Tag: Sports Betting

1.3 Win/Loss Odds & Betting Prices in Trading Card Games

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.

CompetitorWin Probability
C3+S50.733
C7+S30.267
TOTAL1.000

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:

CompetitorWin Probability
C3+S50.8063
C7+S30.2937
TOTAL1.1000

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:

CompetitorWin Probability
C3+S50.813
C7+S30.287
TOTAL1.100

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:

  1. C3+S5 (the favorite) beats C7+S3 (underdog) with a proportional vig.
  2. C7+S3 (underdog) beats C3+S5 (the favorite) with a proportional vig.
  3. C3+S5 (the favorite) beats C7+S3 (underdog) with a disproportional vig.
  4. 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.

Project 2: Trading Card Game Sportsbook Financial Calculator

Websites abound with information about how to translate moneyline prices into the bookmaker’s implied probabilities.

These webpages also include some discussion about how moneyline prices are read and what they mean. A few also have discussions about the concept of a vig or vigorish (the “over rounding” that a bookmaker does to the probabilities to bake in a profit for itself).

All of these websites speak to bettors.

In keeping with the project to theoretically model a sportsbook that takes bets on trading card game events, we want to model such variables as the house.

If we’re the sportsbook, we want to know the probable outcomes of different over round percentages, splits in the over round, competitor win probabilities, total money wagered on either side, and what our expected financial outcomes should be.

In this project, we accomplish all of these things with an Excel Calculator.

The Excel TCG Sportsbook Financial Calculator

If you’d prefer to see the Excel calculator first and skip (or save for later) the discussion on how it works and why, you can find it below:

Excel TCG Sportsbook Financial Calculator

Breaking Down the Tunable Parameters

The Excel Calculator gives us five parameters that can be changed. These are found under the Assumptions heading.

Let’s take a look at each of them.

Vig

The vigorish, or “over round”, is the markup the house puts on the probabilities it uses to quote the bettors its bet prices.

This can be viewed as the long term profit margin the house expects on the outcomes of events with similar probabilities.

For most popular sporting events, the over round hovers a bit below 5%.

The Excel Calculator allows a vig of between 0 and 25%.

(We should expect that if trading card game bet prices were set, the vig would be on the higher side, as they are more thinly traded.)

Matches Played

Here, the user can set the number of matches played between competitors.

This is a simplification, for illustrative purposes, because the probabilities are identical for each match played (whereas, in real life, we’d wish to go further into developing a Bayesian predictive model to account for changes in win probabilities for either side given a series of N matches).

A more complex model deserves its own project, which will come soon enough.

These are tunable between 1 and 1,000,000 matches.

Vig Split

This allows the user to split the vig between the two competitors.

Often times, the book maker will not apply the vig equally to both, so as to help limit liability on one side of the event. Placing more of the over round weight to one side (one competitor) over another can make that side seem less attractive than the other, enticing bettors to place their money elsewhere.

Only the vig split on Player A is tunable. The vig split on Player B is automatically updated based on the input for Player A.

The Vig split for Player A can be between 1 and 100% (with Player B having the remainder).

Win Probability

The Calculator assumes that we know the win probabilities for either player.

How to arrive at these win probabilities in trading card games, at least, is the subject of another project. Here, we assume that we know them.

Only the win probability for Player A is tunable. The win probability for Player B is automatically updated based on the input for Player A.

The win probability for Player A can be between 0 and 1, inclusive (with Player B having the remainder).

Total Money Wagered

The parameters for total money wagered for either side of the matches played can be set to any amount between $1 and $1,000,000 in even dollar increments.

As noted in the Calculator, the total money wagered on either side is per match played. 

Reading Financial Outcomes

After the five tunable parameters have been set, we can see the financial outcomes of the selected series of matches.

Let’s assume we set our assumptions as follows:

  • Vig: 10%
  • Matches Played: 50
  • Vig Split: 70%/30%
  • Win Probability: 0.674/0.326
  • Total Money Wagered: $24,500/$47,850

Fair Probability & Over Round Probability

The fair probabilities are carried over from the Assumptions we placed in the Calculator.

Note that the sum of these probabilities will always sum to 1. There are fair probabilities, because they reflect our true beliefs about the winner of the matches.

The over round probabilities apply the vig and the vig split to each probability.

Since we placed the vig at 10% and weighted 70% of that vig on Player A and 30% on Player B, the Calculator applies those figures to each side accordingly.

Note that the probabilities sum to 1.1, meaning the fair probability sum of 1, plus the vig of 10%.

Moneyline

The Calculator gives us the moneyline for the players.

As we discussed in the post about setting odds and betting prices for trading card games, these prices use the American moneyline system for bet prices.

We see that Player A has a price of -291 (meaning that a bettor must bet $291 to win $100), while Player B has a price of +181 (meaning that betting $100 will win the player $181; plus the staked $100, in both cases).

Player A, as we should expect from our probabilities, is the favorite (with a negative quoted price), and Player B is our underdog (with a positive quoted price).

Money Wagered & Bettors to Win

The Calculator gives us the total money wagered on both sides for all events (remember, we put $24,500 on Player A and $47,850 on Player B on each of 50 matches).

It also gives us the bet liability for each side, or what bettors stand to win if they’re right.

The Bettors to Win calculation takes the total money wagered for each side and applies the moneyline price for each side to arrive at the liability figures.

Wins

The wins for each side simply applies the fair probability for each player as a proportion of the total number of matches we input.

Since we selected 50 matches, given probabilities of 0.674 for Player A and 0.326 for Player A, we expect Player A to win 34 matches and Player B to win 16 matches.

Financial Outcomes

Finally, given all of our assumptions, we have the expected financial outcomes for our venture.

With 50 matches, assuming our probabilities are correct, we expect to pay out the bettors on Player A a total of $1,109,379 and the winners on Player B a total of $2,191,674.

Our total handle, or the total money wagered by bettors, came to $3,617,500, on which we paid out $3,301,052.

That leaves us, the sportsbook, with a gross gaming revenue (GGR) of $316,448 for a profit margin of 8.7%.

Not too shabby!

Conclusions

Our Calculator allows us to model some basic assumptions about the financial viability of our sportsbook.

We can tune a number of parameters about each competitor and how we choose to price the bets we offer to bettors. We can experiment with how much money we’d need on either side to maintain profitability given these assumptions.

We’ve seen that, if it all works out more-or-less according to plan, the bookmaking business is good to us.

Please let me know if you have any questions or comments in the comments section below!

————

You can find a link to the completed Excel TCG Sportsbook Financial Calculator below:

Excel TCG Sportsbook Financial Calculator

1.2 Modelling Competitor + Strategy Probabilities in Trading Card Games

In the previous post, we discussed our decision to use Bayesian Inference as the preferred method to compute the win probabilities for different players and decks.

We will now expand on how Bayes’ Theorem can help us do this.

We’re going to take for granted that the probabilities here are examples only and that, given this demonstration, we’re also taking for granted that someway, somehow, all the probabilities have been provided to us.

In later parts of this project, we’ll tackle actually getting to the point where we can create such matchups with our own computed proabilities.

The Competitor + Strategy Matrix

Assume we have 8 Competitors (C1…C8) and 8 Strategies (S1…S8).

We might have a table that looks like this:

C1C2C3C4C5C6C7C8
S10.3310.5290.4500.6260.1830.6490.2570.151
S20.5070.2710.5370.2000.2580.5130.1190.330
S30.5120.2420.2870.2530.6130.3930.1790.574
S40.6770.5400.6940.6100.4060.1380.2090.395
S50.6540.2220.4920.6300.5690.1190.1440.397
S60.2970.5440.5340.2620.2430.4320.6960.352
S70.3510.6000.7050.1190.3650.7190.3990.518
S80.4150.3420.4770.1100.2050.4320.4930.134

This table tells us how probable a given Competitor, using a given Strategy is to win, in general. This would be a probability against a disembodied opponent: the probability isn’t conditioned on facing another person or deck. It’s most likely the long term average for the Competitors and their Strategies (wins/losses) expressed as a ratio.

Reading the table, above, we can see that Competitor 2 using Strategy 4 has a .540 probability of winning any game.

But what is this Competitor using this Strategy’s probability of winning against another Competitor and Strategy combination?

Computing Win Probabilities

Last time, we reviewed Bayes’ Theorem in the abstract:

[math] P(A|B)=\frac{P(A)P(B|A)}{P(B)} [/math]

We decided that the win probability for Player A against Player B was a conditional probability, or that this probability was conditioned on the circumstances of these two opponents facing one another.

It wouldn’t make sense for us simply to give a probability of Player A (or Player B) winning in isolation. This is what our table, above, does.

It doesn’t, however, give us a probability of a Competitor + Strategy combination against another Competitor + Strategy combination, which is what we want.

Let’s take the case of Competitor 3 using Strategy 5 (C3+S5) playing against Competitor 7 using Strategy 2 (C7+S2).

How do we calculate the probability that C3+S5 win a match against C7+S2?

We set up our equation thusly:

[math] P(C3+S5|C7+S3)=\frac{P(C3+S5)P(C7+ S3|C3+S5)}{C7+ S5} [/math]

Let’s examine each of the variables in turn.

P(C3+S5 | C7 + S3)

This is the win probability of C3+S5 against C7+S3, which is what we’re looking for.

We won’t know this until the end!

P(C3+S5)

Referring to our table above, we find that the probability of C3+S5 winning any match is 0.492.

Thus, the value of P(C3+S5) is 0.492 in our formula.

P(C7+S3|C3+S5)

This one is a little trickier. This is the probability that C7+S3 win against C3+S5.

And if we think about it, is the opposite of what we’re looking for, P(C3+S5|C7+S3)!

If we knew this now, we’d be done, because to find the other probability, we simply subtract it from 1. This is something we’ll return to shortly.

For now, let’s leave this as a mystery.

P(C7+S3)

This is simply the probability that C7+S3 win any match.

Consulting our table, we see that this is 0.179.

So we’ll use that.

Missing Information

So far, our revised formula looks like this:

[math] P(C3+S5| C7+ S3)= \frac{0.492*P(C7+ S3|C3+S5)}{P(0.179)}[/math]

But what about that pesky P(C7+S3|C3+S5)?

If we could calculate that, we’d know P(C3+S5 | C7+ S3), our desired answer, too.

Let’s find an easier way to make this calculation.

Using Excel and Basic Algebra

Let’s take another look at our Competitor and Strategy table:

We know that C3+S5 has an overall win probability of 0.492 and that C7+S3 has an overall win probability of 0.179.

To make our life easier, we will constrain the probability space to the universe that contains only these two match ups. That is, to say, that 1 (or 100%) will be the combined total probabilities for these two Competitors + Strategies.

We can think of this as a ratio:

[math] Win Probability= \frac{P(C3+C5)}{P(C3+C5) + P(C7+ S3)}[/math]

This pits the probability of C3+C5 against the combined probability of C3+C5 and C7+ S3.

We get the following:

[math]  \frac{0.492}{0.492+0.179}=0.733[/math]

This works because the probability that C3+C5 win any match is a fraction of the combined probabilities that C3+C5 win any match and C7+S3 win any match.

In fact, we can put all of this back into Bayes’ Theorem to prove our case.

Remember, that if the P(C3+C5|P(C7 + S3)) = 0.733, then the probability that the other player win is exactly 1 – this amount, or P(C7= + S3|C3+C5) = 1 – 0.733 = 0.267.

This is because all probabilities must sum to 1 (absolute certainty). All potential outcomes represent the whole of possible outcomes. 

[math]P(C3+S5| C7+ S3)=\frac{0.492*0.267}{0.179} =0.733[/math]

All we have to do is divide Cn+Sn by Cn+Sn combined with Ck+Sk.

This is much easier.

What we’ve done here is normalize the probabilities of the two competitors such that their conditional probabilities sum to 1, like this:

The Expanded Cn+Sn Matrix

To carry this theory into action, we pit each and every combination of Competitors + Strategies (Cn+Sn) against each other in the same way as we did above for C3+C5 against C7+S3.

Our previous win probability table has 8 rows and 8 columns. This one has 64 rows and 64 columns.

Wins? ↓C1+S1C1+S2C1+S3C1+S4C1+S5C1+S6C1+S7C1+S8C2+S1C2+S2C2+S3C2+S4C2+S5C2+S6C2+S7C2+S8C3+S1C3+S2C3+S3C3+S4C3+S5C3+S6C3+S7C3+S8C4+S1C4+S2C4+S3C4+S4C4+S5C4+S6C4+S7C4+S8C5+S1C5+S2C5+S3C5+S4C5+S5C5+S6C5+S7C5+S8C6+S1C6+S2C6+S3C6+S4C6+S5C6+S6C6+S7C6+S8C7+S1C7+S2C7+S3C7+S4C7+S5C7+S6C7+S7C7+S8C8+S1C8+S2C8+S3C8+S4C8+S5C8+S6C8+S7C8+S8
C1+S10.50.3949880670.3926453140.3283730160.3360406090.5270700640.4853372430.4436997320.3848837210.5498338870.5776614310.3800229620.5985533450.3782857140.3555316860.4918276370.4238156210.3813364060.5355987060.3229268290.402187120.382658960.3194980690.4096534650.3458725180.6233521660.5667808220.3517534540.3444328820.5581787520.7355555560.7505668930.6439688720.561969440.3506355930.4491180460.3677777780.5766550520.4755747130.6175373130.3377551020.3921800950.457182320.705756930.7355555560.4338138930.3152380950.4338138930.562925170.7355555560.6490196080.6129629630.6968421050.3222979550.4534246580.4016990290.6867219920.500756430.3657458560.4559228650.454670330.4846266470.3898704360.711827957
C1+S20.6050119330.50.4975466140.4282094590.4366925060.6305970150.5909090910.549891540.4893822390.6516709510.6769025370.4842406880.6954732510.4823977160.457994580.5971731450.5297805640.4856321840.6385390430.422148210.5075075080.48703170.4183168320.5152439020.4474845540.7171145690.6671052630.453894360.445910290.6592977890.8099041530.821717990.7347826090.6627450980.4526785710.5553121580.4711895910.6760.5814220180.7120786520.4385813150.4970588240.5633333330.7860465120.8099041530.5399361020.4135399670.5399361020.6636125650.8099041530.7390670550.7081005590.7788018430.4214463840.5596026490.5070.7705167170.6057347670.4690101760.5620842570.5608407080.5902211870.4946341460.790951638
C1+S30.6073546860.5024533860.50.4306139610.4391080620.6328800990.5932792580.552319310.4918347740.6538952750.6790450930.4866920150.6975476840.4848484850.4604316550.5995316160.5322245320.4880838890.6408010010.4245439470.5099601590.4894837480.4207066560.5176946410.4499121270.7191011240.6692810460.4563279860.4483362520.6614987080.811410460.8231511250.7366906470.6649350650.4551111110.5577342050.4736355230.6781456950.5838084380.7140864710.4409991390.4995121950.5657458560.7876923080.811410460.5423728810.4159220150.5423728810.665799740.811410460.7409551370.7101248270.7804878050.423841060.5620197590.5094527360.772247360.608076010.471454880.5644983460.5632563260.5925925930.4970873790.792569659
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You’ll notice that starting with Row 1, Column 1 (position [1,1]) and following along a perfect diagonal (positions [2,2], [3,3], [4,4], etc.) all the probabilities are 0.5, or dead even. This is because due to the way the table is set up, each Cn+Sn is pitted against itself exactly once.

There are also 448 other instances of the same Competitor facing itself with a different Strategy.

These 512 matchups notwithstanding, our table does give us the win probabilities for every real combination of Cn+Sn.

This is Bayes’ Theorem at work.

We’re on to a start.

Project 1: What if a Sportsbook Offered Odds on Trading Card Games?

Background

I’ve been a lifelong fan of trading card games.

Ever since the Star Wars CCG (Customizable Card Game) in 1995 and later, Pokémon TCG (Trading Card Game) in 1999 (in the U.S.), I’ve been hooked.

Trading card games are games of skill where two competitors construct decks of cards from those available in the game and play against one another.

One player wins and another loses. (Sometimes, there is a draw.) These games are “zero-sum” in this way.

Working in the gambling industry, as I have, for 10 years now led me to ask: “What if a sportsbook placed betting prices on the outcome of trading card game events like they do for professional sports events?”

Basically: what if you can bet on games like Pokémon and Magic: the Gathering or Yu-Gi-Oh?

What would this take to make work? What are the theoretical concepts than underpin such an endeavor? What kind of profit could the sportsbook expect?

I attempt to answer these and more during the course of this project.

I aim for this project to change and evolve as its proceeds, knowing that the final conclusions I draw may be very different from my starting assumptions.

I hope also to get some comment from readers to help improve what’s being done here.

This project is both a demonstration and also some food for thought.

Objectives

My objectives for this project are:

  1. Demonstrate how Bayesian Inference can help us construct a predictive model for two-player, winner-take-all events (card games).
  2. Demonstrate how, given the probabilities assumed by these inferences, odds and betting prices by a fictional sportsbook (“TCGBook”) can be set.
  3. Model the outcomes of fictional and real matchups in a trading card game tournament setting.
  4. Model the profit and loss of our fictional sportsbook (“TCGBook”).
  5. Open these ideas to the public for comment, critique, and improvement.

Limitations

Before starting on this quest to model our TCGBook endeavor, it is important that I acknowledge a few key limitations.

We compare our subject, trading card games, to the tried-and-true professional sports leagues on which our sports betting idea and models are largely based.

Data Availability

The data for card game events can be very hard to come by.

Most of the data sources are compiled by fans of the games and not the hosts or producers of the games themselves. The “big dogs”, as it were, do not wish to disclose their proprietary information. Or at least, not all of it. Maybe they never thought to or they not in a place to do this regularly.

The fans that do this tireless service for us should be acknowledged for their efforts, both for this project, and more importantly, for the fandom and playerbases of these games.

That being said, much of the data that we would like to have is simply unavailable or is, at best, incomplete.

In real sports betting, sportsbooks are able to rely heavily on data aggregators to compile every conceivable bit of data about sports, events, scores, goals, fouls, players, training, coaches, etc. This isn’t the case for trading card games. The interest and size of the market just isn’t the same. It’s much smaller.

We would love to see data on each major tournament, broken down by each round. We would love to see player data reported with unique ID keys to keep variations of a player’s name or misspellings from confusing the data. We’d love to see local, sanctioned tournament data, too. But these are not realities.

We will work within these limitations and show that, at least conceptually, our idea is possible.

We’ll focus only on the widely available data, namely that from major tournaments and the highest ranked players and best known strategies.

Nature of Trading Card Game Events

Trading card game events don’t work like professional sports matches.

In professional sports matches, we know which team will play against which team and on what date. This allows the sportsbook advance knowledge of these events and gives it time to compute odds and set prices. Season schedules for any major sport are announced well ahead of time.

This is not the case for card game tournaments.

At local tournaments, anyone can show up with a deck to sign up to play. At major events, any number of qualified players can show up (or not show up). Add to this the possibility of any given strategy (i.e. deck of cards) being used by any competitor, and the matchups are simply unknowable ahead of time.

In this project we will simply ignore this as a problem. We will make the assumption that the odds are set sometime in advance of the event taking place (maybe just minutes before). Making this assumption allows us to proceed to demonstrate our ideas.

Feasibility of Taking Bets

This project isn’t a serious attempt to find a way to start taking bets on trading card games.

This may or may not be legal in any jurisdiction, and what is proposed in this project is not legal advice nor an inducement to try and make this work outside of the law.

To complicate matters, the participants of many card game events are under the legal age to gamble in many places.

Nowadays, most jurisdictions (at least in the U.S.) allow betting on college sports, where the expectation is that competitors are least 18 years of age.

Whether or not taking bets on such events would fly with gaming regulators is not considered here. This is about proving a concept (and having fun while doing it).

Don’t take anything in this project too seriously as far as making money at gambling on trading card games goes.

This is a big “what if” sort of project.

Assumptions

With our objectives in mind, and our limitations outlined, we’ll make the following assumptions for this project:

  1. All probabilistic modelling will be based on Bayesian (not Frequentist) inference.
  2. We will briefly discuss, but largely ignore, the outcome of ties. We care only about win probabilities (and consequently, not win probabilities).
  3. Win probabilities are expected to describe the win probability of matches; that is, “best two-out-of-three” matches in which the first competitor to win two games, wins the match. (This is the circumstance which often contributes to a draw between players: a time limit for the match it met with neither player having a decisive, tie-breaking win).
  4. The outcomes we seek are not only probabilistic, but also commercial: this is about setting bet prices for potential bettors. As “the house”, we expect to make money in the long run. Our models, odds, and prices will reflect that desire.
  5. As mentioned previously, we assume that we know who is playing and which deck they are using before the match. We know the identities of players and the decks they each use beforehand, thus, giving rise to our probabilities for each player to win and the consequent bet prices for each side of the match.
  6. While I will take time to explain many of the theories and logic behind each step we take in this project, I will assume that readers have some familiarity with the mathematics of probability, statistical inference, the software systems we’ll use, and the games we are speaking about. Feel free to ask in the comments if you’re unsure about something!

Segments

The project is broken down into the following segments, each with its own dedicated page: