# Can playing lotteries be rational?

Suppose you have to choose between:

• a. getting 1\$
• b. getting a ticket of the lottery L(p) which gives you 1'000'000'000\$ with probability p and 0 with probability (1-p), with p such that the expected gain is less than 1\$

could it be rational to play?

One could think that the 1\$ you get by choosing a doesn't actually change your life in any relevant way, while the choice of b could actually change your life completely, althought with a very small probability, so b seems rational. But this would amount to ignore the size of the probability p (provided it is positive), and this means also we are ignoring the expected value of the utility. So do p and expected utility actually matter for a rational decision?

• Is this just about the utility value of money? I think you're right that utility and money don't scale proportionally, and you're right that a million dollars seems like more than a million times better than one dollar, but I don't see why this is an argument against expected utility as a mechanism in rational choice. We might think that the choice is really between a 1 "Utile" outcome and a billion "utile" outcome, with the expected utilities (though not the expected monetary payoff) ultimately equalling out in the case of the lottery win. – Paul Ross Sep 28 '12 at 22:07
• I think it's more tricky: actually the argument above applies independently of the utility function choosed because it doesn't depend on the value of p: it's saying that it is rational to prefer the "possibility" of a great positive change Vs almost no change. There is no utility function that would work in this way for any value of p. – Marco Disce Sep 29 '12 at 6:08
• I think @PaulRoss has it right. It's not just the value of "p" that changes here, it's to do with the "utility" of each dollar. If we posit that every dollar that less than doubles your worth is worth 1/10 utility points, and every dollar after that is worth 1 (and perhaps that every dollar after 10 times your worth is worth 10), then p can be quite low to have this make sense here. – Ryno Sep 29 '12 at 19:04
• (BTW, I'm not suggesting that this is the actual utility spread, just that this is an extreme spread that would make the function work) – Ryno Sep 29 '12 at 19:06
• @ Ryno: wait, the challenge is to provide a utility function such that for every p the lottery L(p) described above is preferred to 1\$. Your example works only with some values of p, not for arbitrary small values of p. – Marco Disce Sep 29 '12 at 20:22

The comment of Marco have given me another idea of the example of rationality of lottery playing.

Let's assume I'm the prisoner on the island, and I'm getting 10\$ a day. 5\$ a day is enough to buy everything I need (food, accomodation etc.) and there's little to be bought on that island. I can't eat twice as much. So the additional 5\$ a day have practically no value for me.

I could store them, but it has no sense - because the only valuable think I could buy for a lot of money is freedom, and possibility to move to the city, which costs 10.000.000\$. It's easy to calculate that even saving 5\$ a day wouldn't be enough to get freedom before death.

So, the only chance to be free is to play a lottery. The value of freedom is extremally high, and the value of additional food which I don't need very low. In that model, playing a lottery is rational even with very low propability of win.

• My example is similar: Imagine you owe \$100,000 to a loan shark, and he will kill you tomorrow unless paid in full. If you only have a portion of the money, and can't get it, and can't get away, playing the lottery for a slim chance of winning is better than definitely getting your head bashed in. Similar scenarios could be dreamed up for expensive medical care, etc. So... there are rational circumstances, but they tend to be exceptional. – kbelder Jan 7 '13 at 19:28
• Upvoted because it's the only post that actually answers the question. – medivh Jun 23 '13 at 22:30

Strictly speaking, the answer to your question depends on what you define to be rational. "Rationality" is not a utility theory; it is essentially just a subjective judgment regarding the quality of your reasoning about such things. For example, if it is held that the best way to lose weight is by eating lots of tofu burgers and nothing else (regardless of whether it's actually true or not), then it is most rational eat lots of tofu burgers if you want to lose weight.

Likewise, if you deem it an immense moral good to win \$1 lottery tickets, even when your net gains end up negative, than it is rational to buy lottery tickets.

Call me fastidious, but I just wanted to make that distinction clear. What you perhaps mean to imply then is whether the lottery is rational given the average person's desire of maximizing gains and minimizing losses ("do p and expected utility actually matter for a rational decision?"). At this point, this question no longer becomes a philosophy question and is purely mathematics and your own personal views of what is "rational". Depending on how much "weight" you place on various outcomes and investments, any given lottery may or may not be rational.

This person who won 4 lotteries seems to think the lottery can be very rational....

• "According to Forbes, the residents of Bishop, Texas, seem to believe God was behind it all." – Sklivvz Oct 4 '12 at 23:45

As stoicfury noted, it's hard to define "rational" in a non-question-begging way. But let's recall why we believe the expected utility hypothesis: because any person who follows some basic rules will act as though they are maximizing the expectation of some utility function.

On your thought experiment: sure spending \$1 will cost me very little, but it will almost certainly gain me nothing. Deciding I'd rather have the small gain for certain that an (almost zero) possibility of a huge gain does not seem "irrational".

Pascal's mugging is a more extreme example of your thought experiment, which you might find interesting.

• Would you consider it irrational to prefer the lottery L(p) (described in my post) over 1\$ for any p>0? Which of the basic rules would it violate? – Marco Disce Sep 30 '12 at 15:13
• Say U(\$10<sup>10</sup)=x. The EU of winning is px; the EU of getting \$1 is (let's say) 1. For any p < 1/x, getting the dollar is preferred. For p > 1/x, the lottery is preferred. – Xodarap Sep 30 '12 at 19:06
• Ok, so apparently it cannot be rational to prefer L(p) for any p. Am I wrong? It's not clear to me which of the axiom of rationality it does violate. – Marco Disce Oct 1 '12 at 12:22
• Correct, there must be some p for which the lottery is not preferred. I don't know that this violates any individual axiom, but it violates the EU hypothesis (which can be proven from those axioms). – Xodarap Oct 1 '12 at 12:47
• Actually it's not hard to find an utility function where L(p) will we preferred for any p: Just use an utility function which is not monotonous in the amount of dollars. If winning \$1 is considered worse than both winning nothing and winning a lot, then L(p) is the better choice independent of p because no matter whether you win, the result will be better (according to your utility function) than winning exactly one dollar. – celtschk Oct 1 '12 at 20:36

Two scenarios.

"a. getting 1\$ b. getting a ticket of the lottery L(p) which gives you 1'000'000'000\$ with probability p and 0 with probability (1-p), with p such that the expected gain is less than 1\$" AND b. having a real causal relation with said consumer rethinking some useless expense worth 1\$ (say, a candy bar which brings short-term joy but long term decline in health). I think this is getting closer to "real world" ecological rationality. I know I could induce myself to think in accordance with this "AND" clause. If we think about real causal relations (inducing a habit of saving for a lottery ticket), this seems to make sense.

I think lottery playing can, in practice and in a real ecological situation, be rational. We are never rational consumers to begin with, and the 52 dollars saved per year from not spending on lottery brings us very little well-being (or added chance for well-being or survival). Whereas a lottery win would drastically improve your well-being and chances of survival (e.g. the medical care example above, but a more "realistic" example would be the relief of stress from day to day work, which significantly increases risk of death -- a dollar saved per week arguably does not relieve stress at all!).

There have been actual cases where people noted the expected value of lottery tickets exceeded face value. Here's a famous case: