Given our conditional probabilities of unique Competitor + Strategy (Cn+Sn) combinations from last post, in this post we’ll look at pricing these probabilities as bets.
We are, after all, trying to model a sportsbook that would take bets from bettors on trading card game events.
Let’s now look at how we’d make money doing this.
Differences Between Odds and Probabilities
Probabilities, as discussed previously, are values we give to uncertain outcomes.
If we say that Player A has a “probability of 0.492 to beat Player B”, we mean that, we lend 49.2% of our belief that Player A, will, in fact, beat Player B. This leaves 50.8% of our belief outside of this outcome, meaning that this amount of our belief is placed with Player B winning, the two players drawing, or something else (a player is disqualified? the event is shut down due to an emergency?).
“Odds” are often used with “probability” interchangeably in everyday speech. But for our purposes, they mean different, but similar things.
To say that Player A has 4:5 odds to beat Player B means that we, in fact, assign a 44.4% probability that Player A beats Player B.
This is because, as we saw with conditional probabilities and the concept of “limiting the probability space”, the total space of the 4:5 odds is 9.
In other words, when we say that Player A has 4:5 odds to beat Player B, we are saying that in a hypothetical 9 matches between these two players, we expect Player A to win 4 matches and Player B to win 5 matches.
That’s where the 4:5 comes from.
A little simple algebra does the trick:
[math] 4:5 \:odds = \frac{4}{4+5} = 0.4444 [/math]
If this looks like the probability formula we used in Excel to find the conditional probabilities of one Competitor + Strategy combination against another… it is!
We’re going to see this kind of conditional probability formulation show up quite a bit throughout this project.
Converting Probability to Odds
To convert a probability to odds, we use this simple formula:
[math]Odds = \frac{Probability}{1-Probability}[/math]
For example, given our above example, if Player A was to have a 0.492 probability to win against Player B, we convert this probability to odds like so:
[math]Odds \: (P|B) = \frac{.492}{1-0.492}=\frac{.492}{.508}=\frac{123}{127}[/math]
The resulting fraction, [math]\frac{123}{127}[/math] is pretty ugly, so we can “normalize” it by setting the denominator (127) equal to 1:
[math]\frac{123}{127}=\frac{x}{1}≈\frac{0.9685}{1}[/math]
The odds that Player A beat Player B are 0.9685:1, 9.685:10, 96.85:100, or 968.5:1000, etc.
This means, given say, 1,969 games between Player A and B, given the present state of our knowledge, belief, and the circumstances of them playing against one another, we believe that Player A would win about 969 of those games.
Converting Odds to Probability
To convert odds to probability, we perform the opposite operation:
[math]Probability(A|B)=\frac{Odds(A)}{Odds(A)+Odds(B)}[/math]
To reconvert our above odds of 0.9685:1, we plug these values into our formula and find:
[math]Probability(A|B)=\frac{0.9685}{0.9685+1}≈0.492[/math]
Given the calculations we’ve done, we’ll have to settle for very approximate equivalency (≈), or the number of decimal places we carry these calculations out will become unmanageable! (This is much less of an issue when using Excel or R to do the calculations, as we’ll see soon enough.)
The “Vig” (or, Sportsbook Odds are Not Fair Odds)
Sportsbook are in business. This means that they are out to make a profit from the service (betting prices) that they offer to their customers.
Bookmakers bake their profits into the betting prices they offer.
As the saying goes, “you can’t beat the house.” (I live in Las Vegas, a town built on this simple, but largely disregarded, truth.)
The vig or vigorish is the marking up (or “over rounding”) of betting prices. Bookmakers intentionally set the probabilities on either side of a bet higher than the real probability when setting these prices. The result is that the prices, when combined, sum to more than a probability of 1.
As we discussed previously, it’s not possible for probabilities to sum to more than 1. The entire concept of probability is that 1 means absolute certainty the outcome will happen and 0 means absolute certainty the outcome will not happen.
Sportsbooks ignore this rule which is how we intend to make money from the bets players place. Regardless of which individual players win or lose any given bet, as the house, we’re out to always win in the long run.
When we set our betting prices, we’re going to explore to methods to “over round” the probabilities of a given match.
Setting Bet Prices
Let’s start by returning to a previous example: we pit two player and deck combinations—C3+S5 and C7+S3—against one another.
We previously determined that C3+S5 had a 0.733 probability of beating C7+S3, which means that C7+S3 has a 0.267 probability of beating C3+S5.
| Competitor | Win Probability |
|---|---|
| C3+S5 | 0.733 |
| C7+S3 | 0.267 |
| TOTAL | 1.000 |
So far, so good.
As the bookmaker, we’re going to over round these to bake in our expected profit.
Proportional Vig
The easiest thing to do is to increase both sides of the contest by a proportional amount.
Let’s say we increase both sides by 10%. We just multiply each probability by 1.1, like so:
| Competitor | Win Probability |
|---|---|
| C3+S5 | 0.8063 |
| C7+S3 | 0.2937 |
| TOTAL | 1.1000 |
Easy. Now our betting prices are overpriced on both sides, equally.
But there’s another way.
Disproportional Vig
If we want one side to be higher priced than the other (perhaps we have too much bet liability on one side and we want to make the other more attractive) we can apply the over round in a disproportional way.
Let’s say that C7+S3 is a favorite underdog, and lots of players are placing bets on that side. If C7+S3 wins, we could get wiped out as the sportsbook on this one game, so we hedge our bets by increasing the vig by 80% on the side of C3+S5 and 20% on the side of C7+S3, like so:
| Competitor | Win Probability |
|---|---|
| C3+S5 | 0.813 |
| C7+S3 | 0.287 |
| TOTAL | 1.100 |
As we’ll see later, this can drastically affect our profitability, depending on how the game concludes.
From Vig to Prices
Once we’ve set up our vig (either proportional or disproportional), we can turn these new (unfair) probabilities into prices.
For these examples, and the examples used throughout the entire project, we’ll be using American moneyline odds, where a positive (+) price means that the quoted side is an underdog and a negative price (-) means that the quoted side is a favorite.
In the American system, a +price means that you will win this amount for every $100 bet, while a -price means you must bet this amount to win $100 (more about that in a bit).
There are two equations we’ll use to set bet prices for two outcomes.
P(C7+S3|C3+S5)
Since C7+S3 is our underdog, we’ll calculate the probability of this player + deck combination winning, first.
The moneyline for this will be positive (+), so we use the following formula:
[math]x=\frac{100}{Probability}-100[/math]
“X” in this case, is the price we’ll quote to bettors.
Plugging in our probability that C7+S3 win, we get for the proportional vig:
[math]\frac{100}{0.2937}-100 ≈+240[/math]
Or for the disproportional vig:
[math]\frac{100}{0.287}-100 ≈+249[/math]
This means that, given a $100 bet for C7+S3 to win, a player would win either $340 total (including the $100 stake) if the sportsbook uses the proportional vig, or $349 total (including the $100 stake) if the sportsbook uses the disproportional vig.
P(C3+S5|C7+S3)
Now we turn to pricing bets for the favorite, C3+S5, who will have negative (-) moneyline prices.
For a favorite with a negative (-) moneyline price, we use the following formula:
[math]x=-\frac{100*Probability}{1-Probability}[/math]
“X”, again, is out desired bet price.
Plugging in our probability variable for a proportional vig, we find:
[math]-\frac{100*0.8063}{1-0.8063}≈-416[/math]
For the disproportional vig, we get:
[math]-\frac{100*0.813}{1-0.813}≈-435[/math]
This means that to win $100 a bettor must bet $416 for the proportional vig or to win $100 a bettor must bet $435 for the disproportional vig.
We can see the effects on both sides between the proportional and disproportional vig variations.
Long Run Example of Profitability
Let’s put all of this together and find out how much money we expect to make.
Let’s imagine four different outcomes:
- C3+S5 (the favorite) beats C7+S3 (underdog) with a proportional vig.
- C7+S3 (underdog) beats C3+S5 (the favorite) with a proportional vig.
- C3+S5 (the favorite) beats C7+S3 (underdog) with a disproportional vig.
- C7+S3 (underdog) beats C3+S5 (the favorite) with a disproportional vig.
To keep things simple, we’ll imagine that in each case, 100 bettors each place a moneyline on each side of the match.
This means that in every case we’ll examine below, there will be 100 bets on each side, regardless of price.
C3+S5 Wins, Proportional Vig
C7+S3 Wins, Proportional Vig
C3+S5 Wins, Disproportional Vig
C7+S3 Wins, Disproportional Vig
Summary of Profitability
With these simple examples, we can see that the book maker’s unfair pricing has given it quite a cushion.
If C3+S5 wins in either case, the sportsbook loses no money, as shown above.
If, however, the underdog C7+S3 wins in either case, the sportsbook makes money.
The moneyline is priced in such a way that a bettor on the underdog winner wins far less than the bettor on the favorite paid for their “sure win”. And the sportsbook is indifferent to the bettor on a favorite to win because, it’s just a wash.
Given 1000 games with the outcomes above, we’d expect the following:
For a proportional vig:
For a disproportional vig:
Notice that in every case, our disproportional vig makes us (a little) more money.
While the pricing mechanics of real events is much more complex, this simple illustration tells us that we have the right stuff, in theory, to make money on trading card game bets.
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