Mathematical Considerations From the Bookmaker's Point of View

This post is prompted by the line analysis undertaken by @Crickett1414 on stream with Grass Guy and Ghost Wins. I'm trying to analyse it in hopes of finding new angles with it. This post will be a little bit all over the place and I will probably be correcting my derivations in the next day or so as there's a good chance I've made a few mistakes here and there and I may even be outright wrong on a few points. For now, I just want to get my thoughts out and I'll refine it later: this is very much an experimental post.

Betting from a bookmaker's point of view can be considered by reversing the bet from the bettor's point of view. Whereas a bettor at a bookmaker can only enter at market at a given price set by their counterparty the bookmaker, a bookmaker can only enter when they have found a counterparty, that is, the bettor. This would be the equivalent of someone setting a limit order on an exchange and having someone fill that order at market. The primary difference is that the bookmaker operates as maker and the bettor as taker, whereas on an exchange, someone can be a maker on one transaction but a taker on another. This only happens with bookmakers under certain circumstances where they give bettors leeway to invent a bet but this is a tiny fraction of a bookmaker's overall volume. If a bookmaker offers a bet at -110 odds and the bettor places $100 on this, he risks $100 for ~$91 rewards. Conversely, the bookmaker is risking ~$91 to make $100 reward: if the bettor wins, he loses ~$91 and if the bettor loses, he gains $100. It is clear to see that a bettor taking a bet at -110 is equivalent to a bookmaker making a bet at +110 for himself.

To take the simplest example possible, consider a market for the outcome of the Superbowl coin toss. Let us suppose that the coin is perfectly unbiased (that is, it is a true 50:50 outcome on heads and tails) and that the bookmaker, as a show of good faith decided to offer fair market odds, that is to say, each side is +100, or 50c. We now go about computing the expected value of all bets placed. Partition money on heads and money on tails as A and B. The expectation value of this market's profit is the probability of heads times the return in the case of heads plus the probability of tails plus the return in the case of tails. In this case, it is 0.5(B1/0.5-B-A)+0.5(A1/0.5-A-B), which is 0. This is because the odds are perfectly aligned with the probabilities, which means relative variance tapers off to zero as sample size tends towards infinity. Note that since we are considering expected value from the perspective of the bookmaker, the outcome heads means heads is a loss for him while he keeps the money placed on tails.

What if we introduce an overround to the coin toss? BetOnline had both sides as -102 for this year's Superbowl but for simplicity, let's set a 2% overround such that you have 51c on both sides, which would be ~-104 each way. From the point of view of the bookmaker, he has 100c-51c=49c each way, or ~+104 The expectation value is now 0.5(B1/0.49-B-A)+0.5(A*1/0.49-A-B), which becomes B/0.98-B/2-A/2+A/0.98-A/2-B/2 or B(1/0.98-1)+A(1/1.02-1) and then (A+B)(1/0.98-1), giving ~0.019(A+B) where A+B is your total market volume. The constant factor is the vig and represents the profit derived from the bookmaker from overpricing the odds.

The next question is what if we remove the overround but vary the probabilities. Make a and b the probabilities; since a+b=1, b=1-a. The expectation value is now a(B1/0.5-B-A)+b(A1/0.5-A-B) or a(B1/0.5-B-A)+(1-a)(A1/0.5-A-B). This expands to aB-aA+A-B-aA+aB and then reduces to 2aB-2aA+A-B and followed by (A-B)(1-2a). We see that there are two scenarios where this is zero; one is when a=1/2, which is the odds truly reflecting the underlying probabilities and also when the amount of money on heads equals the amount of money on tails. This makes sense as probabilities aligning with odds indicates fair value and money partitioned in ratio of odds balances the books. The bookmaker is profitable when both of the expressions in brackets have the same sign, that is when there is more money on heads than tails and yet tails is the more likely outcome or when there is more money on tails than heads and yet heads is the more likely outcome.

We can generalise this to include the overround by setting normalised percentage odds p and q related by p+q=1+r where p is the percent odds of heads and q is the percentage odds of tails and r is the overround. The bettor sees odds p and 1+r-p on heads and tails but the bookmaker sees odds 1-p and p-r. Now we have as our expectation value a(B1/(1-p)-B-A)+(1-a)(A1/(p-r)-A-B), which become aB(1/(1-p))-aB-aA+A/(p-r)-A-B+aA/(p-r)-aA+aB and then aB/(1-p)-aA(1/(p-r))+A(1/(r-p)-1)-B. For a coin toss, set p = 0.5 + r/2 and we get aB/(0.5-r/2)-aA(0.5-r/2)+A(1/(0.5-r/2)-1)-B. This is (A-B)(1-a/(0.5+r/2))+A(1/(0.5-r/2)-2) and finally (A-B)(1-a/(0.5+r/2))+A(r/(0.5-r/2)).

To make sense of the above, if even money comes in on both sides, the first term resolves to 0 and the second term then gives you a guaranteed positive term so long as r>0. On the other hand, if we let 1-a/(0.5+r/2)=0, we have a/(0.5+r/2)=1 and a=0.5+r/2. Thus, if the real probability matches the overrounded probability, the first term is zero and once again, the bookmaker pockets the remainder. Let us furthermore set a=0.5+r and the expectation value becomes (A-B)(-r/2/(0.5+r/2))+A(r/(0.5-r/2)) which gives (A+B)(r/2/(0.5+r/2)).

To bring this into focus a bit, we'll pinpoint the bookmaker's two sources of edge. One is the overround and the other is getting more money on the overpriced side (from the bettor's point of view) than the underpriced side. Since bookmakers in practice cannot perfectly balance the books, they use a combination of overround and getting money on the wrong side of the tracks (as far as the public is concerned) to make their profit. As we can see from the derivations above, we have two expressions corresponding with each of these sources of edge from the bookmaker's point of view.

What implications does this have for setting an moving lines? A bookmaker who understands the betting public's betting habits can spike the lines by setting them deliberately askew from the true underlying probabilities knowing that they will take the bait Since the betting public likes to bet on favorites, the bookmaker can set the line higher than it should be to entice the public to take the favourite, knowing the true line should be lower. The bookmaker will move the line up in order to reduce his risk and to keep the books somewhat balanced. Sometimes the bookmaker will move the line down even though public money is piling on the favourite: that generally indicates sharp action coming in on the underdog and the bookmaker will take that as a sign that the underdog is undervalued and thus they have breathing room to move the line down as still have the public on the wrong side of the tracks. It would not make sense for him the move the line down if he believed it would place himself on the wrong side of the tracks in the process.

Going back to the bookmaker's perspective, the public market buying the favourite is equivalent to him limit buying the underdog. As public money piles on, he sees that he can actually get filled on the underdog for cheaper, so he will do so in order to reduce his risk and get better entries. Simultaneously, he is limit buying the favourite any time someone buys the underdog. If he moves the underdog price lower he makes the favourite price more expensive for himself to buy but once again, he doesn't mind so long as he is getting more money on the side he believes is overpriced for the public. When sharps hammer the underdog, the bookmaker must take caution about moving the favourite line up as this will simultaneously make it more expensive for him to buy the favourite. If the favourite has been sufficiently underpriced while the line moves up, there is a risk the public will start betting on the underdog too and that's when the bookmaker can find himself exposed badly if the underdog covers.

When the bettor sees the line set by the bookmaker, he is engaged in a game of poker against the bookmaker. The bookmaker has his own models to estimate the probability and yet he sets lines intended to tempt public action a certain way. The bettor does not know which way the bookmaker is trying to entice him and it may even turn out the line the bookmaker wants him to take is the profitable one for the bettor. Nonetheless, the bookmaker can form a middle through line movement as if the line comes down and the public starts betting the other way, the bookmaker can benefit from the public being wrong both ways if the game lands around the middle of the spread movement. This can be relevant to games where one side starts as the favourite and then the other team becomes the favourite. If one team then wins by exactly one point, this can be an extremely profitable game for the bookmaker.

I may add more to this post as I think of it.