Also, why one account keeps running dry when you try to arbitrage two markets.

I was thinking about how inter-market arbitrage might affect one's account balances & the total amount of money on the markets arbitraged.

## The arbitrage. 

Let's say I have accounts on prediction markets A and B, and I have discovered an arbitrage opportunity for an event that has a price/probability of a on A and b on B such that a>b.

This price difference allows us to arbitrage the markets by buying shares of NO on market A and buying shares of YES on market B. So currently our accounts would look like:

Cost:

Market A: (1-a) dollars spent

Market B: b dollars spent

Let the "true" probability of the event happening be `p`. Then, when the market resolves, we profit on:
- Market A: `(1-p)` dollars, in expectation
- Market B: `p` dollars, in expectation

So we can expect to make
- `(1-p) - (1-a) = a-p` dollars on market A
- `p-b` dollars on market B

The arbitrage works because `a-p + p-b = a-b` which is positive, and there is no risk because we always get `1` dollar back regardless of whether the event happens or not.

## The flow of money

If the two markets' probability estimates of the events average to the true probability of the event (such that `(a+b)/2 = p`), then we can expect our profit to be split evenly between market A and B, since if `a+b=2p`, then `a-p = p-b`.

If the markets are skewed such that A is more correct than B, then we can expect to make less money on A than B, as `a-p < p-b` when `2p < a+b`. [^1]

So in effect, by performing the arbitrage, we are draining money from the less accurate prediction market, and feeding into the more accurate one.

If we were to try to continue the arbitrage, we will have to withdraw money from B and feed it into A, **draining money from the less accurate market and putting money into the more accurate one.**

[^1]: Without loss of generality, the same applies if market B had a close-to-true probability estimate than A