Lotteries, Rationality and Pascal’s Wager
Whenever a lottery’s in the news, one thing you can be sure of is that you’ll see a lot of snarky comments about people being gullible or bad at maths, and generally allowing the commenter to feel superior to the irrational drones who buy lottery tickets, which invariably sets me off on one of my hobby-horses. I don’t play the lottery, but I think this attitude is unhelpful and inaccurate, and here’s why.
First, a quick rundown of the statistics. The expected return on a gamble (because let’s be honest, that’s what a lottery is) is a weighted calculation of the probability of different winning outcomes multiplied by the values of those outcomes. In normal circumstances, the expected return for a lottery is easily determined from the proportion of each ticket which goes towards the prize fund. That will always be less than 100% of the stake, for reasons which should be obvious, so if you started with a fixed fund and played every week, sooner or later you’d run out of money.
When a lottery’s big news, it usually means (as in the recent Mega Millions case) that there’s a large “rollover” jackpot on offer. Even though the normal odds are weighted against ticket buyers, the addition of a large sum from the previous week’s prize fund can tip the balance towards the players. The odds of winning don’t change, but the rewards do. The precise calculations vary from lottery to lottery, but the addition of an effective subsidy from the previous draw improves the expected return dramatically. It may even be possible, if you restrict yourself to rollover draws, to play the lottery without running out of money from your hypothetical fixed fund.
That’s not the full picture, though, because no one plays in the long-run. Even the longest-lived person will still be a short-term player in gambling terms, because of the extremely long odds against winning a big prize. That doesn’t change the maths, but it does make it important to consider the value you place on different outcomes.
If you could buy a £1 ticket that gave you odds of exactly 1 in 1 million of winning £1m, most people would probably buy it, even though the expected return is only to break even. But if you asked someone to quantify the pleasure (or utility in economics-speak) they’d get from £1m, they’d be incredibly unlikely to value it a million times higher than having £1, as that would value each pound equally, which people don’t do in real life. You can test this yourself by considering how you’d feel about finding a pound coin if you were a) completely broke, or b) a millionaire. The utility gained from £1 is diminished the richer you are.
But utility cuts both ways. The reason why people would buy into that £1m draw and play lotteries with much worse expected returns is that gambling can also bring great pleasure even without a win, either through the possibility of a big win, or (though this doesn’t apply directly to lotteries) through the thrill of competition. We often spend money on intangible things which give us pleasure without being accused of irrationality, and if sufficient value is placed on the utility gained in this way, it may be considered rational to play the lottery after all.
All of which brings me (at last) to Pascal’s Wager. It’s received a lot of justified criticism on several different fronts, and is often regarded (not least by me) as a notorious pitfall to avoid in argument, rather than an attitude to aspire to. But I wonder if it’s possible to get some understanding of belief by approaching Pascal’s Wager from a different angle – in the way I analysed the rationality of lotteries.
Let’s leave the question of heaven and hell to one side – what’s the utility of belief in this life, or in lottery terms, is the playing as important as the winning? Is it possible that – for some people at least – belief in God would make sense whether it was true or not, as long as they believed it to be true? Could there be people for whom the utility gained in this life through belief exceeds the utility lost as a result of time and money invested in those beliefs?
I think there may be something in this, but it’s worth distinguishing between two sorts of rationality – the economic sense of maximising satisfaction or utility, and the scientific, logical sense of drawing justified conclusions from the available evidence. A given belief may fulfil either, both or neither definition, but I’m looking at this in a purely economic sense.
I’m the sort of person who cares about propositional truth, so I’ve always been puzzled by people who believe things while giving no indication that they’re all that bothered about whether their beliefs are actually true. Thinking about lotteries and railing against the facile mockery of people who play them suggests that this approach may not be as strange as I previously thought.
Maybe it’s rational after all, just not necessarily my sort of rationality.