Wednesday, February 19, 2014

Tournament Seeding: A basic look at how player positions change tournament results.



I have not updated this blog in a long time and I decided it’s been too long.  So time to finally put another entry in!  Last time I took a look at the simplest double elimination bracket you could come up with and how the bracket was affected by uniformly changing player 1’s chance of winning a single game.  You can see the results from that study at this link

http://loyalsol.blogspot.com/2013/04/winning-major-just-how-good-do-you-have.html

It took me a while to get back to this both because I’m still working on my PHD time is precious and also that I wanted to update and optimize the code some more before proceeding to make larger studies easier to do.  The previous version of the code required me to recompile every time I wanted to change the bracket size which made large scale studies tedious. So I took the time to generalize the code into a version where I could control all the parameters from a simple control file at run time.  To do so, I had to essentially gut a good portion of the code and then go through the debugging process to ensure the result I was getting was accurate.   Took some time, but finally got back to where it should be.   At any rate hope you guys like this. 

Set up:

This time I want to up the level of complexity a little.  So this time we are going to examine another common topic that comes up and that is seeding.   If you go to a local tournament you might have a situation come up once in a while where two of the better players have to play each other fairly early in the bracket or you might have the person who has to face the top player in the first round go “Ah man! Do I have to play him in the first round?!!”   But this leads to a question how much does playing the top player early affect the outcome of the tournament?  So to begin to look at this I set up a tournament bracket with two unique players which will be the top players and the remaining slots were filled by “field” players.   This is the simplest “seeding” scenario you can have.   Player 1 and Player 2 were set to have a 90% chance to beat any of the field players while Player 1 had a 40% chance to win over Player 2 head to head.  So basically in this scenario Players 1 and 2 are well above the rest of the players in terms of skill, but Player 1 has a slight disadvantage against Player 2.   In this scenario the focus of the study is how the chances of winning the tournament varies with how far apart the two players are positioned.


Fig 1.  Illustration of how distance is defined. Since the first round the two players can fight each other in winners bracket is in round 2, this would be defined as a starting distance of 2.  In a 32 man bracket 5 is the maximum distance the two players can be separated by.
The starting distance is defined as the round the two players will meet in winner's bracket assuming both players win all of the matches up to this point.  This provides a nice clean way to turn 32 possible starting positions into simply 5.

Results:

When I first started these calculations I had a couple initial ideas what I might expect, but when I actually performed the calculations I found some very interesting results.  


Fig 2.  The probability plots for Player 1 and Player 2's chance of winning the 32 man bracket as a function of the starting distance. 


This first result I noticed was the overall win rate of Player 1 in general shoots down heavily just by having Player 2 in the Bracket.  To compare Player 1’s chance of winning a tournament with a 90% win rate against the field in the absence of a tough matchup is between 70% and 93%, but the addition of a single disadvantaged match up was enough to drop the overall win rate well below 30%.  In this case the match up isn’t even that horrible, but because of the way the double elimination format works player 1 will have to beat player 2 twice and more than likely one of those times will be in grand finals. 

While the first result was interesting (but not completely shocking) the most interesting result was tied to the seeding.  The graph is shown in terms of the first round the two players could meet in assuming both players won all their matches in winners prior to that.   So for instance if Player 1 was in the 1st slot and Player 2 was in the 3rd slot, the first round the two could possibly meet in is the 2nd round.   The graph shows two very interesting trends.   For player 2 intuitively his chances of winning the tournament progressively go up as the two players are placed further and further apart, but for Player 1 it is the exact opposite!  Player 1 has the highest chance of winning the tournament by facing Player 2 early in the tournament.  Player 1’s worst chance is if he is on the opposite side of the bracket from Player 2.  

So how can we explain these results? Well it’s actually not that difficult.  The best chance for Player 1 to win the tournament is to knock Player 2 into losers early so he has to run the whole losers bracket which gives Player 2 the highest chance of being knocked out of the tournament.   If the two players are on opposite sides of the bracket so that they can only meet in winners finals or in the losers bracket, this causes Player 1’s chances to drop more because now if he meets Player 2 in winner’s bracket Player 1 will have to play a 3 out of 5 set instead of a 2 out of 3 which extends Player 2’s advantage.  

Now the thing to note here; however, is that advantage is very small.   The best chance of winning and worst chance in a 32 man bracket only differ by about 2% with the largest drop being from the winner’s finals drop. In terms of who wins the tournament seeding makes a very small difference for player 1.  For player 2 the difference increases by about 2.3% from worst seeding to best seeding. Overall seeding has an effect on who wins the tournament, but the effect is very minimal.   

The seeding makes very little difference toward who will win the tournament.  So the question becomes   “Does seeding make any difference at all?” Well in, it actually does.  Because in a standard 32 man tournament the top 3 positions are usually paid out, so while Player 1 has the best chance of winning it all by playing Player 2 early does he have the best chance of actually getting top 3 consistently by playing player 2 early? 


Fig 3.  The chance of Player 1 placing in the Top X or higher.  For the X-axis a value of X=4,means Top4 or higher. 

Table 1.  Tabulated probabilities of getting TopX or higher for all starting distances.

Well the answer to this is that playing Player 2 early hurts Players 1’s overall chances of getting into the money positions.  The same effect that helps Player 1 “win it all” is the same effect that lowers his chances of placing in the Top spots.  Losing to Player 2 early forces you to run through all of loser’s bracket which increases your chances of getting “scrubbed out”.   As a result player 1 won’t place in 2nd as consistently as when the bracket has the players seeded as close as possible.  Player 1’s chance of placing in the money goes down from 98% to 83% from the furthest possible seeding to the closest.   This is also demonstrated by the chance of player one entering grand finals from the winner’s side of the bracket. 


Table 2.  Player 1 and Player 2's chance of making it to grand finals as a function of starting distance. Listed as the chances of making it from Winner's Bracket, Loser's Bracket, and the sum of those two giving the total chance of making it.

 Now of course we also need to take into account the fact that you get far more money for 1st than you do for 2nd or 3rd so we also want to take a look at the average and not just the overall placing percentage. Fortunately this is actually very easy to estimate.  Let's assume we have a normal 70/20/10 tournament split which is standard for a lot of tournaments. For a 32 man/$5 entry tournament the total pot will be $160. As shown by the graph, seeding player 1 further away from player 2 will result in a much more consistent amount of cash even when you consider player 1 has a higher chance of getting 1st by playing Player 2 sooner. And actually when you adjust for the average income you realize that once again the best position for Player 1 is either opposite side of the bracket from Player 2 or where the two players will meet in the round right before winner’s finals.   While the latter position has a slightly higher average, the opposite position has the most consistent chance of getting some money even if it means less of a chance of getting 1st.

Fig 4. On the left Player 1's average expected winning based on starting position and on the right Player 1's chance of finishing in any of the three money positions.


Conclusion:

The first message to take away from this was that the overall win rate dropped significantly for Player 1 by simply introducing a player who has a small advantage over Player 1.  This once again shows that the double elimination bracket coupled with 2 out of 3 and 3 out of 5 sets tends to exaggerate advantages players have over each other.  In this case you can see that a 6-4 advantage is enough to shift the chance of winning a tournament into a 75%-24% split between Player 1 and Player 2. (Note: 75% and 24% don’t add up to 100% because the field consumes a small percentage).  It goes to show that a single disadvantaged match up in the tournament can bite you big time if you are forced to play it. 

The results from this simulation paint a very interesting picture.  On one hand by seeding the two players as far away as possible both players have a much higher chance of placing either 1st or 2nd and it also ensures that the better player will have the best chance at winning.   If you are the underdog you will get paid more consistently by being seeded as far away as possible, but you also have the worst chance of getting 1st place by being seeded that far.    If your goal is a consistent placing then being seeded far away is what you want, but if your goal is to get first place (say to win a large prize that goes with it) you actually want to face the top player before you get to winner’s finals where you have a better chance of winning a 2 out of 3 set instead of a 3 out of 5 in addition to the other player having to play more loser’s bracket matches.  In terms of funding using a standard 70/20/10 tournament split, the highest average is also to play the top player right before winner’s finals.    The top player overall benefits the most from seeding as his odds of winning and odds of placing high only go up with being put further and further apart. 

In my next article I will add some more details to this study such as bracket size effects,  how things change upon the addition of a third player, etc.  I plan on getting this next one out much faster. Fortunately the bulk of the coding is done so all I need to do is small tweeks here and there.