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RichardKennaway comments on Open Thread June 2010, Part 2 - Less Wrong
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How many lottery tickets would you buy if the expected payoff was positive?
This is not a completely hypothetical question. For example, in the Euromillions weekly lottery, the jackpot accumulates from one week to the next until someone wins it. It is therefore in theory possible for the expected total payout to exceed the cost of tickets sold that week. Each ticket has a 1 in 76,275,360 (i.e. C(50,5)*C(9,2)) probability of winning the jackpot; multiple winners share the prize.
So, suppose someone draws your attention (since of course you don't bother following these things) to the number of weeks the jackpot has rolled over, and you do all the relevant calculations, and conclude that this week, the expected win from a €1 bet is €1.05. For simplicity, assume that the jackpot is the only prize. You are also smart enough to choose a set of numbers that look too non-random for any ordinary buyer of lottery tickets to choose them, so as to maximise your chance of having the jackpot all to yourself.
Do you buy any tickets, and if so how many?
If you judge that your utility for money is sublinear enough to make your expected gain in utilons negative, how large would the jackpot have to be at those odds before you bet?
The traditional answer is to follow the Kelly criterion, is it not? That would imply
where n is the number of tickets. This implies you should buy n such that (€1)*n = Wf*, where W is your initial wealth.
Edit: Thanks, JoshuaZ, for pointing out that the Kelly criterion might not be the applicable one in a given situation.
OK, I have a question! Suppose I hold a risky asset that costs me c at time t, and whose value at time t is predicted to be k * (1 + r), with standard deviation s. How can I calculate the length of time that I will have to hold the asset in order to rationally expect the asset to be worth, say, 2c with probability p?
I am not doing a finance class or anything; I am genuinely curious.
So am I - I'm only aware of the Kelly Criterion thanks to roland thinking I was alluding to it. I haven't worked through that calculation.
I knew about Kelly, but not well enough for the problem to bring it to mind.
I make the Kelly fraction of (bp-q)/b to work out to about epsilon/N where epsilon=0.05 and N = 76275360. So the optimal bet is 1 part in 1.5 billion of my wealth, which is approximately nothing.
The moral: buying lottery tickets is still a bad idea even when it's marginally profitable.
Yes, and note that Kelly gets much less optimal when you increase bet sizes then when you decrease bet sizes. So from a Kelly perspective, rounding up to a single ticket is probably a bad idea. Your point about sublinearity of utility for money makes it in general an even worse idea. However, I'm not sure that Kelly is the right approach here. In particular, Kelly is the correct attitude when you have a large number of opportunities to bet (indeed, it is the limiting case). However, lotteries which have a positive expected outcome are very rare.So you never approach anywhere near the limiting case. Remember, Kelly optimizes long-term growth.
That raises the question of what the rational thing to do is, when faced with a strictly one-time chance to buy a very small probability of a very large reward.
Well, no - you shouldn't buy one ticket. And according to my calculations when I tried plotting W versus n by my formula, the minimum of W is at "buy all the tickets", so unless you have €76,275,360 already...