Magic Math – Is a Team GP Better EV than an Individual GP?

A year ago, I analyzed the expected value (EV) of a Grand Prix as a function of your win rate, number of byes, and tournament size. That article focused on individual-format events. But with the team Grand Prix in Detroit coming up, it’s a good time to do the same for three-player team format events. In particular, I was curious to see whether a pro player would be more likely to make the elimination rounds as a member of a pro team at a team Grand Prix or as an individual player at a regular Grand Prix.

The Probability that a Team Wins a Round

Let’s start with a quick question. If each individual player in your team wins 60% of their matches, what is the probability that your team wins the round? This question made the rounds at Grand Prix Florence a couple of months ago. The answer, as is explained in this text coverage piece, is 64.8%. So it’s a little higher than 60% due to the best-of-three structure of team matches.

More generally, if you have three identical players on your team, each with the same individual match win percentage, you’ll find the following:

This graph indicates that good players have an even better chance of winning a match in a team event. But on the other hand, there are no byes at team Grand Prix, which is a downside for the top players. Moreover, team Grand Prix only have 5 rather than 6 rounds on Day 2. So there are conflicting factors, and it’s not a priori clear whether a team GP is better EV than an individual GP or not.

Model and Assumptions

To facilitate a proper comparison with individual-format Grands Prix, I will adhere to the same simplifying assumptions as I made in my previous previous work in which I determined the EV of an individual format Grand Prix. The main assumptions are:

The probability of winning a match is stationary: The team match win rate remains the same throughout the Swiss rounds of the tournament.

There are no draws: You can only win or lose a match.

A Top 4 in a team Grand Prix is worth \$1,625 per player: This number is the sum of all Top 4 payouts evenly distributed among all the Top 4 participants. For a justification of this and the previous two assumptions, please refer to my individual format article.

A team consists of three players with the same individual match win percentage, and the potential benefit of communication is disregarded: The assumption of identical players seems reasonable because players tend to team up with people of the same skill level. The no-communication assumption effectively means that I can apply the formula for the match win probability as described above. I view it as a reasonable simplification because the benefit of communication tends to be minor and it can help both teams equally.

The EV of any record is estimated via historical data: I used GP San Jose 2015 (656 teams), GP Portland ’14 (649 teams), and GP Kyoto ’13 (578 teams) to estimate the EV of a large-sized tournament with approximately 1,900 players. I used GP Barcelona ’14 (527 teams), GP Nashville ’14 (460 teams), and GP Florence ’15 (431 teams) to estimate the EV of a medium-sized tournament with approximately 1,400 players. (Although 1,900 and 1,400 are not divisible by three, these were the numbers I covered in my analysis of individual-format Grand Prix.) For each GP size, I put the final standings for the 3 corresponding GPs in one list and subsequently took the average, rounded to the nearest dollar, of the winnings for every possible record with no draws. This yielded the estimates for expected monetary winnings shown in the table below.

The probability of achieving a certain record is based on the convolution of two binomial distributions: I actually forgot to include the 7-2 or better cutoff to Day 2 in my previous article, but I might as well include it now for a minor accuracy boost. (At small team GPs, a few teams can make it in at 19 points, but since draws are not allowed in my model I’ll peg the cutoff at 7-2.)

Expected Monetary Winnings

Here are the outcomes:

To adequately compare these numbers to individual format Grand Prix (cf. the numbers that came from my analysis in my previously referenced article) I have to factor in byes. In doing so, I’ll use what I consider to be an optimistic estimation of the number of byes held by a typical a player with a certain match win probability. That is, I dish out byes generously, which may overestimate the EV of an individual format GP.

Conclusion

This was a fun exercise, but before concluding, I want to stress that I made a lot of simplifying assumptions along the way. This was deliberate, but it means that the numbers I obtained should be considered ballpark estimates to be taken with a grain of salt. Nevertheless, upon comparing the tables, I can reach an interesting conclusion.

The model indicates that team tournaments are (slightly) better EV than individual format tournaments if your individual match win percentage is at least 50%. This holds both in terms of expected monetary winnings and in terms of the chance of going X-2 or better for the Pro Tour invite. For instance, a Gold-level pro with an individual match win percentage of 65% can expect to win approximately \$446 at a regular GP and \$472 as a member of a team with three players of equal skill.

Since I used an (in my eyes) optimistic estimation of the EV of an individual format GP, these conclusions should be fairly robust. So, all in all, the variance-reduction effect of playing three matches per round overshadows the lack of byes even for pro teams.

With all that said, there’s one final thing that the model doesn’t account for: Team Grand Prix are simply a lot of fun. If you take that into consideration, then it’ll always be good EV to grab two friends and make the trip!