Thurn and Taxis tiebreak system is as follows:

Each game you play that you finish first or second, you score points as follows:

• Win in first game played: 1500 points plus (your score/second place score)
• Win not in first game played: 1000 points plus (your score/second place score)
• Second place, 4-player game: 100 points plus 10 * (your score/winner’s score)
• Second place, 3-player game: 50 points plus 10 * (your score/winner’s score).
• Top 16 scores qualify for the finals.

This scoring system was designed to fulfill the following goals:

1. Playing never hurts

People should always be encouraged, not discouraged, from playing a game. So I never want to put people in the situation of “I qualify now, but if I play again, and do badly, I might not qualify”. So playing an additional game can only help, not hurt, your qualification score.”

2. Reward achievement, not attendance.

The best players should qualify for the semifinals. Showing up for a heat and finishing last doesn’t show you are a good player; it just shows that you showed up. So points are awarded only for finishing at least average, which in a game with 3 or 4 players, means 1st or 2nd. If you play a heat and finish 3rd or 4th, it doesn’t hurt, but it doesn’t help, either.

3. Scores should only be counted as score differentials.

In many games, a game may be high scoring for everyone or low scoring for everyone, depending on how the game goes. A Thurn and Taxis or Saint Petersburg game with many turns will have higher scores for everyone, but doesn’t mean that everyone is a better player. So the absolute score should never be used in the qualification formula, since it gives an incentive to aim for a high-scoring game, which can be in conflict with the main objective of scoring higher than the other players.

4. Score differentials should be measured by ratios.

If scores are higher, we expect score differentials to be higher, too. So winning 15 to 10 is a more impressive achievement than winning 20 to 15, and should get more tiebreak points. So the tiebreak always measures the ratio of your score to another’s score, rather than a difference. This has the added benefit of producing far fewer ties; since an 18-17 win is just slightly better by ratio than an 18-19 win, exact ratio ties are very rare except when both compared scores are the same.

5. Compare to other good players where possible.

Tournaments will always include some beginners who play poorly and finish with very low scores. The difference in skill, and therefore in score between the best and the worst players can be quite large, and the luck of having a very weak player at your table should not be rewarded. Some luck of the draw is unavoidable, but if we say that it’s unlikely that there will be more than two very weak players at a table, their effect on qualification is minimized if the first player is only compared to the second player, and vice versa, rather than comparing to all players or to an average.

Also, to the extent that one can make plays that target particular other players, I’d rather the first and second player target each other, rather than target the players who are losing, which I think makes a more fun experience for all.

6. Count closeness of seconds as more important than big firsts.

A close second means you did well against a player who won, while a win far ahead of second only means you did well against a player who came in second. There’s less evidence you did well against a good player, so you get fewer tiebreak points.

7. First in a three player game counts as much as first in a four player game, but second in a three player game counts for less than in a four player game.

If first didn’t count as much for three players as four, a player in a three player game would be at a severe handicap in qualifying. A second in a three-player game is easier to achieve than a second in a four-player game, so it counts for less. The fact that you can’t get full credit for a second balances out the fact that it’s easier to get a first.

8. For simplicity, incorporate the HMW (heats: most wins) rules into the formula

Technically, the point formula should only apply if the list of precedence rules for qualification reaches the ‘GM specified tiebreak rule’. But I think it’s easier to understand a rule that says “Add these up; highest score qualifies” than one that says:
“First see who has most wins; then see who has a win in their first heat entered, then…then add these numbers up and see who is highest”
So the size of the bonuses for win in first heat entered, other win, and second place ensure that the standard HMW rules will be followed, with the tiebreak points only mattering in comparing people with the same number of firsts and same number of seconds, and both or neither having a win in first heat entered.

If your tournament is HSW (heats: single win), rather than HMW, so that a win in the first heat entered is more important than two wins, you can achieve this by modifying the points for win in first heat entered to be:

Win in first game played: 5000 points plus (your score/second place score)

Feedback on this scoring system is welcome; email me at andy.latto@pobox.com.