# Measuring the Non-Transitivity in Chess
## 3 Theoretical Formulations and Methods
### Non-Transitivity Quantification
#### 3.3.1 Payoff Matrix Construction
Finally, there are several avenues for possible improvements, which we leave for future work:

* In Section 3.3.1, missing data between match-ups of bins are filled using Elo formulas; that is, Equations (2) and (3). However, the Lichess ratings use the Glicko-2 system [20], which has a slightly different win probability formula that also considers the rating deviation and volatility of the rating. Therefore, a possible improvement would be to additionally collect the rating deviation and volatility of the players in every game, and apply the Glicko-2 predictive formula to compute the win probability, instead of Equation (2). * Non-transitivity analysis could also be carried out using other online chess platforms, such as Chess.com [30], in order to provide more comprehensive evidence on the spinning top geometry of the real-world strategy space. Furthermore, the non-transitivity analysis could also be carried out using real-world data on other, more complex, zero-sum games, such as Go, DOTA and StarCraft. * Other links between non-transitivity and human progression can be investigated. One possible method would be to monitor player rating progression from the start of their progress to when they achieve a sufficiently high rating in a chess platform, such as Lichess. Their rating progression can then be compared to the non-transitivity at that rating and in that year. However, player rating progression is typically not available on online chess platforms and, thus, would require monitoring player progress over an extended period of time. Conceptualization, R.S. ; methodology, R.S. ; software, R.S. ; validation, R.S. ; formal analysis, R.S. ; investigation, R.S. ; resources, R.S. ; data curation, R.S. ; writing--original draft preparation, R.S. ; writing--review and editing, R.S. and Y.Y. ; supervision, Y.Y. and J.W. ; project administration, Y.Y. and W.J. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable

The python codes to replicate the experiments in this paper are available at [https://anonymous.4 open.science/r/MSc-Thesis-8543](https://anonymous.4 open.science/r/MSc-Thesis-8543) (accessed on 13 March 2022). README.md contains all the instructions to run the codes. Finally, the data sets can be downloaded from [[https://database.lichess.org/](https://database.lichess.org/)]([https://database.lichess.org/](https://database.lichess.org/)) (accessed on 13 March 2022) for Lichess and [[https://www.ficsgames.org/](https://www.ficsgames.org/)]([https://www.ficsgames.org/](https://www.ficsgames.org/)) (accessed on 13 March 2022) for FICS. The authors declare no conflict of interest. ## Appendix C Theorem Proofs

**Theorem A1**. : _With Algorithm 1, the probability of any object in \(M\) being sampled into \(F\) is \(\frac{d}{m}\)._

Proof. : For any object in \(U\) to end up in \(F\), they must first get picked into \(G\). As \(H_{i}\cap H_{j}=\emptyset\)\(\forall i,j\in[k]\), \(i\
eq j\), the probability that any object is sampled into \(G\) is \(\frac{h-1}{h}\frac{C_{d-1}}{C_{d}}=\frac{d}{h}\). Then, given that they are in \(G\), they must get sampled into \(F\). The probability that this happens is \(\frac{d\times k-1}{4\times C_{d}}=\frac{1}{k}\). Hence, the probability that any object from \(U\) gets sampled into \(F\) is \(\frac{d}{h\times k}=\frac{d}{h}\). **Theorem A2**. : _Let \(C\) be the Nash Clustering of a two-player zero-sum symmetric game. Then, \(NPP(C_{i},C_{j})\geq 0\)\(\forall i\leq j\), \(C_{i},C_{j}\in C\) and \(NPP(C_{i},C_{j})\leq 0\)\(\forall i>j\), \(C_{i},C_{j}\in C\)._

Proof. : Let \(C\) be the Nash Clustering of a two-player zero-sum symmetric game with a payoff matrix of \(\mathbf{\mathcal{M}}\) and strategy space \(S_{g}\) for each of the two players. For any \(k\in[|C|]\), let \(\mathbf{p_{k}}\) be the NE solved to obtain the \(k^{th}\) cluster; that is, \(\mathbf{p_{k}}=\mathrm{Nash}(\mathbf{\mathcal{M}}|\cup_{i=k}^{|C|}C_{i})\), where \(\mathbf{p_{k}}\in\Delta_{|C|}\).