For the 2017 season, the Hearthstone Championship Tour (HCT) switched to Swiss format. While this change was requested by many players during the previous seasons, the implementation elicited a bunch of criticism. To be honest, I do not think it was as bad as the loudest critics said, but I think a proper examination into a fair and exciting Swiss tournament format for Hearthstone is warranted.

The HCT Winter 2017 Swiss format

The European tournament consisted of 8 players from the Tavern Hero qualifiers and the top 64 point earners from the Winter season – with all players tied at #64 making the cut. Overall, the tournament had 88 participants.

For these 88 players, the Swiss tournament consisted of 7 rounds, after which the top 8 advanced to a single-elimination bracket in order to solve who would be the top 4 players who get to represent Europe in the global Winter Championships and who would be the regional champion.

Running these figures through a Swiss triangle calculator shows that the expected result would be one player a 7-0, 5 players at 6-1, and 14 players at 5-2. The actual figures were almost identical with one player at 7-0, 5 players at 6-1, and 13 players at 5-2. (There can be small discrepancies in Swiss results because some players end up paired with weaker records and upsets in these games change the overall results a little)

With a top 8 cut, it is clear that some spots would be decided by tiebreakers. The tiebreaker system was divided into three levels:

Tiebreaker #1 (T1): Win/Loss Sum

Tiebreaker #1 represents the performance of opponents that a player has played over the course of the tournament. Players that played against stronger opponents will be ranked higher within the tournament. The formula to calculate the number is: Total the number of points each opponent contributes.

Opponents contribute +1 point for each win they accumulate throughout the tournament and -1 for each loss they accumulate throughout the tournament. Each single opponent may never contribute less than -3 points to a player’s first tiebreaker. A bye does not contribute any points towards a player’s tiebreaker.

Tiebreaker #2 (T2): First Tiebreaker Sum

Tiebreaker #2 represents the performance of the opponents that all of a player’s opponents played. Players that played against opponents who consistently played against stronger opponents throughout the tournament will be ranked higher. The formula to calculate the number is:

Total the sum of Tiebreaker #1 (T1) for all opponents that player played.

Tiebreaker #3 (T3): Timing

Tiebreaker #3 represents the importance of the rounds in which you lost. Players losing in later rounds will be ranked higher within the tournament. The formula to calculate this number is:

The sum of the squares of the rounds in which you lost.

With this setup, the functionality of the tiebreakers is of crucial importance, as they decide two of the top-8 spots.

The actual tiebreakers turned out to be pretty close, here are the tiebreakers of the top players at 5-2:

Neirea 23, 76, 85
GreenSheep 17, 57, 61
Twink 17, 49, 65
M1SA 15, 57, 41
Farm4fame 11, 58, 65

The final spot went to GreenSheep over Twink on the second tiebreaker!

We’ll analyze tiebreakers a bit more later. For now, let’s take a look at an alternative way to arranging tiebreakers.

Magic: The Gathering Swiss tournament rules as per January 2017

Magic: The Gathering is an inevitable comparison for any card game. They’ve been around since the early 1990s and created the whole category. So, how does Magic run tournaments?

Magic also uses the Swiss format. There is one major difference in that in Magic, it is possible to draw a game, and in the later rounds of Magic Swiss tournaments, players often intentionally draw (ID) their games to ensure that both progress to the top cut playoffs.

In Magic, the number of rounds for Swiss tournaments are specified as follows:

8 players: single elimination, 3 rounds
9-16 players: 5 rounds, top-4 playoffs
17-32 players: 5 rounds, top-8 playoffs
33-64 players: 6 rounds, top-8 playoffs
65-128 players: 7 rounds, top-8 playoffs
129-226 players: 8 rounds, top-8 playoffs
227-409 players: 9 rounds, top-8 playoffs
410+ players: 10 rounds, top-8 playoffs

In the system used in Magic, all x-1 records are safe and make it to the playoffs up to a tournament size of 799 players. The HCT Winter 2017 tournament used a round count compatible with the system used in Magic, so all x-1 records were safe and it was some spots for x-2 records that were decided by tiebreakers.

The tiebreakers in Magic work slightly differently than on the Hearthstone Championship Tour (although in Hearthstone, Dreamhack has used the same first two tiebreakers as Magic):

Opponent’s match-win percentage
Game-win percentage
Opponent’s game-win percentage

Opponent’s match-win percentage

A player’s match-win percentage is that player’s accumulated match points divided by the total match points possible in those rounds. If this number is lower than 0.33, use 0.33 instead. The minimum match-win percentage of 0.33 limits the effect low performances have when calculating and comparing opponents’ match-win percentage.

In Magic, players receive 3 points for a match win and 1 point for a draw. In Hearthstone, where there are no draws, it is more simple to count match wins as 1 point each. There is some difference between match-win percentage and win/loss sum (the HCT tiebreaker), but this mostly affects the bottom scores – match-win percentages of 33% or less – and the effect of byes. When it comes to the first tiebreaker, players are unlikely to have such scores on their opponents if they are near the top of the overall tiebreakers.

Game-win percentage

In HCT, your game-win record has no effect on your tiebreakers. In Magic (and in Dreamhack tournaments), it does.

Similar to the match-win percentage, a player’s game-win percentage is the total number of game points he or she earned divided by the total game points possible (generally, 3 times the number of games played). Again, use 0.33 if the actual game-win percentage is lower than that.

Fair tiebreakers

Let’s examine the tiebreakers of the top 5-2 players (finishing 7th-11th in Swiss) in HCT Europe Winter Playoffs in a bit more detail, also including some figures used in Magic for tiebreakers:

Neirea:

2-3 vs sjow 6-1
1-3 vs Shtanudachi 6-1
3-2 vs GreenSheep 5-2
3-0 vs Anakth 6-1
3-2 vs Grayj 4-3
3-2 vs Juristis 4-3
3-1 vs BoarControl 5-2

Win/loss sum 23, opponents’ match-win percentage 73.5% (36/49), game-win percentage 58.1% (18-13)

GreenSheep:

3-0 vs SomiTequila 4-3
2-3 vs Anatkh 6-1
2-3 vs Neirea 5-2
3-0 vs farm4fame 5-2
3-2 vs noSalvati0n 4-3
3-0 vs Kalàxz 4-3
3-2 vs Dizdemon 5-2

Win/loss sum 17, opponents’ match-win percentage 67.3% (33/49), game-win percentage 65.5% (19-10)

Twink:

2-3 vs Pavel 6-1
3-1 vs Kycoo 4-3
3-1 vs CelticGuard 3-4
1-3 vs Shtanudachi 6-1
3-2 vs Weghuz 5-2
3-1 vs tholwmenos 5-2
3-2 vs TonyBanger 4-3

Win/loss sum 17, opponents’ match-win percentage 67.3% (33/49), game-win percentage 58.1% (18-13)

M1SA:

3-1 vs Tyler 4-3
3-1 vs Juristis 4-3
1-3 vs sjow 6-1
2-3 vs pokrovac 7-0
3-1 vs erikbeck 3-4
3-0 vs FibEli3 4-3
3-2 vs Bunnyhoppor 4-3

Win/loss sum 15, opponents’ match-win percentage 65.3% (32/49), game-win percentage 62.1% (18-11)

Farm4fame:

2-3 vs Anakth 6-1
3-1 vs Powder 4-3
3-2 vs Lifecoach 5-2
0-3 vs GreenSheep 5-2
3-0 vs ikealyou 5-2
3-2 vs Un33D 1-3
3-0 vs Xzirez 1-3

Win/loss sum 11, opponents’ match-win percentage 55.1% (27/49), game-win percentage 60.7% (17-11)

The difference between the win/loss sum and match-win percentage is usually small. GreenSheep and Twink would have been tied with either metric. Only Farm4fame would have had a different (better) tiebreaker if match-win percentage was used, as two of his opponents went 1-3 and had a match-win percentage below the lower limit of 33%.

However, it is notable that there were players with 2-5 and 1-6 records in the tournament, and the lower limit of win/loss sum at -3 punished the tiebreakers of their opponents quite hard. As only the best 5-2s make the top cut, it is rational to drop out at 0-2, or 1-3 (like Xzirez) for example: these records give -2 win/loss, whereas a player who continues from there and finishes at 2-5 gives -3 win/loss, hurting his opponents’ tiebreakers by continuing.

This is undesirable: win/loss tiebreaker should never be worse than the logical drop-out point, which in this case was -2 (Thanks to Pascoalabear on Twitter for pointing this out to me). While the effect is negligible on the best first tiebreakers, it can affect the second tiebreakers of any player, if the second tiebreaker is the performance of the opponents’ opponents.

There is also a potential issue with byes (none of the players in our sample had any). In the HCT system, a bye is +0, whereas in Magic, byes are ignored in tiebreaker percentage calculations. This can either benefit or hurt a player, but for all top spots, where each opponent generally contributes positively to the tiebreaker, a bye would mostly hurt a player’s win/loss sum tiebreaker.

The second tiebreakers of HCT and Magic are completely unrelated. In Magic, they measure your own performance within a match, and in HCT they measure the performance of your opponents’ opponents.

In this particular tournament, GreenSheep would have advanced over Twink with either metric.

It is understandable why there is such a difference though: the multi-deck format used in Hearthstone. In Magic, you play with one deck (and sideboard). In Hearthstone, you play with multiple decks, and can choose strategies that target specific decks (remember Pavel’s reverse sweeps last year). In a multi-deck format, game scores are not as reliable a measure, so it is reasonable to go for the next step. However, game-win records still seem to be a better choice than timing for the final tiebreaker.

Conclusion: The tiebreakers used in the tournament were reasonable. At minimum though, the lower limit of the win/loss tiebreaker should be set to the logical drop-out point (in this case -2). Adopting a win-percentage approach would be a better improvement as it would improve bye handling as well.

Fair number of rounds

In Magic, the number of rounds is set so that x-1 record is always safe for the playoffs. This is extremely desirable. In fact, Hearthstone is in a better position than Magic, because there are no draws in Hearthstone. In Magic, players draw matches intentionally to secure playoff spots, and this cannot happen in Hearthstone.

What could happen is people intentionally losing games in order to affect who else qualifies. Number of rounds is a crucial factor to prevent opportunities to do this.

The biggest effect of intentional losses is at the top. When x-1 record is safe, and x-2 is not, there are hardly any opportunities to lose intentionally while still retaining certain qualification for yourself. If the number of rounds were increased so that x-2 or worse records would be safe, there would be more opportunities to lose intentionally while still qualifying.

Fair prizes: reducing the motivation for collusion

While players who cannot qualify themselves have limited ability to affect who qualifies, they can still affect it. For example, if CelticGuard had won DrHippi on the last round of Swiss (he lost 2-3), Twink would have qualified to the playoffs over GreenSheep! At that point, neither CelticGuard nor DrHippi had a chance to qualify. Furthermore, neither of them had any additional prizes to earn. These situations should be as rare as possible (although this particular one, with both players at 3-3, is nigh impossible to prevent).

The prize pool for HCT Europe Winter Playoffs was particularly bad, at least as far as it is public, we still don’t know for sure. The common version is that there was a $100 participation prize, and other than that the top-4 qualified for the Championships and the guaranteed prizes there, whereas the 5th-8th finishers received $5000. This means that the prizes at 5-2 record were from $100 to $5000, depending on tiebreakers.

In Magic, the prizes at Pro Tours increase steadily with the top 64 (of around 400) getting money. Granted, some of the money spots are decided on tiebreakers: at Pro Tour Aether Revolt, the 48th-89th finishers all had the same number of points and were decided on tiebreakers.

There is also another way that is sometimes used in Swiss tournaments: a specific record can have a prize pool attached to it which is then divided by all the players who achieve that record. (It can also be a set sum for any players who finish with a specific record, although in this case the overall prize pool can change a little depending on match results, there is often one more or one fewer players at a certain record as not all pairings can be against players with the same record)

For example, the Winter Playoffs could have had a certain prize pool for 5-2 records in addition to the top 8. This would ensure that more players have something to play for even if they are not going to make it to the top 8.

There would still be two gray areas: a 4-2 paired with a 3-3 in the final round, where it would be logical for the 3-3 to concede, and any players with 3-3 or worse records on the final round who had played the top players and could affect their tiebreakers. The issue cannot be completely removed, but a wider prize distribution can remove most of the motivation to intentionally skew results.

Balancing fairness and excitement in the playoffs

Both HTC and Magic have been using a top-8 cut for final single-elimination playoffs. To be clear, Swiss does not need a single-elimination playoff stage in order to find a winner. The only reason for that is excitement, and that is a crucial part of any sport as well. In a purely Swiss tournament, the end result is a consequence of all the matches played, but there are not that many big matches that have a considerable impact alone. Nonetheless, such important matches are exciting for the viewers, and that is why Swiss is often combined with separate playoffs in some other format.

There was some outrage at the fate of pokrovac at the Europe Winter Playoffs: he won the Swiss part 7-0, but lost on the first single-elimination playoff round and was eliminated from the top 4 and thus from the Winter Championship. Is this exciting enough to balance out the feeling of unfairness?

Magic uses a slightly different system: the top 2 from Swiss get seeded directly into semi-finals, and positions 3 and 4 are seeded into top 6. This gives the best Swiss performers an advantage, and could be a reasonable compromise for Hearthstone as well.

There are many other top-8 playoff formats that are designed to give the higher seeds an advantage, such as the McIntyre final eight system (that gives most advantage to #1 and #2), AFL final eight system (that gives most advantage to top 4), and the Super League system (that gives most advantage to top 4). All of these systems are also designed to provide the chance for as many possible pairings for the grand final as possible, an advantage over a more simple system. Adopting one of these systems would have the advantage that top-4 spots are not given outright based on Swiss, but the higher seeds still have an easier time making it to that top-4 spot and the Championship. As it favors higher ranks progressively and especially the very top, the McIntyre final eight system seems particularly suitable for Hearthstone.

The McIntyre final eight system

Ken McIntyre was an Australian lawyer who developed multiple playoff systems for various group sizes. One of the systems he developed was for top 8 finishers, and it was used in top-level sports in Australia for several years.

There are several desirable features in the system when it comes to Hearthstone Playoffs that determine 4 spots for the Championship tournament as well as the regional champion:

None of the players qualify to top 4 based on Swiss alone (adopting the bye system used in Magic would mean that the top 2 players from Swiss qualify automatically)
26 out of the 28 possible combinations of players can appear in the regional final (only 1 vs 7 and 2 vs 8 are impossible at that stage)
No repeats of games are possible until the regional final (same players can only meet on the first round and then again in the regional final)
The higher finishers in Swiss have an easier path to the finals: players finishing first and second are never eliminated if they lose on the first round of the playoffs – given how there is often one undefeated player in Swiss, specifically rewarding the very top places makes sense. Alternative systems that protect the top seeds either place the top seeds directly into top 4 or unconditionally protect the entire top 4 from early elimination, spreading the protection wider. The McIntyre system gives higher seeds a better chance while limiting the best protection to top two only.

Potential criticism of the system:

The main criticism against the system has been the way home advantage is handled. In Hearthstone, there is no such thing, so this line of criticism does not apply.
Another criticism has been that there can be meaningless matches on the first round of the playoffs if everything goes as seeded. If #1 and #2 seeds win their matches, players #3-#6 will all advance to the second round. However, upsets are not that unusual in a game like Hearthstone, so the system would likely produce a more varied playoff chart most of the time. Furthermore, this can be used in production to create redemption storylines: “OK, he lost the #4 vs #5 match, is he out?” “No, there is another upset and he gets a second chance!” The rest is history.
Finally, it is possible for the finals to take place between two players who already met on the first round of the playoffs. Is it wrong for the winner of that match to have to face the same opponent again, and possibly lose?

How does it work?

Round 1:

1st qualifying final: 4th vs 5th
2nd qualifying final: 3rd vs 6th
3rd qualifying final: 2nd vs 7th
4th qualifying final: 1st vs 8th

The two lowest-ranked losers are eliminated from the finals, while the two highest-ranked winners progress straight to round 3.

Round 2:

1st semi-final: 4th highest-ranked winner vs 2nd highest-ranked loser
2nd semi-final: 3rd highest-ranked winner vs 1st highest-ranked loser

The two losing players are eliminated, the two winners progress to round 3.

Round 3:

1st preliminary final: 1st highest-ranked winner (from round 1) vs winner of 1st semi final
2nd preliminary final: 2nd highest-ranked winner (from round 1) vs winner of 2nd semi final

The two losing players are eliminated, the two winners progress to round 4.

Round 4:

Regional final: winner of 1st preliminary final vs winner of 2nd preliminary final

Example (from Wikipedia):

At the Qualifying Finals, team 1 won and so went straight through to the Preliminary Finals. Team 4 also went through to the Preliminary Finals, because they won while teams 2 and 3 lost. However, teams 2 and 3 were not eliminated, but played again in the Semi Finals, because two teams ranking lower than them on the ladder also lost. Those two losing teams ranking lower than them, teams 5 and 8, were eliminated.

Percentage chances of winning the tournament (assuming a 50% win rate):

1st seed: 18.75%
2nd seed: 18.75%
3rd seed: 15.625%
4th seed: 12.5%
5th seed: 12.5%
6th seed: 9.375%
7th seed: 6.25%
8th seed: 6.25%

As can be seen from these odds, the system favors the top finishers from Swiss, but less so than alternative systems that protect the top seeds. In the Magic system, for example, 1st and 2nd seeds have a 25% chance to win the tournament under the same assumptions. Likewise, the Magic system gives the 1st and 2nd seeds a 100% chance to qualify into the top 4, whereas the McIntyre system gives them a 75% chance to do so.

Conclusions: a fair and exciting format

Based on the discussion above, I would like to see the HCT use the following system:

Number of rounds in Swiss set so that x-1 record is safe for the top-8 cut, but x-2 is not. Any players with an x-2 record who make the top cut do so via tiebreakers.

Tiebreakers based on percentages, not absolute figures. Byes ignored in calculation. #1 Opponents’ Match-win percentage, #2 Match-win percentage of the opponents’ opponents, #3 Game-win percentage

Prize pool extended to outside top-8 based on match record: all players with at least x-2 record receive a prize.

Use the McIntyre final eight system for the top-8 playoffs.