SimonM comments on Prediction Markets aren’t Magic

[SimonM]

So the first question is: “how much should we expect the sample mean to move?”.

If the current state is ${\mu }_n$, and we see a sample of $x$ (where $x$ is going to be 0 or 1 based on whether or not we have heads or tails), then the expected change is:

$E\left(\right({\mu }_n-{\mu }_{n+1}{)}^2)=\frac{1}{(n+1{)}^2}E({\mu }_n-x)=\frac{1}{(n+1{)}^2}({\mu }_n^2-2{\mu }_{n}E(x)+E(x^2\left)\right)$

$\cdots =\frac{1}{(n+1{)}^2}({\mu }_n^2-2{\mu }_n{\mu }_n+{\mu }_n^2+{\sigma }_x^2)=\frac{{\sigma }_x^2}{(n+1{)}^2}\sim O\left(\frac{1}{n^2}\right)$

In these steps we are using the facts that ($x$ is independent of the previous samples, and the distribution of $x$ is Bernoulli with $p={\mu }_n$. (So $E\left(x\right)={\mu }_n$ and $Var\left(x\right)={\sigma }_x^2={\mu }_n(1-{\mu }_n)$).

To do the proper version of this, we would be interested in how our prior changes, and our distribution for $x$ wouldn’t purely be a function of ${\mu }_n$. This will reduce the difference, so I have glossed over this detail.

The next question is: “given we shift the market parameter by $O\left(1/n^2\right)$, how much money (pnl) should we expect to be able to extract from the market in expectation?”

For this, I am assuming that our market is equivalent to a proper scoring rule. This duality is laid out nicely here. Expending the proper scoring rule out locally, it must be of the form $f\left(x\right)=c+O\left(x^2\right)$, since we have to be at a local minima. To use some classic examples, in a log scoring rule:

$q\log \left(p\right)+(1-q)\log (1-p)=\left(q\log \right(q)-(q-1\left)\log \right(1-q\left)\right)+\frac{(p-q{)}^2}{\left(2\right(q-1\left)q\right)}+O\left(\right(p-q{)}^3)$

in a brier scoring rule:

$q(1-p{)}^2+(1-q)p^2=q(1-q{)}^2+(1-q)q^2+(q-p{)}^2$