Wikipedia:Reference desk/Archives/Mathematics/2021 September 7

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

September 7

Betting formula

What is the formula for calculating what the optimal bet is in the simplified scenario where the odds of the event are known and the total prize pools on both sides of the bet are known? e.g. 500 points have been bet on A and 800 points have been bet on notA (which implies a ~38% chance of A happening), so if you think that notA happens 90% of the time, you would expect to profit by betting on notA, so if you bet 200 points on notA you would win $500*{\frac {200}{800+200}}=100$ points 90% of the time and lose 200 points 10% of the time, resulting in an expected profit of $100*0.9+-200*0.1=70$ points. My searches so far have only found the Kelly criterion#Gambling_formula, which appears to be the special case where the total prize pool is infinite. Iffy★Chat -- 20:37, 7 September 2021 (UTC)

Interesting problem. Note: The following is probably not the proper way to do it since it gets very messy and I might have made a mistake somewhere, but it's what I've done.

To simplify the problem, let's say a total of 1 point has been bet already (you can obviously just scale up by e.g. 1300 in your case). Given $a$ bet on option A (in your case 5/13), $1-a$ bet on option B (in your case 8/13), and $P$ as the probability that A is correct (in your case 1/10), the expected value of a bet of $k$ on A is $k\left({\frac {P(1-a)}{a+k}}-(1-P)\right)$. (Here, we have $0<a<1,0<P<1$). Taking the derivative with respect to $k$: Product rule (and chain rule for the second term) gives ${\frac {P(1-a)}{a+k}}-(1-P)-k\left({\frac {P(1-a)}{(a+k)^{2}}}\right)$. We want to find the zeroes of this, so multiplying by $(a+k)^{2}$ yields $(a+k)P(1-a)-(a+k)^{2}(1-P)-kP(1-a)=aP(1-a)-(a+k)^{2}(1-P)$, so we want those two to be equal. $a$ and $P$ are constant, so we need $a+k={\sqrt {\frac {aP(1-a)}{1-P}}}\implies k={\sqrt {\frac {aP(1-a)}{1-P}}}-a$ is a critical point (and in this case, will be a local maximum). Now, note that this value of $k$ is only positive if ${\frac {P(1-a)}{1-P}}>a\iff P-aP>a-aP\iff P>a$ (after a bit of manipulation), so that's the only case in which this idea works. (This is common sense -- as you stated, you would expect to profit by betting on the option if your predicted probability is more than what the prize pool implies.)

So, if $P>a$, you should bet ${\sqrt {\frac {aP(1-a)}{1-P}}}-a$ on option A, and if $P<a$, we can simply reverse the roles of A and B, i.e. you should bet ${\sqrt {\frac {(1-a)(1-P)a}{P}}}-(1-a)$ on B. In your case, the second applies, so you end up with ${\sqrt {\frac {(8/13)\cdot 0.9\cdot (5/13)}{0.1}}}-{\frac {8}{13}}={\frac {6{\sqrt {10}}-8}{13}}\approx 0.844$, or after scaling back up by 1300, $600{\sqrt {10}}-800\approx 1097.$ (We can check that with a bet of 1097 on B, you would expect to win $0.9\cdot 500\cdot {\frac {1097}{800+1097}}+-1097\cdot 0.1\approx 150.53$ points.) Of course, this relies on your predicted probability $P$ is accurate. eviolite (talk) 00:44, 8 September 2021 (UTC)

Edit: added some stuff on the expected results.. also a courtesy ping @Iffy: eviolite (talk) 00:48, 8 September 2021 (UTC)