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.
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.