Generalized Kelly betting
[tl;dr: It’s a mess, don’t go there]
[Thanks to Diff for proof reading this post]
Kelly betting has the very nice property that if a gambler is betting according to a given world model, and the amount of money the gambler starts out with equals to the prior probability of that model, then after each round of bets, this gamblers money will equal the current posterior probability.
The problem with Kelly betting is that it relies on only being given one bet at a time, and that the previous bet will be evaluated before you are asked to bet on a new question. Compare this to the situation faced by the traders in a Logical Inductor, where there are always multiple bets every round and the traders don’t know when any bet will be settled.
I have (almost) calculated the generalized Kelly criterion, in the case of two dual outcome simultaneous bets, with general market odds and general gambler beliefs. The only remaining part of this calculation is a cubic equation.
Send me an e-mail (linda.linsefors@gmail.com) if you want my notes. There is no guarantee that I will read all blog post comments.
Solving this last equation for general market odds, is in principle not very hard. You can find the general solution to cubic equations on Wikipedia. Except in this specific case the equation is:
solve for .
The probability that I will get this right by hand is near zero. Maybe someone with Mathematica, or a similar tool could help me?
For the notation, there are two simultaneous and independent bets. Each bet has two outcomes, denoted by index .
And the ′ superscript denotes the second bet. By definition
is just a help variable I made up. It does not have a super clear interpretation, but just happens to be the key to calculating everything else.
As mentioned above, I don’t have the final formula for the generalized Kelly criterion, in the case of two dual outcome simultaneous bets, with completely general market odds and gambler beliefs, not until someone solves the above equation. What I currently do have is the special case where the market probabilities for both statements is 50%, and this special case already shows that the nice property of money representing probability, is not preserved.
The Bayesian updating factor for hypotheses , given two independent observations and , and , should be
The update factor for the amount of money that a gambler following hypothesis has, given the above circumstances, is
Tl;dr: The problem is that we have no way to bet on joint outcomes. If we add bets on joint outcomes, then the market is complete, we can combine the two outcomes into a single joint outcome, and Kelly criteria should work. To properly break Kelly, we need bets which resolve at different times.
This hits on a critical point which is fundamental to mathematical finance, but virtually unknown outside of it: complete markets. A “complete market” is one in which we can place any possible bet on whatever random variables are involved.
For instance, if we have a stock market with nothing but a single stock, and we’re betting on the stock’s price in the next time-step, then that’s an incomplete market: we have no way to place a bet which pays $1 if the price ends up within some window, and $0 otherwise. On the other hand, if we add in the full option chain (call options at every possible price), then the market is complete. We can pick a portfolio of options to make any possible bet on the stock’s price next timestep.
Mathematically, incomplete markets are a mess. You can’t get the bet you actually want, so you’re stuck trying to approximate it with the available bets, and that approximation gets messy.
On the other hand, if you do have complete markets, then you can combine everything into a single random variable and just use the Kelly criterion.
Can you recommend me a good textbook that covers these things? I know basic econ (Krugman and Wells) and a bunch of probability and game theory.
Chapter 6 of Cover & Thomas’ “Elements of Information Theory” gives good info on the Kelly criterion, how to derive it, and the relations between prices/probabilities and entropy/rate of return.
For math finance, the class I took back in college used Shreve’s “Stochastic Calculus for Finance II”. I wouldn’t necessarily recommend that just to learn about this, but it’s a good source for brownian motion, some basic measure theory, and the core theory of asset pricing.
Typically complete markets come up in discussing the fundamental theorem of asset pricing. The first part of the theorem says that any arbitrage-free set of asset prices has a “risk-neutral measure”, i.e. a market-implied set of probabilities. The second part says those probabilities are unique iff the market is complete—if some bets can’t be placed, then there are multiple possible market-implied probabilities. Any book which covers the fundamental theorem should have at least some coverage of complete markets.
Finally, if you’re looking for something more applied, Hull’s “Options, Futures and Other Derivatives” is the usual starting point.
Thank you!
I think that for Kelly betting to work out in the case of multiple propositions, you need to have a combinatorial prediction market that effectively lets you bet on every possible joint outcome—especially in the realistic case where the propositions you’re looking at aren’t independent. The good news is that if you do have a combinatorial prediction market, then the property of money representing probability is maintained even if only some propositions are revealed, as described in the last section of my unfortunately overly long blog post on the topic.
[Edit: see also johnswentworth’s LessWrong comment which makes a similar point and which I didn’t see before making my AlignmentForum comment]