I'm having a hard time choosing a good strategy for this problem: assume that you have $m$ money that you can bet on $n$ mutually exclusive outcomes, all with unknown probabilities, and that each outcome $i$ has a known payoff $P_i$. If each $P_i$ is big enough, you can construct a vector of bets $b$, such that for each outcome $i$,
$b_i\geq 0$,
$\sum_k b_k=m$, and
$P_i*b_i\geq \sum_k b_k$
(that is, whatever the outcome, you won't lose money).
Given no knowledge of the probability of each outcome, what would be a "good" strategy to choose each bet? In other words, what would you choose as your goal function?
So far I've tried maximizing $\sum_i (P_i*b_i)$; however this (and all linear goal functions) result in a strategy in which you only win money on the outcome with the smallest payoff, and recover your money on every other. It would probably be nice to make money on every possible outcome. What strategy would you choose and why?
You can scale the bets so you get the same payoff regardless of what the outcome is. So if you are willing to bet $b_{tot}$, let $$b_i=\frac {b_{tot}/P_i}{\sum_j 1/P_j}$$ In each case you get $b_iP_i=\frac {b_{tot}}{\sum_j 1/P_j}$, so you make a profit if $\frac 1{\sum_j 1/P_j} \gt 1$
Example added: Suppose there are three outcome with payoffs $2,4,5$. As $\frac 1{\frac 12+\frac 14+\frac 15}=\frac 1{0.95}\approx 1.0526 \gt 1$ we can win. If we have $100$ to bet, we make sure we get $105.26$ back, so bet $52.63,26.315, 21.052$