But if we flip it around, we can also say that for any fixed time-horizon, there exists a number of bettors such that the EV-maximizers dominate the market throughout that time with arbitrary probability.
The number of bettors isn’t the relevant parameter here. The relevant parameter is what fraction of the bettors are Kelly vs EV. However you set it up, the fraction of money in the hands of EV bettors will decrease over long time periods with high probability. If we have some fixed time-horizon, as long as that time horizon is fairly long, EV-maximizers will only dominate the market throughout that time with high probability if the market is essentially all EV-maximizers at the beginning.
An analogy: if one species has higher reproductive fitness than another, will that species eventually dominate? The math for Kelly betting is identical to the usual setup for natural selection models.
The point with having a large number of bettors is to assume that they all get independent sources of randomness, so at least some will win all their bets. Handwavy math follows:
Assume that we have n EV bettors and n Kelly bettors (each starting with $1), and that they’re presented with a string of bets with 0.75 probability of doubling any money they risk. The EV bettors will bet everything at each time-step, while the Kelly bettors will bet half at each time-step. For any timestep t, there will be an n such that approximately 0.75t of EV bettors have won all their bets (by the law of large numbers), for a total earning of 0.75t2tn=1.5tn. Meanwhile, each Kelly bettor will in expectation multiply their earnings by 1.25 each time-step, and so in expectation have 1.25t after t timesteps. By the law of large numbers, for a sufficiently large n they will in aggregate have approximately 1.25tn. Since 1.5tn>1.25tn, the EV-maximizers will have more money, and we can get an arbitrarily high probability with an arbitrarily large n.
Ah, I see. The usual derivation of the Kelly criterion explicitly assumes that there is a specific sequence of events on which people are betting (e.g. stock market movements or horse-race outcomes); the players do not get to all bet separately on independent sources of randomness. If they could do that, then it would change the setup completely—it opens the door to agents making profits by trading with each other (in order to diversify their portfolios via risk-trades with other agents). Generally speaking, with idealized agents in economic equilibrium, they should all trade risk until they all effectively have access to the same randomness sources.
Another way to think about it: compare the performance of counterfactual Kelly and EV agents on the same opportunities. In other words, suppose I look at my historical stock picks and ask how I would have performed had I been a Kelly bettor or an EV bettor. With probability approaching 1 over time, Kelly betting will seem like a better idea than EV betting in hindsight.
Thanks, that way to derive it makes sense! The point about free trade also seems right. With free trade, EV bettors will buy all risk from Kelly bettors until the former is gone with high probabiliity.
So my point only applies to bettors that can’t trade. Basically, in almost every market, the majority of resources are controlled by Kelly bettors; but across all markets in the multiverse, the majority of resources are controlled by EV bettors, because they make bets such that they dominate the markets which contain most of the multiverse’s resources.
(Or if there’s no sufficiently large multiverse, Kelly bettors will dominate with arbitrary probability; but EV bettors will (tautologically) still get the most expected money.)
The number of bettors isn’t the relevant parameter here. The relevant parameter is what fraction of the bettors are Kelly vs EV. However you set it up, the fraction of money in the hands of EV bettors will decrease over long time periods with high probability. If we have some fixed time-horizon, as long as that time horizon is fairly long, EV-maximizers will only dominate the market throughout that time with high probability if the market is essentially all EV-maximizers at the beginning.
An analogy: if one species has higher reproductive fitness than another, will that species eventually dominate? The math for Kelly betting is identical to the usual setup for natural selection models.
The point with having a large number of bettors is to assume that they all get independent sources of randomness, so at least some will win all their bets. Handwavy math follows:
Assume that we have n EV bettors and n Kelly bettors (each starting with $1), and that they’re presented with a string of bets with 0.75 probability of doubling any money they risk. The EV bettors will bet everything at each time-step, while the Kelly bettors will bet half at each time-step. For any timestep t, there will be an n such that approximately 0.75t of EV bettors have won all their bets (by the law of large numbers), for a total earning of 0.75t2tn=1.5tn. Meanwhile, each Kelly bettor will in expectation multiply their earnings by 1.25 each time-step, and so in expectation have 1.25t after t timesteps. By the law of large numbers, for a sufficiently large n they will in aggregate have approximately 1.25tn. Since 1.5tn>1.25tn, the EV-maximizers will have more money, and we can get an arbitrarily high probability with an arbitrarily large n.
Ah, I see. The usual derivation of the Kelly criterion explicitly assumes that there is a specific sequence of events on which people are betting (e.g. stock market movements or horse-race outcomes); the players do not get to all bet separately on independent sources of randomness. If they could do that, then it would change the setup completely—it opens the door to agents making profits by trading with each other (in order to diversify their portfolios via risk-trades with other agents). Generally speaking, with idealized agents in economic equilibrium, they should all trade risk until they all effectively have access to the same randomness sources.
Another way to think about it: compare the performance of counterfactual Kelly and EV agents on the same opportunities. In other words, suppose I look at my historical stock picks and ask how I would have performed had I been a Kelly bettor or an EV bettor. With probability approaching 1 over time, Kelly betting will seem like a better idea than EV betting in hindsight.
Thanks, that way to derive it makes sense! The point about free trade also seems right. With free trade, EV bettors will buy all risk from Kelly bettors until the former is gone with high probabiliity.
So my point only applies to bettors that can’t trade. Basically, in almost every market, the majority of resources are controlled by Kelly bettors; but across all markets in the multiverse, the majority of resources are controlled by EV bettors, because they make bets such that they dominate the markets which contain most of the multiverse’s resources.
(Or if there’s no sufficiently large multiverse, Kelly bettors will dominate with arbitrary probability; but EV bettors will (tautologically) still get the most expected money.)