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In the modern version of Pascal's wager argument, we assume an infinite value to ending up in heaven, and hence an infinite expected value for choosing to believe in god.

I do not understand why this should be the case. In general, it is a bad idea to place a bet on a highly unlikely option even if the rewards are great.

For example, if there is a lottery that will give a quadrillion dollars to one in quadrillion people and the cost of a lottery ticket is half a dollar (for a net expected value of 50 cents), you might as well not apply and save 50 cents because this event is so unlikely it might as well be impossible.

Indeed, it is only optimal to choose paths which maximize expected value if you have a large number of trials (a quadrillion in the lottery case). In Pascal's case there is only one trial in your hand.

Why should it be the case that we should still choose to maximize expected value (even though it is infinite)?

Gerard
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  • It is logical to maximize expected **utility**. That is how we define a rational actor in economics. But there is also a law of diminishing returns: the utility of a quadrillion dollars is less than two-quadrillion times the utility of 50 cents. But that questions whether it is possible to have infinite utility. Even if the driving input could go to infinity, humans' real appreciation of everything we actually know of starts linear, becomes nearly logarithmic for most of its range and then levels off and asymptotically tops out. – hide_in_plain_sight Feb 19 '20 at 21:37
  • There seems to be a greatest possible appreciation to be gained from money, sex, food... and perhaps even of divine glory. – hide_in_plain_sight Feb 19 '20 at 21:37
  • @Conifold I'm afraid not. Answers to that question detail many well-known objections to the wager, none of them address what I mentioned in my question. – Gerard Feb 21 '20 at 10:41
  • @hide_in_plain_sight I saw an interesting response to this in the edx course I'm currently following. If the utility as a function of number of days in heaven is an increasing function tending to a limit, incremental changes in utility become vanishingly small. In particular, there exists an n such that if you are promised n days in heaven, the value of any number of additional days is vanishingly small (smaller than the value of a snickers bar, say). This is clearly counter-intuitive -- we cannot imagine choosing a snickers bar over an eternity in heaven. – Gerard Feb 21 '20 at 10:45
  • @hide_in_plain_sight Regardless of whether the above argument convinces you, your comment about the law of diminishing returns is quite tangential to my question. – Gerard Feb 21 '20 at 10:47
  • I do not think your interpretation of the wager as maximizing the expected value is right. Sayeth Pascal:"*If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.*" Translated to modern terms, this is not expected value, this is minimax, he is [minimizing maximal regret](http://plowing.blogspot.com/2010/09/minimax-regret-and-pascals-wager.html). Not that it works that way either. – Conifold Feb 21 '20 at 11:50
  • So you are saying that Pascal said that the expected value of not believing in god is zero. 1) This is evidently not true and 2) the modern version of the argument does not make this assumption (https://plato.stanford.edu/entries/pascal-wager/#ArguExpe) – Gerard Feb 22 '20 at 04:20
  • What is the "expected value of not believing in god"? Expected value is the weighted average over all options. I am saying that Pascal did not refer to it at all, and neither do modern reconstructions of his argument that try to follow his text. SEP's "maximum expected utility" version does not. Then again, given God's infinite reward it really makes no difference which decision rule is used, expected utility, minimax, maximin, or something else remotely reasonable, God will come out on top. *If* Pascal's reward matrix is accepted your particular objection is moot. – Conifold Feb 24 '20 at 07:45

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I'm admittedly new to philosophy (and as such cannot yet refer to widely-accepted posts that would easily answer this or clear this up more concretely), but I'll give this a shot regardless. Pascal's wager (and yes, you are referring to the modern version) is, I think, different from the scenario you described. Ultimately, the Wager is not concerned with the palliative benefits or mental ease that accepting it might grant, but with what is literally the most important decision that could perhaps be made. You do not have the option to refrain from making the bet; you have to choose to live in one way or another. Yes, blindly assuming that a belief in God will lead to benefits untold and unimaginable is a bit sketchy.

That said, virtually every recorded statement of belief (the acceptance of this Wager, in some form or another) expects "good" things to occur, rather than the opposite, upon the adoption of this lifestyle. This is where your example with the lottery again diverges from the wager, as the money, regardless of how astronomically un-winnable it may seem, is, at the end of it, a definite positive. You wouldn't receive a quadrillion dollars and suddenly spontaneously combust in agony, you would definitely benefit. Lacking some sort of "message" from the divine, one could really only make judgements based upon collective agreement that yes, "God is good."

There are ultimately two questions here: whether the "reward" is even accessible or likely to exist (whether God exists), and whether the "reward" is even worth it (you get to go heaven, Paradise, whatever you want to call it).

I hope that this was not just an utter waste of ten minutes of typing, and I think if this question could be answered more coherently that would be awesome.

  • I'm not concerned with the many gods objection to the wager here (although it is legitimate). I was simply questioning the premise that we must make decisions to optimize expected value. The objection is that making choices based on that can lead one to choose paths which have a very low probability of yielding extremely high gains, which does not seem to be rational in practice. In the context of the wager, this will mean we choose the life of a theist and follow the related rituals and practices in order for a shot at an extremely improbable life in heaven. – Gerard Feb 20 '20 at 05:53
  • @Gerard We are definitely free to make decisions in regard to any particular standard (or none at all), regardless of how optimized it might seem. I think the issue is that there isn't another path that presents level of value anywhere near what heaven supposedly offers, so on the scale it weighs far more than it perhaps "rationally should." It's not about "extremely high" gains, it's about literally infinite gains. There is also the issue that regardless of the level of the gains enjoyed within this life, there may yet be a level of deprivation afterwards that renders those gains meaningless. –  Feb 20 '20 at 06:04
  • Let's ignore the idea of infinity for a minute and look at the example in my question where the gains are very large and not infinite. Even though the expected value is larger if we play the lottery it is common sense (because of the very small probability of winning) that we will lose and end up wasting money. The rational decision seems to be not to play. – Gerard Feb 21 '20 at 10:50
  • The point is that decisions which maximize expected value only yield gains if we have an opportunity to take them again and again and not just once. But there are a large number of decisions that we can take only once in our lives. My point is, perhaps a rational person should give more importance to the probability of success rather than the expected output value for these. – Gerard Feb 21 '20 at 10:52