## 3 Methods Applied
### Thurstone Method with Ties  
The Thurstone method [23] is a stochastic method in the sense that it supposes that the performances of players (in this case, teams) are random variables. This assumption can be reflective of reality. If we keep a close watch on matches, we can also observe some fluctuations in the performances of the individuals and teams. The strengths of the teams are the expectations for the random variables. These random variables are denoted by \(\xi_{i}\), \(i=1,2,3,\ldots,n\), where \(n\) is the number of objects (teams) to evaluate, and the expectations are denoted by \(m_{i}\), \(i=1,2,3,\ldots,n\), respectively. The actual result of a match depends on the actual values of the random variables, or, more precisely, on their differences. Since in football there are three possible results, win, tie, and defeat, we also allowed three options for the result of a comparison: better (win), draw (tie), and worse (defeat). The differences in performances are \(\xi_{i}-\xi_{j}=m_{i}-m_{j}+\eta_{i,j}\), where \(\eta_{i,j}\) are supposed to be independent, identically distributed random variables with the cumulative distribution function \(F\). If \(F\) is the standard normal (Gauss) cumulative distribution function (\(F=\Phi\)), then the model is the generalization of the Thurstone model. If \(F\) is the logistic cumulative distribution function, then the Bradley-Terry model with ties is used. In this paper, we used Gauss distribution. If the result of a match between teams \(i\) and \(j\) is a tie, then the difference \(\xi_{i}-\xi_{j}\) is close to zero. The bounds of the tie are represented by \(-d\) and \(d\), where \(d\) is a positive parameter estimated from the data. If the result of the match is a win, then the difference is above the parameter \(d\). If team \(i\) suffers a defeat to team \(j\), the difference is below \(-d\) (see Figure 1). Of course, if \(i\) beats \(j\), then \(j\) suffers a defeat to \(i\). The probabilities of the defeat/tie/win results for the matches between teams \(i\) and \(j\) can be written as follows:  
\[p_{i,j,1}=P(\text{team $i$ is defeated by team $j$})=P(\xi_{i}-\xi_{j}\in I_{1})=\Phi(-d-(m_{i}-m_{j})) \tag{1}\]  
\[p_{i,j,2}=P(\text{the result is a tie})=P(\xi_{i}-\xi_{j}\in I_{2})=\Phi(d-(m_{i }-m_{j}))-\Phi(-d-(m_{i}-m_{j})) \tag{2}\]  
\[p_{i,j,3}=P(\text{team $i$ wins over team $j$})=P(\xi_{i}-\xi_{j}\in I_{3})=1- \Phi(d-(m_{i}-m_{j})). \tag{3}\]  
Figure 1: The possible results and the intervals belonging to them. Let \(\mathbf{A}\) be a three-dimensional data matrix with size \(n\times n\times 3\) and with elements \(A_{i,j,k}\)\(i=1,2,\ldots,n\), \(j=1,2,\ldots,n\), \(k=1,2,3\). \(A_{i,j,k}\) is the number of matches when team \(i\) has result \(k\) against team \(j,i<j\); \(k=1\) stands for a defeat, \(k=2\) for a draw, and \(k=3\) for a win. If \(i\geq j\), then let \(A_{i,j,k}=0\) for all \(k=1,2,3\). The probability of the results provided by data matrix \(\mathbf{A}\) in the function \(\underline{m}=(m_{1},m_{2},\ldots,m_{n})\) and \(0<d\), supposing the independence of the sample elements, is  
\[L(\mathbf{A}|\underline{m},d)=\prod_{k=1}^{3}\prod_{i=1}^{n-1}\prod_{j=i+1}^{ n}p_{i,j,k}^{A_{i,j,k}}. \tag{4}\]  
where \(L\) is the likelihood function. The maximum likelihood estimation of parameters \(\underline{m}\) and \(0<d\), denoted by \(\underline{\hat{m}}\) and \(\hat{d}\), is the \(n+1\) dimensional argument, where the function \(L\) reaches its maximal value [49]. For Gauss distribution and for the case of more than two options, sufficient conditions for the existence and uniqueness of the maximal value of (4) are provided in [27], which was generalized in [28]. In the following, we formulate the statement for the three options. Let us define the graph \(G_{TH}\) as follows: let the vertices be the teams and let the nodes \(i\) and \(j\) (\(i<j\)) be connected if \(0<A_{i,j,2}\) or \(0<A_{i,j,1}\cdot A_{i,j,3}\) (i.e., teams \(i\) and \(j\) have either drawn at least once, or both teams have beaten each other at least once).