A while ago, I analyzed the expected value (EV) of a Grand Prix as a function of a player’s match win rate. Today, I want to do the same for Pro Tours.

# Model and Assumptions

I will use a mathematical model similar to the one I used for my Grand Prix analysis. The model is an abstraction of reality in which I purposefully disregard some of the messy details of a real Pro Tour for simplicity, elegance, and mathematical tractability. The simplifying assumptions underlying my model are as follows:

**Every competitor has a stationary match win percentage for Day 1 and a slightly lower one for Day 2**. In my GP work, I assumed a stationary match win percentage throughout the entire tournament, but I wanted to make the model a bit more realistic. On Day 2, the level of competition is typically a little higher, and I capture that by setting the Day 2 match win percentage P2 of a player who has a Day 1 match win percentage P1>50% to P2 = 0.8 * P1 + 0.2 * 50%. The 80-20 weighing is calibrated on my own Pro Tour records from the past 3 years.To illustrate what this formula means: in my model, a good player who has a 60% match win percentage for all matches on Day 1 will have a 58% match win percentage for all matches on Day 2. And a player who has 55% on Day 1 will have 54% on Day 2. But a player who has a Day 1 match win percentage below 50% will stay at the same match win rate for Day 2.

**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. Moreover, today’s Pro Point payout largely disincentives last-round intentional draws. The exception is that an ID from 12-3 into 12-3-1, which is not allowed by my model, will often put a player into the Top 8. For that reason, my model may slightly underestimate monetary EV, but I can live with that.**A Top 8 is worth $15,625 and 21.5 Pro Points**: This number is the rounded sum of the Top 8 payouts divided by 8. 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, the very best players may still have a higher win rate and they may also be more likely to get to choose play or draw in the Top 8, but these complicating features are abstracted away for simplicity.**The EV of any record is estimated via historical data**: I used the last four Pro Tours (*Magic Origins*,*Battle for Zendikar*,*Oath of the Gatewatch*, and*Shadows over Innistrad*) to determine the historical average of the monetary winnings of any particular record without draws.

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

Number of wins in the Swiss | Expected monetary prize | Pro Points |

8 or fewer | $0 | 3 |

9 | $0 | 4 |

10 | $979 | 6 |

11 | $1,962 | 10 |

12 | $8,269 | 17 |

13 or more | $15,625 | 21.5 |

Players with a 12-4 record could make Top 8 with good tiebreakers. Players with a 13-3 or better record always made Top 8 in my data set.

# The Probability of Achieving a Certain Record

So now that I have estimated the expected winnings for various records, I still need to figure out how likely it is to obtain a certain number of wins. In this section, I’ll explain how that works. Feel free to skip ahead to the next section if you don’t care about convolutions of binomial distributions.

Let’s start with an example. Suppose that your match win rate on Day 1 is 60%, which under my model implies that your match win rate on Day 2 is 58%. You need a 4-4 record or better on Day 1 to advance to Day 2. How often will you end up with exactly an 11-5 record?

To start, let’s focus on a specific Day 1 record, let’s say 6-2. One way to reach that is by winning the first 6 rounds and then losing the last 2 rounds. This scenario happens with probability 0.6^6 times 0.4^2. But you could also lose the first 2 and subsequently win 6 in a row, or have any other ordering of 6 wins and 2 losses (all of which occur with the same probability). The number of such orderings is 28, as given by the binomial coefficient 8 choose 2. Combining all of this, I find that a 6-2 record on Day 1 occurs with probability 0.6^6*0.4^2*28=0.209 (or, if you prefer, 20.9%).

After entering Day 2 with a 6-2 record, you need to win 5 matches and lose 3 matches to end up with an 11-5 record at the end of the tournament. Given that your match win rate is 58% and that there are 56 ways to order 5 wins and 3 losses, another straightforward application of the binomial distribution yields that a 5-3 record on Day 2 will occur with probability 0.58^5*0.42^3*56=0.272.

Multiplying the two, a 6-2 into 11-5 record happens with probability 0.209*0.272=0.057. But that’s just for the specific 6-2 record on Day 1. I also need to determine the numbers corresponding to 4-4, 5-3, 7-1, and 8-0, and add them all up. The result is a convolution of two binomial distributions, and the general formula can be stated as follows.

# Expected Prize Winnings

You are now ready to determine the monetary EV in US dollars at a Pro Tour as a function of your Day 1 match win rate.

So what does this mean?

- An average Pro Tour competitor with a win percentage of 50% would have an expected value of $637 and an extremely high variance. The variance is not shown in the graph, but its easier-to-interpret square root, also known as the standard deviation, is $2,110. This serves to highlight the top-heavy nature and randomness in Pro Tour winnings. For example, a player who wins $0, $0, $0, $5000, $0, $0, $0, and $0 over the course of 2 Pro Tour seasons would have a standard deviation and expected value in this data set very close to the above-described random variable.
- A player whose Day 1 win percentage is 58%, which corresponds to my own overall win percentage over the last four Pro Tours, would have an expected value of $1,697 (and a standard deviation of $3,606) according to my model. Applying this to my own position as a Hall-of-Famer, I would have a net expected profit of several hundreds of dollars if I attend the 2017 Pro Tours in Nashville or Kyoto (I have to pay for my own flight and accommodation and don’t receive any appearance fee). I highly appreciate the opportunity to have a “free” holiday (under humongous variance) and the invitation to compete in an awesome tournament. But given that the travel, preparation, and tournament takes about 2 weeks of my life for every Pro Tour, I clearly can’t (and don’t) rely on the Pro Tour to pay my monthly bills.
- Seth Manfield, currently ranked #1 in the world, posted an exceptional 68% win percentage overall in the last four Pro Tours (“Overall” includes Day 1 and Day 2). Such an extraordinary win rate may not be sustainable for anyone in the long term, but according to my model, a player with a Day 1 win percentage of 68% would have an expected value of $4,416 (and a standard deviation of $5,579) per Pro Tour.
- As a quick model validation, the total prize payout to an average-sized field of 380 players, all with a 50% match win rate, would be $242K in my model. This is close to the actual total prize payout of $250K, indicating that my model is at least a reasonable representation of the real thing.
- Suppose that you are a very strong player. You would have a 55% win percentage with just 3 days of testing and a 60% win percentage with 10 days of testing. According to my model, that extra week spent on preparation would lift your expected winnings from $1,204 to $2,102, which (assuming 8 hours of testing per day) would come down to an hourly wage of $16.04. This doesn’t yet encompass the potential extra benefits associated with reaching certain pro levels, but it also doesn’t take into account expenses such as the rent of a testing house for a week. I’m confident that the players who are smart and dedicated enough to be able to attain such a high win percentage could earn more in a “regular” job, but in the end, many pros are in it mostly for the love of the game and the thrill of competition.

# Odds of Getting 11-5 or Better

An 11-5 or better record guarantees an invite and airfare to the next Pro Tour.

So an average Pro Tour competitor with a 50% Day 1 match win probability will have a 10.4% chance to post a good enough result to qualify and earn airfare for the next Pro Tour.

# How About Pro Points?

If you pick up enough Pro Points in one season, then you can reach a threshold for the Pro Players club that comes with various benefits. For instance, 33 Pro Points in one season earns Gold level, which provides invites and expenses-paid air travel to all Pro Tours as well as 3 byes at every individual Grand Prix. Once you score 50 points in one season, you become Platinum, which yields an appearance fee of $3,000 per Pro Tour. Pick up a few more, and you can qualify for the World Championship with large cash prizes and prestige.

Let’s start with a breakdown of the expected number of Pro Points per Pro Tour.

Unfortunately, the expected value doesn’t tell me that much when it comes to Pro Points. There is a lot of variance, and to determine the probability of getting a certain number of Pro Points in a season, I’d need to have the full Pro Point probability distribution and convolute it four times.

I’ll spare you the exact details, but let’s focus on a player with a match win percentage of 60% on Day 1 and 58% on Day 2. This is the level that a Gold+ pro player in their prime tends to attain, in line with the stats listed in the various Hall of Fame inductee profiles. For such a player, my no-draw model yields the following distribution.

Pro Points earned in one Pro Tour | Percentage probability |

3 | 35.06% |

4 | 16.66% |

6 | 18.47% |

10 | 15.10% |

15 | 6.33% |

18 | 4.19% |

22 | 2.09% |

26 | 1.05% |

30 | 1.05% |

This gives an 8.38% probability to make Top 8, indicating that such a player will make 1 Pro Tour Top 8 per 12 Pro Tours (i.e., three seasons) on average in the long term.

Convoluting this distribution four times, you get the following.

To interpret this graph, consider a pro player who picks up 18 points via Grand Prix, the World Magic Cup, and/or the World Championship. For Gold level, this player would have to score at least 15 additional points at Pro Tours, which happens with 93.5% probability according to this graph. For Platinum level, a minimum of 32 extra points from Pro Tours are required, which corresponds to a 33.4% probability. In other words, even very strong players are far from guaranteed to clinch Platinum level. But any percentage point they add to their win rate or any additional eligible point picked up at a Grand Prix greatly increases their Platinum odds.

# Concluding Thoughts

If you were ever curious about how much an invitation to the Pro Tour would be worth in monetary terms, then I hope this article has provided some insight. A few quick hits:

- A Pro Tour competitor with a Day 1 win percentage of 50% (68%) has an expected value of $637 ($4,416) per Pro Tour.
- It’s not realistic to expect to be able to live off of Pro Tour winnings alone—it’s just one source among many. Pro players generally rely on extra income from Platinum, the World Championship, sponsorship deals, content production, and/or a regular job.
- A pro player with a 60% Day 1 win percentage makes one Top 8 per 12 Pro Tours on average and (depending on how many points they pick up at Grand Prix events) is still often an underdog to clinch Platinum.

I need to stress that I deliberately made a lot of simplifying assumptions, so the numbers I obtained are ballpark estimates to be taken with a grain of salt. Although the prize money payout distribution is changing a bit for next year, the total cash purse stays roughly the same, so my numbers will remain reasonable approximations. Another thing to keep in mind is that the payout at a Pro Tour is top-heavy, so the amount of money you may win at a Pro Tour has huge variance.

All things considered, the life of a Magic pro isn’t a gold mine in expectation. Even if the very best Worlds-bound Platinum pros can win a nice amount of money in a year, they would be foolish to do this in the expectation to get rich. But the monetary EV is just one aspect. Most of them are in it for the love of the game, the competition, and the experience that a Pro Tour provides. I still have the dream of winning a Pro Tour one day, hoisting a trophy together with my team after breaking the format with an incredible deck. And that, as they say, would be priceless.

## Discussion