If utility is logarithmic in wealth, the Kelly Criterion tells me the right size of stake to put on a given bet, given the odds offered, my subjective probability and my wealth. In practice, in the real world, what’s the right number to plug into the “wealth” part of the equation? My current savings? My yearly salary? The value of my home minus the money owing on it?
The “wealth” part of the equation is the total amount you’re willing to gamble with. If you have money set aside for frivolities like food, then that wouldn’t be part of your wealth as far as the Kelly Criterion is concerned.
The general principle with gambling is never to bet more than you’re willing to lose. Kelly betting is optimal in the sense that over the long run, no other system will outperform it. In the short run, it’s quite volatile and you can get very low.
Not so much in response to your specific question, but when trying to figure out what I can afford, I actually take a pretty simple approach: my liquid assets (mostly in the bank) plus things I could easily liquidate (stocks, etc.) minus a “rainy day fund” (this has varied in size over the years, but tends to sit at between 2 and 10 thousand USD, based on how hard I think it would be to get a job or find housing in the event that I lost one or both). Things like 401K and HSA are omitted; they’re already earmarked for specific things and mean I don’t have to worry about keeping other funds back for those purposes. Assets that are technically resalable but either not worth the effort or of high utility to my daily life (my computer, my car, etc.) are also omitted, though in a pinch I would of course sell them too.
The result is the money I can afford to spend. I can use it on video games, or vacations, or gifts for people, or a new car (at which point I would sell the old one), or fighting malaria. I can trickle it away on living expenses if I decide to quit my job and pursue hobbies or whatever (I would start looking for a new one once I got within “expected job hunt time * cash outflow rate” distance of the bottom of this wealth, though I could dip into the rainy day fund if needed).
I can also invest it into riskier things than a savings account, like stocks… or into any other kind of betting.
The answer probably depends on what your utility is nearest to being proportional to the logarithm of.
Most likely your utility (so far as you have one) looks like other_stuff + f(wealth) where wealth = g(annual_income, liquid_assets, other_assets) or something of the kind, where f and g are functions about which we don’t know very much. It’s probably OK to assume that g is just a linear combination of its inputs. So it seems like there are two things to do.
Figure out what g looks like by imagining various possible states-of-wealth and ordering them by preference. E.g., would you rather an income of £50k/year and assets of £500k of which £100k are easily accessible, or an income of £200k/year and no assets, or no income and assets of £1M of which £200k are easily accessible? Etc.
Figure out what f looks like by imagining various gambles. E.g., suppose you have no assets; would you prefer a salary of £30k or of (£25k or £40k, equiprobably)?
And then you can try plugging the result into the Kelly formula, seeing how over-risky it feels, and (if you are so inclined) correcting for excess risk aversion not already factored into f.
No, it is true if you want to have the highest expectation value of utility in your life, and the answer to ciphergoth’s question is you use all your wealth.
If instead of wanting to maximizing utility, you might prefer to minimize the probability that your utility will fall below a certain level. In this case, the bad tails of the distribution of Kelly criterion strategies matters to you and the expected utility does not. You might come up with some modification of Kelly criterion that meets this criterion of avoiding really bad outcomes with high probability. Or you might find some entirely different criterion or policy that meets the avoidance of bad outcomes that you are trying to achieve.
The OP’s original question hides a lot of complexity in its “in the real world” clause. In the real world, are we expected-utility maximizers? Or are we low-utility-probability minimizers? Or are we something else? Until we know we can’t evaluate investment/betting strategies.
If utility is logarithmic in wealth, the Kelly Criterion tells me the right size of stake to put on a given bet, given the odds offered, my subjective probability and my wealth. In practice, in the real world, what’s the right number to plug into the “wealth” part of the equation? My current savings? My yearly salary? The value of my home minus the money owing on it?
The “wealth” part of the equation is the total amount you’re willing to gamble with. If you have money set aside for frivolities like food, then that wouldn’t be part of your wealth as far as the Kelly Criterion is concerned.
The general principle with gambling is never to bet more than you’re willing to lose. Kelly betting is optimal in the sense that over the long run, no other system will outperform it. In the short run, it’s quite volatile and you can get very low.
The amount which you can afford to lose.
Not so much in response to your specific question, but when trying to figure out what I can afford, I actually take a pretty simple approach: my liquid assets (mostly in the bank) plus things I could easily liquidate (stocks, etc.) minus a “rainy day fund” (this has varied in size over the years, but tends to sit at between 2 and 10 thousand USD, based on how hard I think it would be to get a job or find housing in the event that I lost one or both). Things like 401K and HSA are omitted; they’re already earmarked for specific things and mean I don’t have to worry about keeping other funds back for those purposes. Assets that are technically resalable but either not worth the effort or of high utility to my daily life (my computer, my car, etc.) are also omitted, though in a pinch I would of course sell them too.
The result is the money I can afford to spend. I can use it on video games, or vacations, or gifts for people, or a new car (at which point I would sell the old one), or fighting malaria. I can trickle it away on living expenses if I decide to quit my job and pursue hobbies or whatever (I would start looking for a new one once I got within “expected job hunt time * cash outflow rate” distance of the bottom of this wealth, though I could dip into the rainy day fund if needed).
I can also invest it into riskier things than a savings account, like stocks… or into any other kind of betting.
The answer probably depends on what your utility is nearest to being proportional to the logarithm of.
Most likely your utility (so far as you have one) looks like other_stuff + f(wealth) where wealth = g(annual_income, liquid_assets, other_assets) or something of the kind, where f and g are functions about which we don’t know very much. It’s probably OK to assume that g is just a linear combination of its inputs. So it seems like there are two things to do.
Figure out what g looks like by imagining various possible states-of-wealth and ordering them by preference. E.g., would you rather an income of £50k/year and assets of £500k of which £100k are easily accessible, or an income of £200k/year and no assets, or no income and assets of £1M of which £200k are easily accessible? Etc.
Figure out what f looks like by imagining various gambles. E.g., suppose you have no assets; would you prefer a salary of £30k or of (£25k or £40k, equiprobably)?
And then you can try plugging the result into the Kelly formula, seeing how over-risky it feels, and (if you are so inclined) correcting for excess risk aversion not already factored into f.
The same thing that you want to maximize in the long run.
This is true, but incomplete. If utility is monotonically increasing with wealth, the Kelly Criterion tells you how to size your bets.
That is only true if you are making lots of bets and expect them to be your main source of income. But that assumes away Ciphergoth’s question.
No, it is true if you want to have the highest expectation value of utility in your life, and the answer to ciphergoth’s question is you use all your wealth.
If instead of wanting to maximizing utility, you might prefer to minimize the probability that your utility will fall below a certain level. In this case, the bad tails of the distribution of Kelly criterion strategies matters to you and the expected utility does not. You might come up with some modification of Kelly criterion that meets this criterion of avoiding really bad outcomes with high probability. Or you might find some entirely different criterion or policy that meets the avoidance of bad outcomes that you are trying to achieve.
The OP’s original question hides a lot of complexity in its “in the real world” clause. In the real world, are we expected-utility maximizers? Or are we low-utility-probability minimizers? Or are we something else? Until we know we can’t evaluate investment/betting strategies.
A little knowledge is a dangerous thing.