# Why is maximizing expected value always rational according to Pascal? [duplicate]

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)?

• 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