How much can you expect to earn on average at a Grand Prix tournament? What is the likelihood of earning a qualification for the Pro Tour? And how do these things depend on your match win rate, your number of byes, and the size of the Grand Prix? Let’s do some analysis!

## Model and Assumptions

To answer the questions raised above, I will construct a mathematical model. This model will be an abstraction of reality in which I purposefully disregard some of the messy details of a real Grand Prix for simplicity, elegance, and mathematical tractability. The simplifying assumptions underlying my model are as follows:

**The match win rate is stationary**: The probability of winning a match remains the same throughout the Swiss rounds of the tournament. So, for example, I can consider a good player with a 60% match win rate, and then that 60% will be the same on Day 1 and Day 2. In reality, the Day 2 competition may be a little tougher, but the difference is not that large in my experience. The probability of winning a match also doesn’t depend on the outcome of the previous match, so I ignore the possibility of “in the zone” win streaks or “tilting” loss streaks. A more detailed model could feature a match win rate that is a function of the previous match history, but this would lead to substantially more complicated analysis and a muddled interpretation.

**There are no draws**: You can only win or lose a match. This assumption is not far from reality because draws are rare, especially if you play fast enough. Granted, sometimes you may be able to ID into a money finish or the Top 8, and my model doesn’t allow for that possibility, but these situations are once again rare and can be disregarded without too much impact, especially for medium-sized or large GPs where 12-2-1 is not enough for Top 8 anyway. For small GPs where 12-2-1 does make Top 8, this assumption may lead to a slight underestimation of the EV, but I can live with that.

**A Top 8 is worth $1,713**: This number is the rounded sum of the Top 8 payouts divided by 8. (I only consider monetary winnings, not the travel award.) What this implies is that I assume that the Top 8 is filled with experienced players of a similar caliber and that all Top 8 competitors have a 50% chance of winning any Top 8 match. In reality, Platinum Pro players may still have a higher win rate against the average GP Top 8 competitor, and they may also be more likely to get a better Swiss record so they can choose to be on the play in the Top 8, but adding those features would make the model way more complicated and not substantially more accurate.

**The EV of any record is estimated via historical data : I used 9 Grand Prix held in 2014 to estimate the EV (which stands for Expected Value, by the way, and can be thought of as a long-run average) for tournaments of three different sizes: 900, 1,400, and 1,900 players. I considered adding a “huge” category represented by the 4,500-player GP in Las Vegas, but it was held when GP Trials still gave out 3 byes, so the numbers wouldn’t be accurate for future events.**

Here are the expected winnings I found, rounded to the nearest dollar.

I used GP Buenos Aires (882 players), GP Salt Lake City (893 players), and GP Melbourne (901 players) to estimate the EV of a 900-player GP; I used GP Prague (1,398 players), GP Manchester (1,406 players), and GP Washington DC (1,422 players) to estimate the EV of a 1,400-player GP; and I used GP Sacramento (1,858 players), GP Philadelphia (1,889 players), and GP Chicago (2,011 players) to estimate the EV of a 1,900-player GP.

For each GP size, I put the final standings for the 3 corresponding GPs in one list and subsequently took the average of the winnings for every possible record with no draws. This yielded the numbers shown in the table above.

## Probability of Achieving a Certain Record

So we have now estimated the expected winnings for various records, but we still need to figure out how likely it is to obtain a certain number of wins. Let’s start with an example. Suppose that you have 2 byes and that your match win rate is 60%. How often will you end up with an 11-4 record?

Well, one possibility is that, after your 2 byes, you win the first 9 rounds and then lose the last 4. This scenario happens with probability 0.6^9 times 0.4^4. However, you could also lose the first four and subsequently win nine in a row, or have any other ordering of nine wins and four losses (all of which occur with the same probability). The number of such orderings is 715, as given by the binomial coefficient of 13 choose 4. Combining all of this, we find that an 11-4 record will occur with probability 0.6^9*0.4^4*715=0.184.

More generally, if you have *B* byes and a probability *P* of winning a match, then you will end up with *W* wins with probability:

We now have everything we need to obtain some results. For ease of presentation, I will consider four different match win rates: 40%, 50%, 60%, and 70%. You may interpret 40% as an inexperienced player and 70% as a Platinum-level Pro. (As an indication: I won 66% of my Grand Prix matches in the last season. This is based on a sample size of merely 90 matches, and my historical win rate is a few percentage points higher, but this may provide a useful benchmark.)

## EV for a Small GP (900 players)

Byes clearly help, but their impact shouldn’t be overstated. For example, if you are a player with a 50% match win rate and you win a GP Trial to go from 0 to 2 byes, your expected winnings increase by $50. That’s nice, but you can achieve a bigger jump by adding 10% to your match win percentage. I found the difference between a 60% to a 70% match win rate to be particularly striking.

## EV for a Medium GP (1,400 players)

Most European GPs are around this size, so my model indicates that with my win rate and 3 byes, I can expect to win approximately $500 if I participate in a European GP. Not bad, but it’s important to keep in mind that the top-heavy nature of the payouts leads to a huge variance. For instance, a player with an EV of $500 could easily win $2700, $0, $250, $400, $300, $300, $0, $250, $300, and $250 in ten consecutive GPs, in which case he or she would win less than their EV in all but one of those tournaments.

## EV for a Large GP (1,900 players)

The EV of a 1,900-player GP is still more than half the EV of a 900-player GP, even though there are over twice as many players. The increased prize pool, among other factors, does help a bit here. But looking at these numbers, if I would only be interested in maximizing my monetary winnings minus travel costs, then I’ll still take a large European GP and a cheap flight over a small Australian GP and an expensive flight.

## Odds of getting a 13-2 or better record

These numbers, which obviously don’t depend on GP size, are interesting because a 13-2 or better record guarantees a Pro Tour invite. So, for instance, a good player with a 60% match win rate and 2 byes will obtain a PT qualification once every 17 GPs on average. That’s a nice incentive to attend a GP, and the odds will improve tremendously with every percentage point you add to your match win rate. Perfecting your play and deck choice does pay off, at least in expectation.

Feel free to discuss and interpolate my results, and best of luck at your next GP!

## Discussion