Suppose that I am interested in a market on a betting exchange for the outright winner of some event, with three competitors, $A, B$ and $C$ with corresponding probabilities of winning $a, b$ and $c$. We expect that the sum $a + b + c = 1$.
Now suppose that the probability of $A$ winning changes to some $a'$ (as defined by that bet's current price). We expect $b$ and $c$ to change to $b'$ and $c'$ such that $a' + b' + c' = 1$.
By what mechanism do $b$ and $c$ change? I see two options:
- They move by market forces, since if at any instant $a + b + c < 1$ people will take this arbitrage opportunity which will push $b$ and $c$ up.
- The probabilities sum to one rule is enforced 'externally' by the exchange through some mechanism. I am interested in how such a mechanism could work.
I'm aware that there are probably different ways of doing this, but I'm interested in any ideas people can come up with!