds with a much higher expected utility value, and a much higher than .1% chance of paying off.
(OP obfuscated his question by introducing Pascal’s Wager. The recent billion-dollar lottery payoff was huge … but not as huge as the payoff in going to Eternal Paradise! )
There is a well-known approach (start with “utility function”) introduced more than 250 years ago by Daniel Bernoulli and which has been frequently discussed on this Board. As Voyager’s daughter points out “Eating two [ice cream cones] is probably also good, but not twice as good [as eating one].” Voyager’s daughter went on to point out something that Chronos overlooked in the most recent thread on this topic: “Eating eight is significantly less good than eating one.”
While there is no single universal utility function, Kelly’s Criterion is a common-sense approach, introduced by Bernoulli in his long-ago paper, and is provably optimal in some well-defined mathematical settings.
Whereas the ordinary solution to a gambling problem is to maximize the weighted arithmetic mean of your resultant total bankroll, with Kelly’s Criterion you maximize the weighted geometric mean.
This change will have little effect if wager or payoff are smallish relative to your total bankroll, but makes a big difference when wager or payoff is big.
Try it. For example, with the famous St. Petersburg Paradox gamble but adopting Kelly’s Criterion, even a millionaire should only pay $11 to play this “infinite expectation” game!
Even if infinite rewards exist, is it possible for one individual to experience an infinite amount of rewards (even if you assume that individual exists forever)?
Yes, that is a common form of health insurance. It’s a well known feature of actual American insurance policies as implemented. I’m not familiar with the all possible American health insurance polices, and I’m aware that there are many aspects I don’t know about, but it’s one the interesting characteristics that is often discussed.
Even more interesting to me, as I understand the changes going through the Aus health insurance system at present, this is going to be documented and codified for (in Aus) the first time. You will pay more to get access to some rare expensive treatments. Recently, some Aus health insurance companies have been more or less lying about it, like American health insurance companies do.
I don’t care for Pascal’s wager because it sets up belief in a religion as a punishment/payoff or risk vs. reward transaction, and most arguments against it fall into the same folly. It ignores the countless people who follow a religion because they believe it enhances their life here and now (i.e., they’re not really giving up anything) and also don’t necessarily believe that followers of other religions are doomed to eternal damnation (i.e. are non-exclusive).
I follow the Christian religion. I don’t believe all Muslims or Sikhs or atheists are going to hell. If you could prove to me there is no God, I just might go on living as though there is one anyway, including going to church on Sundays, donating part of my income, sheltering the homeless and feeding the poor.
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