Is this just a lot of words to say “many wagers are constrained such that the Kelly-recommended amount is out of range, so it becomes a binary “bet or don’t”? I don’t see the linearity there—you’re still using logarithmic valuation to make fixed bets (or not, if the minimum is much higher than the Kelly amount).
No. The point of the model where acting like your utility is linear is optimal wasn’t that this is a more realistic model than the assumptions behind the Kelly criterion; it’s just another simplified model, which is slightly easier to analyze, so I was using it as a step in showing why you should follow the Kelly criterion when it is your wealth that constrains the bet sizes you can make. It’s also not true that the linear-utility model I described is still just maximizing log wealth; for instance, if the reason that you’re never constrained by available funds is that you have access to credit, then your wealth could go negative, and then its log wouldn’t even be defined.
Sure, I’m a fan of simplified calculations. But it’s not either-or. Kelly simplifies to “bet your edge” for even-money wagers, and that’s great. It simplifies to “bet the max on +EV wagers” in cases where “the max” is a small portion of your net worth.
It’s great, but it’s not a different model, it’s just a simplified calculation for special cases.
Again, the max being a small portion of your net worth isn’t the assumption behind the model; the assumption is just that you don’t get constrained by lack of funds, so it is a different model. It’s true that if the reason you don’t get constrained by lack of funds is that the maximum bets are small relative to your net worth, then this is also consistent with maximizing log wealth on each step. But this isn’t relevant to what I brought it up for, which was to use it as a step in explaining the reason for the Kelly criterion in the section after it.
I suspect I’m dense, or at least missing something. If ability to make future bets aren’t impacted by losing earlier bets, that implies that you cannot bet more than Kelly. Or are there other ways to not be constrained by lack of funds?
An example of a bet you’d make in the linear model, which you wouldn’t make in the logarithmic (bankroll-preserving) model, would help a lot.
Access to credit, presuming it’s finite, just moves the floor, it doesn’t change the shape. It gets complicated when credit has a cost, because it affects the EV of bets that might force you to dip into credit for future bets. If it’s zero-interest, you can just consider it part of your bankroll. Likewise future earnings—just part of your bankroll (though also complicated if future earnings are uncertain).
It is true that in practice, there’s a finite amount of credit you can get, and credit has a cost, limiting the practical applicability of a model with unlimited access to free credit, if the optimal strategy according to the model would end up likely making use of credit which you couldn’t realistically get cheaply. None of this seems important to me. The easiest way to understand the optimal strategy when maximum bet sizes are much smaller than your wealth is that it maximizes expected wealth on each step, rather than that it maximizes expected log wealth on each step. This is especially true if you don’t already understand why following the Kelly criterion is instrumentally useful, and I hadn’t yet gotten to the section where I explained that, and in fact used the linear model in order to show that Kelly betting is optimal by showing that it’s just the linear model on a log scale.
One could similarly object that since currency is discrete, you can’t go below 1 unit of currency and continue to make bets, so you need to maintain a log-scale bankroll where you prevent your log wealth from going negative, and you should really be maximizing your expected log log wealth, which happens to give you the same results when your wealth is a large enough number of currency units that the discretization doesn’t make a difference. Like, sure, I guess, but it’s still useful to model currency as continuous, so I see no need to account for its discreteness in a model. Similarly, in situations where the limitations on funds available to place bets with don’t end up affecting you, I don’t think it needs to be explicitly included in the model.
Is this just a lot of words to say “many wagers are constrained such that the Kelly-recommended amount is out of range, so it becomes a binary “bet or don’t”? I don’t see the linearity there—you’re still using logarithmic valuation to make fixed bets (or not, if the minimum is much higher than the Kelly amount).
No. The point of the model where acting like your utility is linear is optimal wasn’t that this is a more realistic model than the assumptions behind the Kelly criterion; it’s just another simplified model, which is slightly easier to analyze, so I was using it as a step in showing why you should follow the Kelly criterion when it is your wealth that constrains the bet sizes you can make. It’s also not true that the linear-utility model I described is still just maximizing log wealth; for instance, if the reason that you’re never constrained by available funds is that you have access to credit, then your wealth could go negative, and then its log wouldn’t even be defined.
Sure, I’m a fan of simplified calculations. But it’s not either-or. Kelly simplifies to “bet your edge” for even-money wagers, and that’s great. It simplifies to “bet the max on +EV wagers” in cases where “the max” is a small portion of your net worth.
It’s great, but it’s not a different model, it’s just a simplified calculation for special cases.
Again, the max being a small portion of your net worth isn’t the assumption behind the model; the assumption is just that you don’t get constrained by lack of funds, so it is a different model. It’s true that if the reason you don’t get constrained by lack of funds is that the maximum bets are small relative to your net worth, then this is also consistent with maximizing log wealth on each step. But this isn’t relevant to what I brought it up for, which was to use it as a step in explaining the reason for the Kelly criterion in the section after it.
I suspect I’m dense, or at least missing something. If ability to make future bets aren’t impacted by losing earlier bets, that implies that you cannot bet more than Kelly. Or are there other ways to not be constrained by lack of funds?
An example of a bet you’d make in the linear model, which you wouldn’t make in the logarithmic (bankroll-preserving) model, would help a lot.
Access to credit. In the logarithmic model, you never make bets that could make your net worth zero or negative.
Access to credit, presuming it’s finite, just moves the floor, it doesn’t change the shape. It gets complicated when credit has a cost, because it affects the EV of bets that might force you to dip into credit for future bets. If it’s zero-interest, you can just consider it part of your bankroll. Likewise future earnings—just part of your bankroll (though also complicated if future earnings are uncertain).
It is true that in practice, there’s a finite amount of credit you can get, and credit has a cost, limiting the practical applicability of a model with unlimited access to free credit, if the optimal strategy according to the model would end up likely making use of credit which you couldn’t realistically get cheaply. None of this seems important to me. The easiest way to understand the optimal strategy when maximum bet sizes are much smaller than your wealth is that it maximizes expected wealth on each step, rather than that it maximizes expected log wealth on each step. This is especially true if you don’t already understand why following the Kelly criterion is instrumentally useful, and I hadn’t yet gotten to the section where I explained that, and in fact used the linear model in order to show that Kelly betting is optimal by showing that it’s just the linear model on a log scale.
One could similarly object that since currency is discrete, you can’t go below 1 unit of currency and continue to make bets, so you need to maintain a log-scale bankroll where you prevent your log wealth from going negative, and you should really be maximizing your expected log log wealth, which happens to give you the same results when your wealth is a large enough number of currency units that the discretization doesn’t make a difference. Like, sure, I guess, but it’s still useful to model currency as continuous, so I see no need to account for its discreteness in a model. Similarly, in situations where the limitations on funds available to place bets with don’t end up affecting you, I don’t think it needs to be explicitly included in the model.