## Annex 5 Appendix D (TPR) Tournament performance rating

### Annex 5 Appendix D (TPR) Tournament performance rating

From annex 5 Swiss system, Appendix D.Tournament performance rating we read:

The problem can be greatly simplified by replacing the S-shaped expectation function with a linear relationship between the rating difference (D) and the score percentage (P):

This performance rating can be computed by solving a set of lineair equations.

When computed, it is easy to verify manually.

The rating is equivalent to the recursive Buchholz tie-break.

The above could be implemented as a first step.

Introduction

The tournament performance rating may be used as tie-break criterion between players with the same score, or even as first criterion more important than the final score, to decide about the final result.The easy way to calculate the tournament performance rating is by using the average rating of the opponents but given the theory of probability behind the ELO rating system it is mathematically not fully correct way to do this. The performance belonging to an average rating may be something a bit different from the average performance, especially when there is a large variation in the ratings of the opponents.The tournament performance rating should be calculated on a game by game basis.

The TPR could replace many tie-break criteria. The chance that after a few rounds the TPR for two players will be equal is negligible. However, the TPR is not yet available in, for example, Draughts Arbiter Pro. I think there are good reasons for that.Definition

The tournament performance rating is that rating for which the expected result of the player, calculated opponent by opponent,equals the realized result of the player.

- The tournament performance ratings are unique up to a factor iff the game results at hand are connected and indivisible. That is, there must always be a non-losing results path between any two players. This is not the case if one player has won or lost all his games. This is easy to see; but it can also be much more complicated as in this chess tournament: iv memorial-lobaziewicza 2007. To calculate the TPR, the underlying structure of the strongly connected components must be determined.
- Within a strongly connected component the TPR constitutes the roots of the function We(x) - W, where x is a vector of ratings, We(x) is the expected score given rating vector x, and W is the actual score.

Finding the TPR comes down to solving a system of non-linear equations as noted byBert Zwart wrote: ↑Mon Aug 06, 2007 14:44Wouter,

Heb je aangetoond dat TPR* uniek is? Het lijkt me een vast punt van een niet-lineair stelsel vergelijkingen...

Het is natuurlijk duidelijk dat TPR* niet uniek is, omdat in ratings alleen verschillen van belang zijn. Dus laten we zeggen uniek op een constante na.

**---> In short: determining the "real" TPR is not straight forward.**The problem can be greatly simplified by replacing the S-shaped expectation function with a linear relationship between the rating difference (D) and the score percentage (P):

- P(D) = D / 800 + 50%

This performance rating can be computed by solving a set of lineair equations.

When computed, it is easy to verify manually.

The rating is equivalent to the recursive Buchholz tie-break.

The above could be implemented as a first step.