Reasonable strategy for simple game

Question

I have a very simple probabilistic process which I have to deal with in the software project I'm involved in, yet I can't figure how to do it. I can describe the situation as a money game (in reality it is not, but this is the most simple way to go):

At each moment $t$ my program gets a probability value $P_t$ on input. I have to output the stake $X_t$ ($X_t$ can be, say, from 0 to 10) and then with probability $P_t$ I will win (get $XA$, where $A > 1$) and with probability ($1 - P_t$) I will lose (get nothing, lose my stake $X$).

What strategy can be considered reasonable here? Make stake proportional to $P$? What should I read to understand the topic?

Answer 1

It rather depends on your utility function for wealth.

To maximise your expected return, bet everything when the odds benefit you, i.e. when $AP_t \gt 1$. This also gives you a good chance of losing all your money.

For a safer strategy which adjusts your bet size to the extent the odds are in your favour and avoids losing everything, read about the Kelly criterion, for example at Wikipedia. This is largely based on logarithmic utility.

Answer 2

Updated: Your expected gain is $X_t(P_tA- 1).$ If this is greater than zero, you are winning and should bet all you can. If less than zero, you are losing and bet the least you can. This presumes that the desirability of money is linear and you are far from the ends (losing so you can't play any more or winning so much your opponent quits).