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RobinZ comments on Open Thread June 2010, Part 2 - Less Wrong
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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...