Now we can reasonably simulate games between randomly seeded players in this tournament. Running one simulation doesn’t help much, but it is quite cheap and fast to run many thousands of simulations. Then, we can look at all the players that have lost 12-10 in the first round and see where they fall in the score rankings. For example, we randomly assigned skills to 16 ‘players’, simulated a tournament, and only one player lost 12-10 in the first round. When we compare that player to all other players in the simulation, we can see exactly where they rank. If we do this many thousands of times, then we can see a distribution of where a 12-10 first round loser ranks:
How did Evan rank?
Rank of the First Round 12-10 Loser
This is interesting, but it tells us mostly what our intuition already likely had: we can’t be very confident about where I rank. But as luck would have it, my wife and I ran into the gentleman that beat me later in the cruise, and it turns out he won the entire tournament! Not only that, but his score in the championship match was 12-6! This is some very useful new information, and using the same assumptions and simulation technique, we can know where I likely ranked out of all 16 competitors. My son still thought I was the worst, but my (maybe a little biased) intuition was that I was confidently the 2nd best player in the tournament. I must be, right!? I lost to the champ by a closer margin than the person who lost in the finals. Let’s run the simulation and see the likely ranking of someone who lost 12-10 to an eventual champion, given the champion won 12-6 in the finals. Since we’re collecting the scores of each game in the simulation, we can simulate many tournaments and only use the ones that have this exact case for our inference.
How did Evan rank?
Rank of the first round loser
These results were quite surprising to me! While it was certainly possible that I ranked 2nd, it was more likely that I ranked 3rd (since the highest bar is the third, that was the most common result of all the relevant simulations). It is also very plausible that I ranked anywhere in the top 10, and to my son’s great excitement, not completely impossible that I ranked dead last.
I was happy to point out that it was quite possible that I ranked 1st, even though I lost in the first round. Being more skilled than an opponent simply means that they you are more likely to win each individual point; a less skilled player can certainly get lucky and win enough points to take the game, especially if the skill level is close.