Assume we have an initial bankroll of $B$ dollars and we can spend $3k$ to play a dice game which has payoff $kn$ where $n$ is the outcome of the roll. I believe that the expected payout of this game is $3.5k$, so the player has an edge of $1/6$, and the variance is $1.2k^2$.
Assume we can play only $m$ rounds of this game. If we bet a the same proportion $p$ of our bankroll on each of $m$ bets we can make a trade-off between the mean and variance of our total expected profit. If $p$ is close to zero we have low variance and low mean profit, and if $p$ is close to one our expected profit and the variance are both high. I know of the Kelly Criterion, which leads to the best expected returns in a scenario with two outcomes.
Is there a best choice for $p$? Is there some sort of generalisation of the Kelly Criterion that applies to this situation?
Kelly maximises the product of your outcomes.
Bet 3p of your wealth every time, get back (1-2p)(1-p)(1)(1+p)(1+2p)(1+3p) after six.
This is maximised at $$p=0.1756,3p=0.527$$