Kelly does kinda take nonlinearity of money into account, in the following sense.

Suppose your utility increases logarithmically with bankroll. (I think it’s widely thought that actually it grows a bit slower than that, but logarithmically will do.) Suppose you make a bet that with probability p wins you a fraction x of your bankroll and with probability 1-p loses you a fraction bx. You get to choose x but not p or b. Then your expected utility on making the bet is plog(1+x)+(1−p)log(1−bx) whose derivative w.r.t. x is p1+x−(1−p)b1−bx=p(1+b)−b(1+x)(1+x)(1−bx). You get max expected utility when p(1+b)=b(1+x) or equivalently when x=p(1+b)−bb which is exactly the Kelly bet. So betting Kelly maximizes your (short-term) expected utility, if your utility grows logarithmically with bankroll.

Well, near zero utility ~ log(wealth) would mean infinite negative utility for zero wealth. That seems obviously false—I would hate to lose all my possessions but I wouldn’t consider it infinitely bad and I can think of other things that I would hate more. (So near zero I think reality is less nonlinear than the log(bankroll) assumption treats it as being.) In reality, of course, your real wealth basically never goes all the way to zero because pretty much everyone has nonzero earning power or benevolence-of-friends or national-safety-net, and in any case when you’re contemplating Kelly-style bets I think it’s common to use something smaller than the total value of all your possessions as the bankroll in the calculation.

Kelly does

kindatake nonlinearity of money into account, in the following sense.Suppose your utility increases logarithmically with bankroll. (I think it’s widely thought that actually it grows a bit slower than that, but logarithmically will do.) Suppose you make a bet that with probability

pwins you a fractionxof your bankroll and with probability 1-ploses you a fractionbx. You get to choosexbut notporb. Then your expected utility on making the bet is plog(1+x)+(1−p)log(1−bx) whose derivative w.r.t.xis p1+x−(1−p)b1−bx=p(1+b)−b(1+x)(1+x)(1−bx). You get max expected utility when p(1+b)=b(1+x) or equivalently when x=p(1+b)−bb which is exactly the Kelly bet. So betting Kelly maximizes your (short-term) expected utility, if your utility grows logarithmically with bankroll.I agree with that, but I think that utility is not even log linear near zero.

Well, near zero utility ~ log(wealth) would mean

infinite negative utilityfor zero wealth. That seems obviously false—I would hate to lose all my possessions but I wouldn’t consider itinfinitely badand I can think of other things that I would hate more. (So near zero I think reality islessnonlinear than the log(bankroll) assumption treats it as being.) In reality, of course, your real wealth basically never goes all the way to zero because pretty much everyone has nonzero earning power or benevolence-of-friends or national-safety-net, and in any case when you’re contemplating Kelly-style bets I think it’s common to use something smaller than thetotal value of all your possessionsas the bankroll in the calculation.