Is Playing The Lottery Rational?

We all know playing the lottery is a mug’s game, right?

If I have (say) a one in two million chance of winning a million, with one dollar ticket, then my expected return is 50c. I expect to get 50c, for my $1, only an idiot would take that. Lotteries are for mugs. It’s no wonder poor people play the lottery more than the wealthy. Clearly poor people are more stupid than the middle-classes.

If you don’t play the lottery, chances are you have some variant of that story. Perhaps without the bald-faced bigotry at the end, perhaps not (can you be honest enough with yourself to know?).

Of course it is rubbish. For a very simple reason. The value of money you control is not proportional with its quantity. If you have a billion dollars, you won’t feel 1000x times more wealthy than you will if you have a million. And with good reason: your standard of living won’t be 1000x times as high.

For those on low income, the value of a lottery jackpot is much higher than it is for you. So much higher that the expected value is greater than the cost. It is rational to play.

This isn’t novel or my idea, this has been analysed by many economists and is perfectly unremarkable.

What I was thinking about tonight, however, was Pascal’s Wager. Unlike the lottery, Pascal’s Wager is the opposite way round. It is always worth taking. The normal objections to it (how do you know if you’ve got the right God with the right arbitrary requirements, there is no good reason to believe in heaven or hell, etc) are merely ways to shorten the odds, but the reward is claimed to be so great, infinite even, that tiny chances of winning still give you a positive expected value. If salvation is infinitely great, no probability is small enough for you to pass.

But I suspect why Pascal’s Wager doesn’t work is for similar reasons to the naive lottery analysis. Our perception of value in eternal bliss tapers off, beyond a certain degree of bliss, it really is no more attractive to us than a slightly less blissful state. We accord an infinitely wonderful salvation with no more value than just a stupendously, but finitely, wonderful salvation. And so the small chance of getting the right God, of propitiating it in the right way, and of the afterlife being in any way as claimed, serves to make a lie-in on Sunday morning more rational than going to church.


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14 responses to “Is Playing The Lottery Rational?

  1. Boz

    You just changed my mind about the rationality of playing the lottery


  2. exrelayman

    There is a difference between truly believing and bet hedging. Don’t think bet hedging is an approved way in.

  3. (1) Mug & Dollar
    Interesting that you did not use the British currency of a pound, but you did us the British colloquial expression “mug” – which I had to look up. Chiefly British Slang: a victim or dupe).

    (2) Bald, Bold or Bare
    Next your choice of “bald-faced lie” made me wonder if it was another Britishism. I was familiar with Bold-faced and maybe even Barefaced, but “bald”? It seems all exist, some with older etymologies and some considered more right than others. But I was surprised (but only a little) to learn that some Brits use to wear masks to social occasions where one would rather not be recognized.

    Apparently “Barefaced” is suppose to be ‘more’ correct. But you and I know that usage defines correctness in language, so I move on. Sorry if you already knew all this.

    (3) Rationality
    I think the whole idea of “rationality” in economics is highly debated now. Also, I think you may both enjoy this gentleman’s writing and his posts — he is an anthropologist and did a good piece on economics here.

    (4) Hope
    Though I have been tempted occasionally to think of those who buy lottery as silly, if not dull, I have many friends I love who do it, so it is a hard attitude to maintain. Instead, I think about the lottery as “hope” : When hope is lacking, buying the lottery is a cheap investment to have hopes and dreams for that week while waiting for the draw.

    And I choose not to do it because I don’t want to build that emotional habit — it is not because of the bet being very bad.

    (5) Pascal’s Wager
    Cute! I was wondering where you were going.
    I was puzzled by your analogy because I think of Pascal’s Gambit as a gamble between ALL or NOTHING. And even if the “All” is not infinite or vast, it is always better than “Nothing”.
    So I think the Pascal Wager does work — for some, just as the lottery works for some. Especially because inside the “All” (even if minute), is social status, traditions and much more.

    Did I make a mistake here?

  4. Ian

    Fun comments, Sabio, some responses on the non-lottery stuff below.

    exrelayman, and sabio, thanks for the viewpoints on Pascals Wager and lotteries. This post was not intende to make the most watertight of theological points, so I’m not going to respond point by point. I appreciate the different viewpoints.

    1) I seem to live in a mid-atlantic purgatory. I get paid in $, but tend to be quite colloquial when I speak. A cofounder of mine was Polish, and the cockney rhyming slang used to get to him. “Have a butchers at this…”.

    2) I’d never use bold-faced (did that emerge from typography?) – Bare- and bald- I’d use in slightly different ways, interesting to think about one’s one nuances of expression.

    3) I have a bit of sympathy with he economics=pseudoscience idea, but only a little. I think books like ‘black swan’ miss the point: most economists I actually have spoken too are well aware their science is more meteorology than physics. But it gets presented as something very different when it has passed through the lens of popular discourse. Short term, local and highly controlled processes are easy (and ‘rationality’ is one such component), but the overall dynamics are chaotic. I might have a biased sample, however, since most of the professional economists I met were through the Santa Fe Institute.

    5) No, I don’t think you made a mistake. Pascal’s Wager is normally posed as an afterlife bargain, which is what I was playing along with here. But definitely, the tradeoff changes when you include benefits before death too. And we broadly agree what benefits we think religion provides, I think. I’ve never met anyone who’s said “I heard Pascal’s Wager and thought – good point – so I became a Christian.”

  5. David Evans

    I think the normal objections are not “merely ways to shorten the odds.” Consider: if I am choosing between 2 religions, one of which (A) says that all disbelievers go to a terrible and eternal fate in Hell, while the other (B) says that disbelievers merely earn a spell in Purgatory, I should clearly choose A. If I am wrong and B is the true religion, my penalty will be finite.

    Moral: believe in the most cruel and judgmental God on offer.

  6. Ian

    David, thanks for your comment, and welcome to the blog! That’s a great point, yes. There might also be sets of gods that one could propitiate simultaneously to spread the risk: finding favour with three gods to avoid three shots an purgatory might be better than avoiding one chance at hell!

  7. oxymandias

    If salvation is infinitely great, no probability is small enough for you to pass.

    …unless you think there is an infinite number of possible gods. In which case, the overall ROI (Return on Ideology) is probably undefined.

    Unless you could find a way to define infinite salvation in terms of limits?

  8. Ian

    @oxymandias – Humans are very very efficient God-creators, but we’re a way off creating an infinite number, I’d say. But I agree, were there an infinite number of Gods with mutually exclusive demands on paradise, we’d all be screwed.

  9. oxymandias

    Why should we only consider the possible gods that humans have so far dreamt up? Why not all possible gods?

    Surely any god that is consistent with Pascal’s Wager qualifies. And there’s nothing in the wager to say that the true God would bother to make Itself known to us – It might just set up the rules of salvation and wander off. That would mean very low odds of anyone ever getting into heaven… but there’s nothing in Pascal’s Wager to say that the true God would be bothered by having set humanity an impossible task.

    Absent this limitation, it’s pretty trivial to construct an infinite set of gods. E.g. assign each string of English letters to a god as its name; make each god very jealous, very non-interventionist and very picky about spelling, but (evidently) completely unbothered by whether their name is pronounceable.

  10. oxymandias

    Come to think of that, it’s worse than that. You could equally assign each god a real-valued number as its “name”. That would give you an uncountably infinite number of gods. (That’s an atheist’s nightmare right there…)

    If the value of salvation is only countably infinite, that would mean that salvation had zero value under Pascal’s Wager.

  11. Ian

    Who says that we’re dealing with possible Gods, or just names, or anything abstract here? I get you’re trying to be clever, but yeah I can do Cantors diagonal slash too. So?

  12. hm

    Playing the lottery is not really about winning, but about the excitement you feel when you think about the possibility of winning. Not to different from getting drunk or high, and economic calculations miss the point.

  13. Ian

    Hi hm, and welcome. Also a good point, that whether rational or not is irrelevant for many (perhaps most) players. Thanks.

  14. Are you aware of Carrier’s (no, not him again!) essay on Pascal’s Wager? (apparently not)

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