Beat the Odds

If you're going to operate a profitable betting ring, there are two things that are important to know. An obvious one is the odds of a particular event happening: a good casino wouldn't make any money if they didn't know the odds associated with blackjack or roulette, and adjusting their payouts accordingly is what gives them any profit.

What's also important to know is the number of people who are likely to bet on a particular outcome. This is less important perhaps on a roulette table, but potentially very important when it comes to something I've been investigating a lot recently: sports betting. It makes a certain amount of sense that sports betting companies (such as SportSelect) track their ticket sales to consumers, and in fact the companies can profit just as much by adjusting their payout according to ticket sales as they can according to individual game odds. As it is likely substantially easier for a company to track their sales than it is to predict the future, this is quite likely a critical factor in their payout odds.

The profit made by a company in sports betting can be visualized like this:

Basically the expected profit can be determined based on the chance of an event occurring multiplied by the cost of that event occurring. So an event (such as home victory) with a chance p of occurring, will pay out an amount equivalent to the payout odds x, and the number of tickets sold that chose that event (m). By considering both at once, we can get an 'expected' average profit that takes into account all possibilities - by tweaking the payout factors, a company can assure themselves of continually profiting from the sports betting.

Presumably SportSelect knows the fraction of tickets sold (m and n) very well. They must. It would be negligence on the part of the company not to know what they're selling, how much they're selling, and who they're selling to. Presumably also they have some sort of model that allows them to predict game odds (p and q) with a reasonable amount of accuracy. As they control the values for payout (x and y), they can then have a good sense of control over their profit.

P and m are both factors that relate to the specific event that's being examined. P is most likely intrinsically involved with the relative strength of the teams, and m accounts for whatever factors lead people to purchase lottery tickets betting on certain teams (perceived skill, popularity, etc.). Together, the factor pm more or less accounts for the expected amount of money SportSelect will lose should that event come to pass.

A quick look at the payout odds that SportSelect offers shows a strong trend - the majority of their payout options average between the two events at a payout of 1.7 - combinations such as 1.6 and 1.8, 1.5 and 1.95, etc. Some more unlikely payout combinations are 1.4 and 2.15, 1.3 and 2.45, and 1.25 and 2.65, and these tend to average a little higher, but still within the range of 1.7-1.95. There are very few combinations outside of that, so for the sake of this piece I'll take only these into account.

In order to try to investigate just how SportSelect comes up with their odds, I set up a series of random p and m values to look at some trends. This is what I got at first:

OH MY GOD THAT'S UGLY. Whew. Jeeze. What I have here is the product of p and m on the x-axis, and then the expected profit percentage based on any of the five payout combinations as explained above (the legend lists the average of the two payout values for each of the five sets) after 1000 data points for each. This is really really ugly though.

Part of the reason it's ugly is the relationship between pm and qn - the complementary payout values for the alternate event. For radical values of either p or m, we tend to get qn values that are tremendously different, which gives the ugly values as shown above. Looking at the cluster of points where the majority lie appears to form a series of curves; this is a cleaner version of that graph:

Much better. This is actually rather interesting, if I may say so myself. What we get is different ranges of the pm factor result in different payout values (x and y from before) giving the largest profits to the company. So for events with either very large differences in who people bet on (m) or who is actually likely to win (p), larger payout odds are more likely to result in profits. Smart, eh? In fact, it's quite easy for SportSelect to guarantee a 14-16% profit by estimating (with not necessarily that much accuracy, even) the pm factor. As they ought to know the sales figures (m), then they only need to be reasonably accurate on the actual game odds in order to make a killing.

Assuming that they do in fact take ticket sales into account, an opportunity to perhaps profit does then exist. Take a look at these tables:

This first one is just a representation of the graph above - each colored zone represents a range where a new payout scheme becomes the most profitable, measured against values for p and m (the middle values are pm). If we change it to represent what those colours actually mean, we get:

In general this follows the trend mentioned before - for games where either the sales or the odds are anticipated to be close, we have lower payouts (an average of 1.7 is typically odds such as 1.6 and 1.8, remember), but with games with larger disparities we have higher payouts offered (an average payout of 1.95 would feature 1.25 and 2.65 for different teams, respectively).

If we look at the pm value with an m of 0.32 and a p of 0.5, for instance, we notice something interesting. The pm value is 0.16, so therefore the most profitable payout distribution would be one with an average of 1.775, such as 1.4 for one team and 2.15 for another. However, if we were really sure that the odds were truly 50-50, then betting on the teams with 2.15 odds against would be profitable - 50% of the time on a $1 ticket we'd get $0, and 50% of the time we'd get $2.15, with an expected return of $1.08. Quickly tabulating these results gives the last graph:

Here the green values are where there's money to be made, the gold values break even, and the red values are guaranteed money losses. These are the values of the absolute best bet that can be made for each combination.

So what does this mean? It means that on certain cases, it could be possible to beat the SportSelect betting system. This would have to involve a very high degree of certainty in the actual odds of a given team winning a game (at least as accurate a model as they use would be required), and it would involve them trying to capitalizing on a fairly significant majority of the public purchasing tickets for one team over another (at least 2:1 ratio would be required).

Still, though, numerically it's possible if you have a good enough model and are patient enough. Good luck!