# Measuring the Non-Transitivity in Chess
## 3 Theoretical Formulations and Methods
### Non-Transitivity Quantification
#### 3.3.1 Payoff Matrix Construction
Discretisation was conducted by binning the Elo ratings such that, given two bins \(a\) and \(b\), any two-way match-up where one player's rating falls into \(a\) and the other player's rating falls into \(b\) was treated as a two-way match-up between \(a\) and \(b\). The resulting payoff matrix of an NF game is, thus, the outcomes of all two-way match-ups between every possible pair of bins. As two-way match-ups were considered, the resulting payoff matrices were skew-symmetric. However, when searching for the corresponding two-way match-up between two bins of rating in the data set, there could either be multiple games or no games corresponding to this match-up. Consider a match-up between one bin \(a=[a_{1},a_{2}]\) and another bin \(b=[b_{1},b_{2}]\). To fill in the corresponding entry of the matrix, a representative score for each case is first calculated. Let \(rs_{a,b}\) denote the representative score for the match-up where the white player's rating is within \(a\) and the black player's rating is within \(b\), and \(rs_{b,a}\) denote the other direction. For the sake of argument, consider finding \(rs_{a,b}\). As games are found for bin \(a\) against \(b\), scores must be expressed from the perspective of \(a\). The convention used in this paper is \(1\), \(0\), and \(-1\) for a win, draw, and loss, respectively. This convention ensures that the entries of the resulting payoff matrix are skew-symmetric. In the case where there are multiple games where the white player's rating falls in \(a\) and black player's rating falls in \(b\), the representative score is then taken to be the average score from all these games. On the other hand, if there are no such games, the representative score is taken to be the expected score predicted from the Elo rating; that is,  
\[\mathbb{E}_{s}[a>b]=2p(a>b)-1, \tag{3}\]  
where \(\mathbb{E}_{s}[a>b]\) is the expected score of bin \(a\) against bin \(b\), and \(p(a>b)\) is computed using Equation (2), by setting \(k\) as \(\frac{\ln(10)}{400}\), \(r_{a}=\frac{a_{1}+a_{2}}{2}\), and \(r_{b}=\frac{b_{1}+b_{2}}{2}\). As this score is predicted using the Elo rating, it is indifferent towards whether the player is playing black or white pieces. After computing the representative score for both cases, the corresponding entry in the payoff matrix for the two-way match-up between bin \(a\) and \(b\) is simply the average of the representative score of both cases (i.e., \(\frac{rs_{a,b}+rs_{b,a}}{2}\)). The payoff matrix construction procedure is summarised in Algorithm 2. ```
Input: \(a\), \(b\), \(b\), \(a\), \(b\), \(b\), \(



#### 3.3.2 Nash Clustering

Having constructed the payoff matrix in Section 3.3.1, we can now proceed with non-transitivity quantification. The first method of quantifying non-transitivity is through Nash Clustering [5]. Nash Clustering is derived from the layered game geometry, which states that the strategy space of a game can be clustered into an ordered list of layers such that any strategy in preceding layers is strictly better than any strategy in the succeeding layers. A game where the strategy space can be clustered in this way is known as a layered finite game of skill. **Definition 5**. : _A two-player zero-sum symmetric game is defined as a k-layered finite game of skill if the strategy space of each player \(S_{g}\) can be factored into k layers \(L_{i}\)\(\forall i\in[k]\), where \(\bigcup_{i\in[k]}L_{i}=S_{g}\), \(L_{i}\cap L_{j}=\emptyset\)\(\forall i\
eq j\), \(i,j\in[k]\), such that \(M_{1}(s_{1},s_{2})>0\)\(\forall s_{1}\in L_{i}\), \(s_{2}\in L_{j}\), \(\forall i,j\in[k]\), \(i<j\) and \(\exists z\in\mathbb{N}_{1}\) satisfying \(\forall i<z\), \(|L_{i}|\leq|L_{i+1}|\), \(\forall i\geq z\), \(|L_{i}|\geq|L_{i+1}|\)._

A consequence of Definition 5 is that strategies in preceding layers would have a higher win-rate than those in succeeding layers.