[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:

$a({\beta }_{11}{)}^3+b({\beta }_{11}{)}^2+c{\beta }_{11}+d=0$

$\begin{array}{cc}a& =m_1{m}_1^{\prime }(1-m_1-m_1^{\prime })\\ b& =m_1{m}_1^{\prime }-p_1{m}_2{m}_1^{\prime }-p_1^{\prime }{m}_1{m}_2^{\prime }+2p_1{p}_1^{\prime }{m}_1{m}_1^{\prime }\\ c& =p_1{p}_1^{\prime }[-(m_1+m_1^{\prime })+p_1{m}_2^{\prime }+p_1^{\prime }{m}_2-p_1{p}_1^{\prime }]\\ d& =(p_1{p}_1^{\prime }{)}^2\end{array}$

solve for ${\beta }_{11}$.

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 $x\in \{1,2\}$.

$\begin{array}{cc}{p}_x& =\text{probability of outcome}x\text{, according to gambler.}\\ m_x& =\text{probability of outcome}x\text{, according to the market.}\end{array}$

And the ' superscript denotes the second bet. By definition

$\begin{array}{cc}{p}_1+p_2=1,& p_1^{\prime }+p_2^{\prime }=1\\ m_1+m_2=1,& m_1^{\prime }+m_2^{\prime }=1\end{array}$

${\beta }_{11}$ 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 $H$, given two independent observations $O$ and $O^{\prime }$, and $P\left(O\right)=P\left(O^{\prime }\right)=\frac{1}{2}$, should be

$\frac{P\left(O|H\right)}{P\left(O\right)}\frac{P\left(O^{\prime }|H\right)}{P\left(O^{\prime }\right)}=4P\left(O|H\right)P\left(O^{\prime }|H\right)$

The update factor for the amount of money that a gambler following hypothesis $H$ has, given the above circumstances, is

$\frac{2P\left(O|H\right)P\left(O^{\prime }|H\right)}{P\left(O|H\right)P\left(O^{\prime }|H\right)+P\left(\neg O|H\right)P\left(\neg O^{\prime }|H\right)}$