The Las Vegas Paradox

8 01 2007

It seems safe to say that money (and basically any other good) has a generally diminishing marginal value. This is perhaps one of the biggest justification for redistributive taxation, in which we take a bunch of money unequally from people and give it to people in some much more even distribution, as with social security and some other government programs. (Of course, most programs redistribute things unequally, but still often in a more equal way than the money was originally distributed.)

However, another sort of redistribution sometimes seems justified, and it suggests that the marginal value of money can’t be strictly decreasing. If we took a penny from everyone in the country, and gave the resulting $3,000,000 to one person at random, it seems that it would make that one person tremendously happy at basically no cost to anyone else. And sure enough, people voluntarily engage in this sort of activity all the time, in raffles, and (notably, especially since I just spent a week and a half visiting my parents at their new place in Las Vegas) in slot machines. In fact, in both cases, people willingly take part despite the fact that some of the money is siphoned off either for charity or to the rich people that own the casinos.

Now, perhaps this behavior is just irrational (so that we shouldn’t derive any moral about the marginal utility of money from it). Or perhaps people get some other benefit from the transaction (like the feeling of doing good for a charity in the case of a raffle, or the excitement one gets from occasionally getting small payoffs that one promptly loses again in the slot machine). But at some level, the original game of taking a penny from everyone to give the entire amount to someone chosen at random just intuitively (at least to me) seems reasonable.

However, there may be some sort of argument that it isn’t reasonable. After all, if it was an improvement to overall utility to do that, then some sort of principle of additivity (which I’m willing to question for other reasons however) would suggest that it would be good to do it multiple times. It’s unclear at what point it could go from being good to bad (maybe there’s a sorites in here?) so if it’s good to do it once, then it would be good to do it 300,000,000 times. But at that point, it seems that it could have a severe negative effect on total utility (if some people ended up losing $3,000,000 overall while some other lucky ones won it two or more times – a loss of $3,000,000 is clearly much worse to almost everyone than a gain of that much is good to anyone) or at best a neutral effect (if everyone wins exactly once). So either it was never good to begin with (which just seems implausible to me), or it switches from being good to being bad at some point (though it seems very hard to say where), or else we have to give up some sort of additivity for gambles, though it’s unclear just how.

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9 responses

9 01 2007
mike love

http://youtube.com/?v=Ju_iQucOrH4

you guys should redo mathporn with processing, no?

9 01 2007
Greg F-A

Hi Kenny —

1. When I first moved to Vegas, I had my own kind of ‘Las Vegas Paradox,’ which was an instance of Moore’s paradox:

(LVP) I live in Vegas, but I don’t believe that I live in Vegas.

The manifest truth of this sentence for the first few weeks I was here convinced me, in a very vivid and concrete way, that Moore’s paradox is not really paradoxical.

2. Next time you’re in Vegas, you should give me a call if you want to meet up.

9 01 2007
Kenny

Greg – I figured you were probably out of town at the time (either at the Eastern or with family) but next time I’ll certainly look you up.

10 01 2007
Jordan DeLange

Alastair Norcross was a professor of mine who published a bullet-biting ethics paper where he advocated a form of consequentialism where it was justified to kill a person in order to relieve billions of people who had headaches. Seems like an analogous case to your Las Vegas paradox, although here its causing lots of harm to one to minimally help lots of others, rather than minimally harming lots to help a few.

That said, in this case I would still think the reason we would see overall negative utility in the large-scale case would be because of increased marginal utility for the people who lost a bunch and never won, as compared to the people who won more than once, right? Due to decreasing marginal utility, someone-winning-lots wouldn’t be more well off enough to offset the multiple people who would have lost and never won, I think.

10 01 2007
Michael Greinecker

That logic is incompatible with utilitarianism if utility functions are continuous. But if you have discontinuous utility functions, repeating the thing several times may suddenly lead to non-zero losses for the involved persons.

So the idea is incompatible with utilitarianism the way it is usually conceived of.

10 01 2007
Kenny

Which logic are you talking about – the reasoning that says the gamble can be good, or the reasoning that says that a bunch of good things can add up to a bad one?

It’s true that utilitarianism (and decision theory) presuppose that utility is a real number, and therefore additive. For a variety of reasons, I’d rather suggest that in general it’s only a partial ordering, and that additivity doesn’t quite make sense. Of course, a wide range of circumstances can be modeled well with real values, but just not all.

11 01 2007
Michael Greinecker

Im talking about the reasoning that says there is no cost to anyone but someone gets $3,000,000.

“For a variety of reasons, I’d rather suggest that in general it’s only a partial ordering, and that additivity doesn’t quite make sense.”

But than the justification of “that money (and basically any other good) has a generally diminishing marginal value. This is perhaps one of the biggest justification for redistributive taxation, in which we take a bunch of money unequally from people and give it to people in some much more even distribution, as with social security and some other government programs.”
breaks down.

17 04 2007
K. C.

Isn’t the decreasing marginal utility of money used to justify the progressive nature of the income tax structure, and not redistribution of wealth in general? The logic is that since each dollar is worth less to you, it is not so bad to take away a larger percentage, as your income is increased. The redistribution of wealth is justified on various grounds, like the value of equality, the badness of poverty, the government’s need to provide services. . .

17 04 2007
Kenny

I’m not sure. Decreasing marginal utility might justify redistribution, because the money is worth more to the person you’re giving it to than the person you’re taking it away from. As for the progressiveness, it’s not clear whether decreasing marginal utility justifies that. If the dollar is already worth less to you than the earlier ones, then why should you get less of it? Mightn’t one see that as adding insult to injury? (Of course, the only “injury” here is having so many dollars that additional ones aren’t worth that much to you.) But I guess the idea is that if we need to get $100,000,000 dollars for some government project, it’s better to take it from the people that won’t miss their dollars all that much?

Anyway, it’s two slightly different arguments. Decreasing marginal utility automatically makes equality valuable, and presumably derives (at least in part) from the badness of poverty.

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