The Curse of the Top Seed

If you were going into a tournament as the top seed, what would your thought process be?  Would you expect to win the section?  Or at least have a decent chance of doing so?  Ostensibly.  Furthermore, let’s assume that as the top seed, you play lower rated opponents outside of your own class (Expert, A, B, etc.).  Would this make it significantly easier?  Again, an ostensible conclusion.

Let’s start with a scenario.  This top seed is playing in a U2200 section in a 7-round tournament.  Furthermore, it is known that the player will play against one 1900, three 2000s, and three 2100s, in that order.

To show how likely (or unlikely, for that matter) it is to win a tournament like this as the highest seed, several active players’ performances against lower- and equal-rated players were compiled from uschess.org with the ‘Game Statistics’ tool.  The weighted averages of the group’s probability of beating 1900s, 2000s, and 2100s were used.  These averages were then slightly increased/decreased to give us the potential performance of an “ideal” player that would theoretically be stronger than the average of the given players.

 

DataTable1
The table shows the probability of the result, as described on the left, against each opponent in a 7 round tournament

 

Shown above is a table with the rounded values for the probabilities of winning, drawing, and losing to each rating class that the opponent for that round would fall into.  We see how as the rating of the opponent decreases, the probabilities of winning increase – that’s expected.  On the flip side, the chance of losing a game decreases as the opponent’s rating decreases – also expected.  However, it is interesting to note how the chance of drawing games do not follow such a clear cut pattern; the chance of drawing to a 1900 and a 2000 player is virtually identical, but the probability spikes as the opponent’s rating approaches the rating of our ideal top seed.  In fact, it surpasses the probability of winning such a game.

Since we are looking at the chances of winning a tournament, the only possible scores we will look at include 7/7, 6.5/7, and 6/7.  After getting to 5.5/7, the different paths (loss + draw, three draws) to get to 5.5 points become so large that attempting to calculate the chance of reaching said points is flat out impractical.  Furthermore, the chance of winning a tournament with 5.5 points out of 7 is much less in its own right since many more players are capable of reaching that score.

The number of ways of reaching each distinct score was laid out, and the respective probability of each result was used to calculate the probability of each distinct path of reaching a certain number of points.

 

DataTable2
The only path to obtaining a result of 7-0
DataTable10
The 7 different possible paths of obtaining a result of 6.5-0.5
DataTable6
The 28 different possible paths of obtaining a result of 6-1

 

The combination computations led to 1 possibility for obtaining 7-0 (all wins), 7 possibilities of obtaining 6.5/7, (a draw in any of the 7 rounds), and 28 possibilities of obtaining 6/7 (7 possibilities of losing a game in one of the rounds plus 21 possibilities of two draws in two distinct rounds).  The probability of obtaining 7/7 was the only result of its class, so it was turned into a percent to find the actual percent chance of running the table in such a tournament.  The seven possibilities of obtaining 6.5/7 and their respective probabilities were summed to find the overall probability of obtaining such a score.  Lastly, the 21 different possibilities of obtaining 6/7 and their respective probabilities were summed to find the overall probability of obtaining such a score.  The resulting probabilities and their derivations are displayed in the following tables.

 

DataTable3
The probability of obtaining each result described in the left column
DataTable7
The calculations to find the overall probability of obtaining a result of 7-0; white cells signify a win.
DataTable8
The calculations to find the overall probability of obtaining a result of 7-0; white cells signify a win and yellow cells signify a draw.
DataTable9
The calculations to find the overall probability of obtaining a result of 6-1; white cells signify a win, yellow cells signify a draw, and red cells signify a loss.

 

As we can see, despite being the top seed in the tournament, our ideal player only has about a 0.53% chance to score 7/7.  Scoring 6.5/7 should be much easier, right?  Well, it is 5 times more likely, at about 2.66%, but it is still not overly convincing since that is still only a 1/40 chance.  We see this probability increase almost threefold when the score comes to 6/7, up to 7.28%.  Even still, this is relatively low!  Here we see that our ideal player would theoretically only go 6/7 in approximately one out of every 13.8 tournaments.  So, if the probabilities of a player are so low in regards to obtaining a high score in a tournament, why do we still see a fair amount of high scorers in 7-round tournaments (or the like)?

The answer to this question is similar to that of the birthday paradox.  If you haven’t heard of this, I encourage you to search it up, as it has some fascinating concepts.  The question simply runs, “If there are 23 people in a room, what is the chance of any two people sharing a birthday?”  As absurd as it seems, the answer is 50%.  That is because we haven’t established a specific reference; for example, if we said, “What is the chance that someone in the room has a birthday on January 1st,” then the probability would be exponentially smaller.  However, since no reference was ever given, the number is much less.  We have a similar situation in this case.  If you pick out any single player and calculate their probability of winning the tournament with any of these points, the probability would be as low as we calculated above.  But, if you have, say, 10 of these very strong players in a section, then the probability of one of these players obtaining a score of 6, 6.5, or 7 is increased tenfold from our original probabilities for the single player.  There are also other factors that could potentially increase or decrease a player’s chance at obtaining a high score in a tournament.  If a section is extremely competitive with not many players “playing up,” such as in the World Open, the probabilities decrease.  On the flip side, if there are many lower-rated players in a section, whether they are there because of the choice of playing up or if they have to, such as in grade-based scholastic national tournaments, then the probabilities increase.

Now that you have an idea of how hard it is to actually win a tournament, the next time a parent or friend asks why you didn’t win a tournament even if you were the top seed, you have statistics to defend yourself!  And, as always, thanks for reading!

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