I suggest augmenting the classic “bet or update” with “Kelly bet or update”.
Epistemic status: feels like maybe going too far, but worth considering?
On the one hand, we have a line of thinking in rationalist discourse which says probability is willingness to bet. This line of thinking suggests that many bad thinking patterns and misapplications of probability theory can broadly be discouraged by a culture of betting.
On the other hand, there’s the longstanding discussion of Aumann Agreement and modest epistemology. According to this line of thinking, beliefs should be contagious among honest and rational folk, so long as they believe each other to be honest and rational. One should never agree to disagree; where two disagree, at least one is wrong (irrationally wrong—or dishonest). This line of thinking has been developed extensively by Robin Hanson and others, and criticized heavily in Eliezer’s Inadequate Equilibria, as well as some earlieressays.
These two perspectives are reconciled in the creedobet or update, which has been proposed as a norm for rationalists: in any significant disagreement, you should either come to an agreement (at least of the Aumann sort, where you may not understand each other’s exact reasons, but assign enough outside-view credibility to each other that your final probabilities align), or, failing that, you should bet. The Hansons of the world can take the outside view and update to agree with each other, while the Yudkowskys of the world put their money where their mouth is.
But how much should you bet?
Risk Aversion
A common norm in the rationalist circles I’ve frequented is to bet small amounts. I think there are some arguments in favor of this:
This makes winning feel good and losing hurt, without putting too much on the line. We want opportunities to learn, we want some useful social pressure against overconfidence and other biases, but we’re not out to bankrupt anyone.
Often, bets have to do with things we have control over, such as placing a bet about when you’ll finish an important project. Bets which are small in relationship to the project’s importance help guarantee that no one gains a perverse financial incentive to delay or sabotage an important project. (On the other hand, people usually avoid perverse bets anyway; and, sometimes it is useful to use bets in the opposite way—setting up extra incentives in virtuous directions. So I’m not clear on how big this advantage is.)
However, small bets might also be the result of irrational risk aversion. This seems like the more likely causal explanation, at least.
I was recently reminded that I tend to act like I’m much more risk-averse than Kelly betting would advise, with no justification I can think of.
At this point, I would suggest the reader play around with the Kelly formula a little, and imagine betting that way with your savings as the bankroll. The formula frequently recommends betting a significant fraction of savings, on bets with a real chance of not paying out. If you’re like me, this feels reckless.
Most investors seem to be similar. Even though Kelly betting itself captures fairly extreme risk-aversion (equating an empty bank account with death), “fractional Kelly”, where one bets some percentage of what a true Kelly better would, appears to be much more popular than true Kelly betting.
One interpretation of this is not that investors are more risk-averse than true Kelly, but that they are not so confident of their own probability assessments. This seems sensible in the abstract, but doesn’t line up with the math of fractional Kelly.
Let’s say I’m betting on horse races and I use a mathematical model which I feel 90% confident in (that is, I think there’s a 10% chance I made a dumb mistake in my math; otherwise, I think the model is a good accounting of my subjective uncertainty). I don’t know what the other 10% of my beliefs look like, so I do a worst-case analysis on all my bets, acting like my probabilities of winning are 90% of whatever the model tells me.
My Kelly bets would normally be p−qb/a of my bankroll, where p,q are my probabilities for winning and losing respectively, and a,b are the house’s calculation of those probabilities. Adjusting for my 10% uncertainty in my math, this becomes .9p−q+.1pb/a. This can take me over the line from betting to not betting at all, for example if p=.85 and a=.80; fractional Kelly, on the other hand, never changes willingness to bet, only quantity.
EDIT: Liam Donovan points out that fractional Kelly can be justified by averaging between the market’s beliefs and your own. This makes a lot of sense. But I still want to point out that this implies updating toward the market belief, which is different from having a true best-estimate belief which differs from the market and fractional-kelly betting without updating. I’m fine with tracking an inside-view position plus an outside-view adjustment. But, for example, 25%-Kelly betting implies putting only 25% credence on my inside view. It’s worth asking myself the question, is my credence on my inside view really 25%? Or am I experiencing irrational loss aversion, focusing on the potential losses more than the potential gains? Am I tracking the probability that the market knows better in a sensible fashion?
The Proposal
Replace bet or update with Kelly bet or update. Don’t just “put your money where your mouth is”—rather, put your money where your mouth is in proportion to expected net winnings divided by best-case net winnings. Or else, update.
The Strong Position
From an expected value standpoint, small bets are almost like no bets at all. To put it severely: it’s as if rationalists noticed that VNM implies willingness to bet, and proceeded to bet as little as possible as a symbolic act, not further noticing that VNM also implies willingness to bet substantially.
Mirroring the “probability is willingness to bet” argument I mentioned at the beginning, the purist argument for Kelly bet or update is that “probability is willingness to Kelly bet”. Within a VNM expected value framework, this argument becomes: if you don’t Kelly bet, then you have to either admit that your probability is not what you said it was, or you have to give a good reason why your utility function is not approximately logarithmic.
Now, there could be lots of reasons why your utility function is not approximately logarithmic. Ability to actually buy something you within a fixed time period basically creates lots of discrete step-functions that go into your actual utility.
However, utility logarithmic in money does seem like a pretty good approximation of most people’s values, and I doubt that more detailed analysis will reveal a justification of the extreme risk aversion which accompanies most bets. (If you think I’m wrong here, I’m very curious to hear the reason!)
A Weaker Case
OK, but all said and done, Kelly-betting with my savings still seems too risky.
Here’s an alternative proposal. Make a special account in which you put some amount of money, say, $100. Call this your betting fund. Kelly bet with this. You might have rules such as “I can take money out if I accumulate a lot from betting, but I can never add more”—this means that if you lose the majority of your seed money, you have no choice but to crawl back up past $100 by betting well. (I’m not sure about this, just throwing it out there.)
The size of your betting account compared to its starting seed could be a mark of shame/honor among those who chose to engage in this kind of practice.
This is sort of like fractional Kelly, which I’ve already argued is irrational… but ah well.
Jacobian recently argued that we should Kelly bet more. One thing I noticed after reading that article is that I’m pretty bad at the Kelly formula. I barely ever approach Kelly-like calculations because I don’t really know how; I have to look up the formula every time, and I’m always using different versions of it, and have to double-check the meaning of the variables.
So, it seems like a good exercise for me to at least do the math more often.
FWIW the version that I think I’ll manage to remember is that the optional fraction of your bankroll to bet is the expected net winnings divided by the net winnings if you win.
I’ve found that I remember this formulation, but the difference between “net winnings” and “gross winnings” is enough to make me want to double-check things, and in the few months since writing the original post, I haven’t actually used this to calculate Kelly.
“expected net winnings divided by net winnings if you win” is easy enough to remember, but is it easy enough to calculate? When I try to calculate it, I think of it this way:
[probability of success] × [payoff of success] + [probability of failure] × [payoff of failure] all divided by [payoff of success].
This is a combination of five numbers (one being a repeat). We have to calculate probability of failure from the probability of success (ie, 1-p). Then we perform two multiplications, one addition, and one division—five steps of mental arithmetic.
But the formula is really a function of two numbers (see this graph for a vivid illustration). Can we formulate the calculation in a way that feels like just a function of two numbers?
Normally the formula is stated in terms of b, the net winnings if you win. I prefer to state it in terms of r=b+1. If there’s an opportunity to “double” your money, r=2 while b=1; so I think r is more how people intuitively think about things.
In these terms, the break-even point is just p=1r. This is easy to reconstruct with mental checks: if you stand to double your money, you’d better believe your chances of winning are at least 50%; if you stand to quadruple your money, chances had better be at least 25%; and so on. If you’re like me, calculating the break-even point as 1r is much easier than “the point where net expected winnings equals zero”—the expectation requires that I combine four numbers, while 1r requires just one.
The Kelly formula is just a linear interpolation from 0% at the break-even point to 100% on sure things. (Again, see the graph.)
So one way to calculate Kelly is to ask how much along the way between 1r and 1 we are.
For example, if r=2, then things start to be profitable after p=50%. If p=80%, we’re 3/5ths of the way, so would invest that part of our bankroll.
If r=3, then things start to be profitable after p=33.¯¯¯3%. If p=66.¯¯¯6%, then it’s half way to 100%, so we’d spend half of our bankroll.
If you’re like me, this is much easier to calculate than “expected net winnings divided by the net winnings if you win”. First I calculate 1r. Then I just have to compare this to p, to find where 1r stands along the way from p to 1. Technically, the formula for this is:
p−1r1−1r
That’s at worst a combination of four numbers (if you count the 1), with four steps of mental arithmetic: find 1r, two subtractions, and one division. So that’s a bit better. But, personally, I find that I don’t really have to think in terms of the explicit formula for simple cases like the examples above. Instead, I imagine the number line from 0 to 1, and ask myself how far along p is from 1r to 1, and come up with answers.
This also seems like a better way to intuit approximate answers. If I think an event has a 12% chance, and the potential payoff of a bet is to multiply my investment by 3.7, then I can’t immediately tell you what the Kelly bet is. However, I can immediately tell you that 1037 is less than halfway along the distance from 12% to 100%, or that it’s more than a tenth of the way. So I know the Kelly bet isn’t so much as half the bankroll, but it also isn’t so little as 10%.
Kelly Bet or Update?
I suggest augmenting the classic “bet or update” with “Kelly bet or update”.
Epistemic status: feels like maybe going too far, but worth considering?
On the one hand, we have a line of thinking in rationalist discourse which says probability is willingness to bet. This line of thinking suggests that many bad thinking patterns and misapplications of probability theory can broadly be discouraged by a culture of betting.
On the other hand, there’s the longstanding discussion of Aumann Agreement and modest epistemology. According to this line of thinking, beliefs should be contagious among honest and rational folk, so long as they believe each other to be honest and rational. One should never agree to disagree; where two disagree, at least one is wrong (irrationally wrong—or dishonest). This line of thinking has been developed extensively by Robin Hanson and others, and criticized heavily in Eliezer’s Inadequate Equilibria, as well as some earlier essays.
These two perspectives are reconciled in the creedo bet or update, which has been proposed as a norm for rationalists: in any significant disagreement, you should either come to an agreement (at least of the Aumann sort, where you may not understand each other’s exact reasons, but assign enough outside-view credibility to each other that your final probabilities align), or, failing that, you should bet. The Hansons of the world can take the outside view and update to agree with each other, while the Yudkowskys of the world put their money where their mouth is.
But how much should you bet?
Risk Aversion
A common norm in the rationalist circles I’ve frequented is to bet small amounts. I think there are some arguments in favor of this:
This makes winning feel good and losing hurt, without putting too much on the line. We want opportunities to learn, we want some useful social pressure against overconfidence and other biases, but we’re not out to bankrupt anyone.
Often, bets have to do with things we have control over, such as placing a bet about when you’ll finish an important project. Bets which are small in relationship to the project’s importance help guarantee that no one gains a perverse financial incentive to delay or sabotage an important project. (On the other hand, people usually avoid perverse bets anyway; and, sometimes it is useful to use bets in the opposite way—setting up extra incentives in virtuous directions. So I’m not clear on how big this advantage is.)
However, small bets might also be the result of irrational risk aversion. This seems like the more likely causal explanation, at least.
I was recently reminded that I tend to act like I’m much more risk-averse than Kelly betting would advise, with no justification I can think of.
At this point, I would suggest the reader play around with the Kelly formula a little, and imagine betting that way with your savings as the bankroll. The formula frequently recommends betting a significant fraction of savings, on bets with a real chance of not paying out. If you’re like me, this feels reckless.
Most investors seem to be similar. Even though Kelly betting itself captures fairly extreme risk-aversion (equating an empty bank account with death), “fractional Kelly”, where one bets some percentage of what a true Kelly better would, appears to be much more popular than true Kelly betting.
One interpretation of this is not that investors are more risk-averse than true Kelly, but that they are not so confident of their own probability assessments. This seems sensible in the abstract, but doesn’t line up with the math of fractional Kelly.
Let’s say I’m betting on horse races and I use a mathematical model which I feel 90% confident in (that is, I think there’s a 10% chance I made a dumb mistake in my math; otherwise, I think the model is a good accounting of my subjective uncertainty). I don’t know what the other 10% of my beliefs look like, so I do a worst-case analysis on all my bets, acting like my probabilities of winning are 90% of whatever the model tells me.
My Kelly bets would normally be p−qb/a of my bankroll, where p,q are my probabilities for winning and losing respectively, and a,b are the house’s calculation of those probabilities. Adjusting for my 10% uncertainty in my math, this becomes .9p−q+.1pb/a. This can take me over the line from betting to not betting at all, for example if p=.85 and a=.80; fractional Kelly, on the other hand, never changes willingness to bet, only quantity.
EDIT: Liam Donovan points out that fractional Kelly can be justified by averaging between the market’s beliefs and your own. This makes a lot of sense. But I still want to point out that this implies updating toward the market belief, which is different from having a true best-estimate belief which differs from the market and fractional-kelly betting without updating. I’m fine with tracking an inside-view position plus an outside-view adjustment. But, for example, 25%-Kelly betting implies putting only 25% credence on my inside view. It’s worth asking myself the question, is my credence on my inside view really 25%? Or am I experiencing irrational loss aversion, focusing on the potential losses more than the potential gains? Am I tracking the probability that the market knows better in a sensible fashion?
The Proposal
Replace bet or update with Kelly bet or update. Don’t just “put your money where your mouth is”—rather, put your money where your mouth is in proportion to expected net winnings divided by best-case net winnings. Or else, update.
The Strong Position
From an expected value standpoint, small bets are almost like no bets at all. To put it severely: it’s as if rationalists noticed that VNM implies willingness to bet, and proceeded to bet as little as possible as a symbolic act, not further noticing that VNM also implies willingness to bet substantially.
Mirroring the “probability is willingness to bet” argument I mentioned at the beginning, the purist argument for Kelly bet or update is that “probability is willingness to Kelly bet”. Within a VNM expected value framework, this argument becomes: if you don’t Kelly bet, then you have to either admit that your probability is not what you said it was, or you have to give a good reason why your utility function is not approximately logarithmic.
Now, there could be lots of reasons why your utility function is not approximately logarithmic. Ability to actually buy something you within a fixed time period basically creates lots of discrete step-functions that go into your actual utility.
However, utility logarithmic in money does seem like a pretty good approximation of most people’s values, and I doubt that more detailed analysis will reveal a justification of the extreme risk aversion which accompanies most bets. (If you think I’m wrong here, I’m very curious to hear the reason!)
A Weaker Case
OK, but all said and done, Kelly-betting with my savings still seems too risky.
Here’s an alternative proposal. Make a special account in which you put some amount of money, say, $100. Call this your betting fund. Kelly bet with this. You might have rules such as “I can take money out if I accumulate a lot from betting, but I can never add more”—this means that if you lose the majority of your seed money, you have no choice but to crawl back up past $100 by betting well. (I’m not sure about this, just throwing it out there.)
The size of your betting account compared to its starting seed could be a mark of shame/honor among those who chose to engage in this kind of practice.
This is sort of like fractional Kelly, which I’ve already argued is irrational… but ah well.
Jacobian recently argued that we should Kelly bet more. One thing I noticed after reading that article is that I’m pretty bad at the Kelly formula. I barely ever approach Kelly-like calculations because I don’t really know how; I have to look up the formula every time, and I’m always using different versions of it, and have to double-check the meaning of the variables.
So, it seems like a good exercise for me to at least do the math more often.
Addendum: Calculating Kelly
In the comments, Daniel Filan mentioned:
I’ve found that I remember this formulation, but the difference between “net winnings” and “gross winnings” is enough to make me want to double-check things, and in the few months since writing the original post, I haven’t actually used this to calculate Kelly.
“expected net winnings divided by net winnings if you win” is easy enough to remember, but is it easy enough to calculate? When I try to calculate it, I think of it this way:
[probability of success] × [payoff of success] + [probability of failure] × [payoff of failure] all divided by [payoff of success].
This is a combination of five numbers (one being a repeat). We have to calculate probability of failure from the probability of success (ie, 1-p). Then we perform two multiplications, one addition, and one division—five steps of mental arithmetic.
But the formula is really a function of two numbers (see this graph for a vivid illustration). Can we formulate the calculation in a way that feels like just a function of two numbers?
Normally the formula is stated in terms of b, the net winnings if you win. I prefer to state it in terms of r=b+1. If there’s an opportunity to “double” your money, r=2 while b=1; so I think r is more how people intuitively think about things.
In these terms, the break-even point is just p=1r. This is easy to reconstruct with mental checks: if you stand to double your money, you’d better believe your chances of winning are at least 50%; if you stand to quadruple your money, chances had better be at least 25%; and so on. If you’re like me, calculating the break-even point as 1r is much easier than “the point where net expected winnings equals zero”—the expectation requires that I combine four numbers, while 1r requires just one.
The Kelly formula is just a linear interpolation from 0% at the break-even point to 100% on sure things. (Again, see the graph.)
So one way to calculate Kelly is to ask how much along the way between 1r and 1 we are.
For example, if r=2, then things start to be profitable after p=50%. If p=80%, we’re 3/5ths of the way, so would invest that part of our bankroll.
If r=3, then things start to be profitable after p=33.¯¯¯3%. If p=66.¯¯¯6%, then it’s half way to 100%, so we’d spend half of our bankroll.
If you’re like me, this is much easier to calculate than “expected net winnings divided by the net winnings if you win”. First I calculate 1r. Then I just have to compare this to p, to find where 1r stands along the way from p to 1. Technically, the formula for this is:
p−1r1−1rThat’s at worst a combination of four numbers (if you count the 1), with four steps of mental arithmetic: find 1r, two subtractions, and one division. So that’s a bit better. But, personally, I find that I don’t really have to think in terms of the explicit formula for simple cases like the examples above. Instead, I imagine the number line from 0 to 1, and ask myself how far along p is from 1r to 1, and come up with answers.
This also seems like a better way to intuit approximate answers. If I think an event has a 12% chance, and the potential payoff of a bet is to multiply my investment by 3.7, then I can’t immediately tell you what the Kelly bet is. However, I can immediately tell you that 1037 is less than halfway along the distance from 12% to 100%, or that it’s more than a tenth of the way. So I know the Kelly bet isn’t so much as half the bankroll, but it also isn’t so little as 10%.