# How is the “probabilities sum to $1$” rule enforced in betting exchanges?

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!

• Just something you might find interesting - usually, the bookies have odds such that $a+b+c>1$, so they always have a slight upper hand – John Doe Apr 29 '17 at 13:52
• This question does not make much sense because exchanges are not bookies. Take a look at How does a betting exchange work?. – Rodrigo de Azevedo Apr 30 '17 at 16:18

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.

If participants A, B and C are all competitors in the same contest and there can only by 1 winner then the sum of all 3 contracts should = 1 at any point in time. When the contest ends, the winning contract is equal to 1, the others zero.

While betting is open, if the odds of one competitor is increasing, then the odds of the other competitors must be decreasing because the sum of all three binary events must equal 1.

If the sum is greater or less than 1, an arbitrage exists. If it is less than 1, you could buy all 3 contracts knowing that the winning contract will be equal to 1 when the contest is over. The opposite would be true if the sum is greater than 1--you could sell all three knowing that you will only owe 1 when the contest closes. Either way, you win.

Exchanges don't need to enforce this with a set of rules. Market participants watch for arbitrages continuously and if/when one does exist, orders are entered as quickly and with as much size as possible until the arbitrage no longer exists. For this reason, they rarely exist.

Suppose we have a horse race in which only $3$ horses — Alex, Bob and Charlie — are taking part.

Suppose further that you believe that Charlie will win with probability $0.25$. You then offer odds of $4.0$ on Charlie winning, putting, say, $\$1000$at stake. Assuming that no commissions are charged, your expected profit is $$0.25 \cdot (\4000 - \1000) + 0.75 \cdot (\0 - \1000) = \0$$ Some John Doe decides to take the bet and puts$\$3000$ at stake. Thus, John Doe also believes that Charlie will win with probability $0.25$. Assuming that no commissions are charged, John Doe's expected profit is

$$0.25 \cdot (\0 - \3000) + 0.75 \cdot (\4000 - \3000) = \0$$

The betting exchange then locks your $\$1000$and John Doe's$\$3000$ until the horse race is over and a winner is announced. Suppose that the betting exchange charges a commission of $5\%$.

• If Charlie wins the race, you get $\$3800$and John Doe loses his$\$3000$. You profit $\$2800$. • If Charlie does not win, you lose your$\$1000$ and John Does gets $\$3800$. He profits$\$800$.

Either way, the betting exchange earns a commission of $\$200$only for matching your bet with John Doe's bet. The betting exchange neither backs nor lays, it merely matches backer and layer. The betting exchange's matching engine ensures that it locks enough money to pay the backer or the layer. Suppose John Doe puts only$\$2000$ at stake. How can the betting exchange pay you $\$3800$in case Charlie wins? If the total amount locked is$\$3000$, where would the extra $\$800$come from? Why do the probabilities sum to$1\$? Which probabilities? Some offer odds, others take the odds.