# Tournament Simulation

The McMahon tournament system is very widely used in Amateur Go. Although the overall principles of the system are well established, there are a number features which are left undefined and have been implemented in a variety of ways - see the detailed BGA definition ^{[1]} for example.

**Size of the bar**- There are recommendations by the EGF, BGA, and AGA and these are mostly expressed in terms of a relationship between number of rounds and number of players above the bar. The existing recommendations can lead to situations where the range of player strengths above the bar is too wide - and only a small proportion of the bar population actually has a realistic chance of winning the tournament.

- Is there an optimal size of bar which takes into account the number of players, the number of rounds and the distribution of player strengths? I have presented results in the very technical document
^{[4]}. Please see the discussion page for a summary.

**McMahon group pairing**- The McMahon principles do not lay down how players within the same McMahon group should be paired, and there are a number of different ways of doing this, depending on the perceived goals of the tournament. Second to the aim of obtaining a unique winner, obtaining a correct ranking of the remaining players must be quite a high priority. Random pairing can lead to some upsets and interesting results. Seeded methods are aimed at ensuring that the strongest players don't suffer upsets and only meet each other in the later rounds.

- Are these methods equivalent in the sense of producing similar rankings at the end of the tournament, or similar player ratings in the long term?

**Tie-Breaks**- These are used to select a unique winner when McMahon score alone does not discriminate. The seeding methods underlying the tie-breaks are also used for pairing within a McMahon group and selecting players to be paired across groups where one of them is not even.

- Is there an optimal seeding which satisfies all three functions, or do we need different seeds for each?

**Comparison of pairing algorithms**- Each of the areas mentioned above may affect the long term ranking of players by providing subtle biases which are not felt in any one tournament, but do accumulate over time. Taking a black-box view of a tournament, and assuming that players enter at a correct grade, there is an ordering by grade of the players before the tournament. At the end of the tournament there is a final ordering by McMahon score and associated tie-breakers, and assuming that the pairing methods are fair we can expect that the final ordering correlates well with the initial ordering. We can call this the ranking quality of the pairing algorithm.

- Now modern pairing methods rely on applying some form of optimising engine to determine the pairing, and that engine is constructed round a set of fixed parameters. Each different set of parameters can be considered to form a different pairing algorithm and this leads to the comparison question.

- Is it possible to tune the pairing parameters to optimise the ranking quality?

**Probability of Win**- A good model for probability of win in even and handicap games is needed to support work in tournament simulation. Even game distributions are reported in
^{[3]}, with a current updated plot shown in^{[6]}. A start on handicap distributions is reported in^{[7]}, and an outline can be found in the discussion page.

Obviously we cannot get to grips with these questions by trial and error in actual live tournaments, and will need to pursue the matter through extensive tournament simulation. Thanks to the efforts of Ales Cieply in setting up the European rating system, and then through Aldo Podavini in providing a very functional access to the data in EGD, we now have a truly valuable resource which will enable us to track down many of the issues associated with Go tournaments.

In order to start the process of realistic tournament simulation, I have constructed software which can analyse data in the European Go Database to produce models for the relevant distributions. The results of this analysis are presented in an article "Tournament statistics and simulation"^{[2]}

I am making the software mentioned in the article available to fellow simulators, and if you have any problems accessing it please just email geoff. If you have any other issues with Tournament methods that I have not mentioned, I will be pleased to hear of them. They may influence the next phase, which is the construction of the tournament simulator now underway.

The tournament simulator relies heavily on Monte Carlo simulation methods. It is useful to have special cases where the exact probability distributions are known, so that the accuracy of the simulation can be controlled. These have now been developed in "Exact McMahon Score Distribution" ^{[5]} for very small tournaments of 2 rounds. There is a summary in the discussion page.

^{[1]} http://www.britgo.org/organisers/mcmahonpairing.html

^{[2]} http://www.kaniuk.co.uk/articles/tourstats/tourstats.pdf

^{[3]} http://www.kaniuk.co.uk/articles/pwin/prob-win.pdf

^{[4]} http://www.kaniuk.co.uk/articles/pairing/mcmahon-bar.pdf

^{[5]} http://www.kaniuk.co.uk/articles/pairing/emcmon.pdf

^{[6]} http://www.kaniuk.co.uk/articles/pwin/pwin-even-raw-2014-2018.eps

^{[7]} http://www.kaniuk.co.uk/articles/pwin/hcap-pwin-local.pdf

Geoff Kaniuk