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...
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...