(This was inspired by the following question by Daniel Murfet: “Can you elaborate on why I should care about Kelly betting? I guess I’m looking for an answer of the form “the market is a dynamical process that computes a probability distribution, perhaps the Bayesian posterior, and because of out of equilibrium effects or time lags or X, the information you derive from the market is not the Bayesian posterior and therefore you should bet somehow differently in a way that reflects that”?”)
1. (Kelly betting is asymptotically dominant) Kelly betting is the asymptotically dominant strategy—it dominates (meaning it has more money) all betting strategies with probability approaching 1 as the time horizon goes to infinity. [this is explained in section of 16.3 of Thomas & Cover’s Information Theory textbook]. For long enough time horizons we should expect the Kelly bettors to dominate.
2. (Evolution selects for Kelly Betting) Evolution selects for Kelly bettor—in the evolutionary biology literature people talk about the mean-variance trade-off.
Define the fitness of an organism O as the number of offspring (this is a random variable) it produces in a generation. Then according to natural selection the organism ‘should’ maximize not the absolute fitness E[# of offspring of O] but it should maximize the (long-run) relative inclusive fitness or equivalently the inclusive fitness growth rate.
Remark. That evolution selects for relative fitness—not absolute fitness could select for more ‘spiteful strategies’ like big cats killing each other cubs (both inter and intra-species)
3. (Selection Theorems and Formal Darwinism)
One of the primary pillars of Wentworth’s agenda is ‘Selection Theorems’: mathematically precise theorems that state what kind of agents might be ‘selected’ for in certain situations. The Kelly optimality theorem (section 16.3 Thomas & Cover) can be seen as a form of selection theorem: it states that over time Kelly bettors will exponentially start to dominate other agents. It would be of interest to see whether this can be elucidated and the relation with natural selection be improved.
This closely ties in with a stream of work on Formal Darwinism, a research programme to mathematically if, how and in what sense natural selection creates optimizes for ‘fitness’ see also Okasha’s “Agents and Goals in Evolution”
4. (Ergodicity Economics) Ole Peters argues that Kelly betting (or his more general version of maximizing ‘time-averaged’ growth) ‘solves’ the St. Petersburg utility paradox and points to a revolutionary new point of view in foundations of economics: “Ergodicity Economics”. As you can imagine this is rather controversial.
5. (Kelly betting and Entropy) Kelly betting is intrinsically tied to the notion of entropy. Indeed, Kelly discovered Kelly betting to explain Shannon’s new informational entropy—only later was it used to beat the house at Las Vegas.
6. (Relevance of Information)A criminally-underrated paper by Madsen continues on Kelly’s original idea and generalizes to a notion of (Madsen-)Kelly utility. It measures the ‘relevance of information’. Madsen investigates a number of cool examples where this type of thinking is quite useful.
7. (Bayesian Updating) If we consider a population of hypotheses Hi with a prior ϕ(i) , we can think of an individual hypothesis Hi as a Kelly bettor with wealth ϕ(i) distributing its bets according to Hi. In other words, it bets Hi(x)ϕ(i) on each outcome x in the sample space Ω. It can’t ‘hold money’ at the side—it must bet all its money. In this case, Kelly betting recommends betting according to your internal probability distribution (which is just Hi in this case).
Remark. What happens in the case that the bettors can hold money on the side? In other words, we would consider a more flexible bettor. That’s quite an interesting question I’d like to answer. I suspect it has to do with Renyi entropy and β tempered distributions.
If we consider a collection of realization {x1,...,xn} the new wealth of the Hi will be Hi(x1)⋅Hi(x2)...Hi(xn)ϕ(i). This is if we bet against Nature. In this case, one can only ‘lose’. However, real betting is against a counterparty. In this case it will be betting against the average of the whole market H(x)=∑iHi(x)ϕ(i). If an event E happens the new wealth of Hi will be Hi(x)H(x)ϕ(i).
This is of course the Bayesian posterior.
If we sample from a ‘true’ distribution q, the long-term wealth of Hi will be proportional to ∝ϕ(i)exp(KL(q|Hi))
(This was inspired by the following question by Daniel Murfet: “Can you elaborate on why I should care about Kelly betting? I guess I’m looking for an answer of the form “the market is a dynamical process that computes a probability distribution, perhaps the Bayesian posterior, and because of out of equilibrium effects or time lags or X, the information you derive from the market is not the Bayesian posterior and therefore you should bet somehow differently in a way that reflects that”?”)
[See also: Kelly bet or update and Superrational agents Kelly bet influence]
Why care about Kelly betting?
1. (Kelly betting is asymptotically dominant) Kelly betting is the asymptotically dominant strategy—it dominates (meaning it has more money) all betting strategies with probability approaching 1 as the time horizon goes to infinity. [this is explained in section of 16.3 of Thomas & Cover’s Information Theory textbook]. For long enough time horizons we should expect the Kelly bettors to dominate.
2. (Evolution selects for Kelly Betting) Evolution selects for Kelly bettor—in the evolutionary biology literature people talk about the mean-variance trade-off.
Define the fitness of an organism O as the number of offspring (this is a random variable) it produces in a generation. Then according to natural selection the organism ‘should’ maximize not the absolute fitness E[# of offspring of O] but it should maximize the (long-run) relative inclusive fitness or equivalently the inclusive fitness growth rate.
Remark. That evolution selects for relative fitness—not absolute fitness could select for more ‘spiteful strategies’ like big cats killing each other cubs (both inter and intra-species)
3. (Selection Theorems and Formal Darwinism)
One of the primary pillars of Wentworth’s agenda is ‘Selection Theorems’: mathematically precise theorems that state what kind of agents might be ‘selected’ for in certain situations. The Kelly optimality theorem (section 16.3 Thomas & Cover) can be seen as a form of selection theorem: it states that over time Kelly bettors will exponentially start to dominate other agents. It would be of interest to see whether this can be elucidated and the relation with natural selection be improved.
This closely ties in with a stream of work on Formal Darwinism, a research programme to mathematically if, how and in what sense natural selection creates optimizes for ‘fitness’ see also Okasha’s “Agents and Goals in Evolution”
4. (Ergodicity Economics) Ole Peters argues that Kelly betting (or his more general version of maximizing ‘time-averaged’ growth) ‘solves’ the St. Petersburg utility paradox and points to a revolutionary new point of view in foundations of economics: “Ergodicity Economics”. As you can imagine this is rather controversial.
5. (Kelly betting and Entropy) Kelly betting is intrinsically tied to the notion of entropy. Indeed, Kelly discovered Kelly betting to explain Shannon’s new informational entropy—only later was it used to beat the house at Las Vegas.
6. (Relevance of Information) A criminally-underrated paper by Madsen continues on Kelly’s original idea and generalizes to a notion of (Madsen-)Kelly utility. It measures the ‘relevance of information’. Madsen investigates a number of cool examples where this type of thinking is quite useful.
7. (Bayesian Updating) If we consider a population of hypotheses Hi with a prior ϕ(i) , we can think of an individual hypothesis Hi as a Kelly bettor with wealth ϕ(i) distributing its bets according to Hi. In other words, it bets Hi(x)ϕ(i) on each outcome x in the sample space Ω. It can’t ‘hold money’ at the side—it must bet all its money. In this case, Kelly betting recommends betting according to your internal probability distribution (which is just Hi in this case).
Remark. What happens in the case that the bettors can hold money on the side? In other words, we would consider a more flexible bettor. That’s quite an interesting question I’d like to answer. I suspect it has to do with Renyi entropy and β tempered distributions.
If we consider a collection of realization {x1,...,xn} the new wealth of the Hi will be Hi(x1)⋅Hi(x2)...Hi(xn)ϕ(i). This is if we bet against Nature. In this case, one can only ‘lose’. However, real betting is against a counterparty. In this case it will be betting against the average of the whole market H(x)=∑iHi(x)ϕ(i). If an event E happens the new wealth of Hi will be Hi(x)H(x)ϕ(i).
This is of course the Bayesian posterior.
If we sample from a ‘true’ distribution q, the long-term wealth of Hi will be proportional to ∝ϕ(i)exp(KL(q|Hi))
8. (Blackjack) One application of Kelly betting is bankrupting the House and becoming a 1/2-billionaire.