Previously I looked at the performance of simple strategies in a simplified version of the WTC Pairing Game. The game is played by teams A and B. Team B reveals one player initially. The other team reveals two players, and team B selects a pairing. Each team then takes it in turns to reveal an additional player and select a pairing. It is widely considered an advantage to be team A since their final choice also determines the fifth pairing. However, I was not able to find any broadly applicable effect in favour of team A.

While the pairing game information was not collected for WTC 2015, the Enter the Crucible stream included footage of the pairing. So while this has not yet allowed me to identify strategies that the 12 captains were using, I did spend some time looking for patterns. One fruitful line of investigation was a checkout of what pairings were possible. Five players in two teams can result in 120 pairing combinations. Since I have estimates of player ratings and Mark 2 caster penalties, I was able to estimate the likely outcome for each player permutation. Given that each captain only has themselves and their four players to put into matchups, we should not assess them on probability of victory, but **quantile** of probability of victory. Were captains able to gain matchups that was better than 50% of the possible player combinations?

The following graphics display two types of information. Team A are listed on the right hand side of the plot. Team B are listed along the bottom. The *x*-axis consists of 120 small panels corresponding to each possible combination of the 10 players. The 5 rows of bars correspond to team A. The colours from purple to light green correspond to team B. For each permutation there is an estimated probability of outcome from 0 (team A certain to lose) to 1 (team A certain to win). The fat black line is the median estimated probability of the outcome. The thin black lines are the 10th and 90th percentiles of the estimated probability. These probabilities were calculated by simulation from the player ratings for random list selections for each player. Since the estimates are the average of the best and worst list combinations, list rating will have the biggest effect when there is an advantage between two players over both of their opponent’s lists. The permutations were then sorted from lowest to highest probability of team A winning. We can visually look for patterns in these plots that indicate pairing that would more likely lead to success for a team of interest.

In Round 1, Australia Platypus was playing Ireland Craic as favourites. Team B (Ireland Craic) will do best when Philip Johnston is paired into Dyland Simmer or Jeff Galea. Team A (Australia Platypus) will do best when Dyland or Jeff are paired into Dan or Mike Porter.

In Round 2, Belgium Blonde was playing England Lions as underdogs. Team A (Belgium Blonde) need to get Laurens Tanguy into Paul North. Team B (England Lions) need to get Paul into Dirk Pintjens.

In Round 3, USA Stripes was playing Germany Dichter & Denker. Team A (USA Stripes) need to get Jay Larsen into Robin Maukisch. Team B (Germany Dichter & Denker) need to get Sascha Maisel into Jeremy Lee.

In Round 4, Australia Wombat was playing Canada Goose as favourites. Team A (Australia Wombat) should be aiming to get Chandler Davidson into Aaron Thompson or Ben Leeper. Team B (Canada Goose) should be aiming to get Charles Soong into Ben Leeper.

In Round 5, Australia Platypus was playing Finland Blue. Team A (Australia Platypus) should be aiming to get Jeff Galea into Jaakko Uusitupa and Sheldon Pace into Henry Hyttinen. Team B (Finland Blue) should be aiming to get Jaakko into David Potts.

In Round 6, Australia Wombat was playing Finland Blue. Team A (Australia Wombat) should be aiming to get Joshua Bates into Jaakko and James Moorhouse into Pauli Lehtoranta or Tatu Purhonen. Team B (Finland Blue) want to get Henry into Joshua.

Given these estimates, how did the captains perform? The following plot shows the 10th, 50th and 90th percentiles of the median estimated probability of team A winning. The purple blobs show the quantile of the permutation selected at the event. If the blob is to the right of the median (50th percentile), team A performed better than the opposing captain. If the blob is to the right of the median, team B outplayed team A in the pairing game. For rounds 1, 3 and 6, the pairings are consistent with equally skilled captains or no advantage for team A. For rounds 2, 4 and 5, the captains for team B outplayed the captains for team A. This information suggests that for these six pairing games there is no evidence that team A is advantaged.

All of this analysis is based on crude estimates of caster rating and player rating. Many assumptions are made, and the subset of the data presented here is very small. However, I consider that these results are additional evidence that it is not advantageous to be team A. It is my intuitive belief that although it appears that team A are choosing more matchups, in fact, the choice is limited in its degrees of freedom. While team A may on occasion be able to choose the optimum combination of two pairings, they may have already sacrificed the ability to have good matchups in the other three selections. Team B are in the position of selecting first, and make two selections before team A make their final selection. The additional gain imposed by the rules of the WTC packet of being able to pick tables makes me believe that there is likely a real advantage to being team B.

Am I misreading the description/analysis of the final chart?

It looks to me like in 1, 4 and 6 are indicative of the captains matching one another (not 1, 3 and 6), and also that in 2, 3 and 5 team A’s captain won, not team B’s captain.

Hmm, I think I have not structured this graphic particularly well. Rounds are increasing as you go up the y-axis. So round 1 is the bottom line (Australia Platypus vs Ireland Craic). This is confusing because the plot looks a bit like a table where you’d expect round 1 to be at the top.

And unfortunately the x-axis is also confusing! The three values are all quantiles of team A winning. If the blob is to the left, the probability of team A winning is decreased, so team A lost. If the blob is to the right, the probability of team B winning is decreased, so team A won. This means that although I’ve helpfully labelled the rounds, the plot looks like a tug-of-war, but the outcome is actually the reverse of that!

Ah. So in round 4 (Wombat vs Goose), B out maneuvered A to reduce the odds of A winning from 80% to 78%

Got it.

Yes exactly! The magnitude of the pairing effect estimated here is a convolution of the optimum combination of players for an overall round victory and the casters that each player chose to take. While caster effects do take into account the Mark 2 individual caster pairings, I needed to restrict them to only be allowed to grow by a certain amount based on the number of games observed. If list selection is more important, than the true effects may be more different.