A game-theoretic model for the classification of selected oil companies’ price changes

The probit variable game

The current approach assumes that among $\beta$ values useful for classification (Eq. (3)), we can find some that better optimize individual $\mathrm{\Phi }(x_i\beta )$, providing a better trade-off in the maximization of their sum in $logℒ$. A normal form game $\mathrm{\Gamma }$ is designed, in which players are the variable/attributes $X_j$ in X that choose their ${\beta }_j$ parameters to maximize their marginal contribution to the log-likelihood function, subject to satisfying conditions in Eq. (5).

Formally $\mathrm{\Gamma }=(A,B,U)$ is defined as:

• the set of players $A=\{1,\dots ,d\}$: a player $j\in A$ represents an attribute $X_j$ in X;

• the set of strategy profiles $B\in \mathbb{R}$^d, an element $\beta \in B$ is $\beta =({\beta }_1,{\beta }_2,\dots ,{\beta }_d)$, where ${\beta }_j$ is the strategy of player $j\in A$;

• the payoff function $U=(u_1,\dots ,u_d)$, with $u_j:B\to \mathbb{R}$:

(4) $$u_j(\beta )=\{\begin{array}{cc}\log \log ℒ(X,Y;\beta )-ℒ(X_{-j},Y;{\beta }_{-j})& r(\beta ;X,Y)-r({\beta }^{\ast };X,Y)\ge 0,\\ \log ℒ(X,Y;\beta )& r(\beta ;X,Y)-r({\beta }^{\ast };X,Y)<0\end{array}$$where $X_{-j}$ represents X with attribute $X_j$ removed, and ${\beta }_{-j}$ represents $\beta$ without ${\beta }_j$ and ${\beta }^{\ast }=argma{x}_{\beta }ℒ(X,Y;\beta )$. Function $r(\beta ;X,Y)$ is

(5) $$r(\beta ;X,Y)=|\{i\in \{1,\dots ,N\}|x_i\beta (2y_i-1)\ge 0\}|$$where $|\cdot |$ denotes the cardinality of a set.

The payoff function is designed to maximize the marginal contribution of attribute $j$ to the log-likelihood function if the parameter $\beta$ evaluated improves upon the probit estimated parameter ${\beta }^{\ast }$ (restriction Eq. (5)), and if not, is taken as the log-likelihood function in order to be maximized. The marginal contribution is computed as the difference between $\log ℒ(X,Y;\beta )$, the value of the log-likelihood function, and $\log ℒ(X_{-j},Y;{\beta }_{-j})$, the log-likelihood function computed without taking into account attribute $X_j$ (and consequently ${\beta }_j$) in the data. By maximizing their payoffs, either the log-likelihood will be maximized, or if a better solution can be found, it will be computed based on the marginal contribution to the log-likelihood function.

Restrictions $r(\beta ;X,Y)$ If we consider conditions in Eq. (3), these are equivalent with having $X\beta >0$ whenever $Y=1$ and correspondingly $X\beta <0$ whenever $Y=0$. This condition can be re-written as

(6) $$x_i\beta (2y_i-1)\ge 0,i=1,\dots N$$

If, for a $\beta$, this condition holds, then also Eq. (3) holds, and, even if $\beta$ does not maximize the log-likelihood function, it provides a good classification of the data. But condition (6) may not hold for all $i$ even for ${\beta }^{\ast }$, therefore searching for a $\beta$ value such that (6) holds may be useless. However, there might exist $\beta$ values such that the number of instances for which the condition Eq. (6) holds is greater than or equal to the corresponding number computed for ${\beta }^{\ast }$. Among these values, we may find better classification models that are based on probit. Because we are only interested in such values, for all other situations, all players seek to maximize the log-likelihood function in Eq. (4).

Nash equilibrium (NE) The Nash equilibrium of a game (Maschler, Solan & Zamir, 2013) is a strategy profile such that no player has an incentive for unilateral deviation. This means that while all other players maintain their strategies unchanged, none of the players can improve their payoff by only changing their strategies. For game $\mathrm{\Gamma }$, the Nash equilibrium represents a $\beta$ value such that each variable $X_j$ cannot contribute more to the log-likelihood function if all other variables maintain their choices. Finding a NE for game $\mathrm{\Gamma }$ is not trivial. In this approach, we use a differential evolution algorithm (Bilal et al., 2020), adapted to approximate the Nash equilibria for game $\mathrm{\Gamma }$ in the following manner.

Probit equilibrium differential evolution (PrEDE)

Differential evolution (DE) is a stochastic search and optimization method that evolves a population of potential solutions to the problem (Storn & Price, 1997). It can be adapted to compute the Nash equilibria of a game by using the Nash ascendancy relation (Lung & Dumitrescu, 2008) during the selection phase of the search. We further adapt the DE to approximate the NE of game $\mathrm{\Gamma }$, and we call this DE version Probit Equilibrium Differential Evolution (PrEDE). The outline of PrEDE is presented in Algorithm 3.

Population and initialization PrEDE population consists of individuals $\beta \in \mathbb{R}$^d that are possible parameters for the probit model and strategy profiles for game $\mathrm{\Gamma }$. Because we are searching in a neighborhood of the probit parameters, the initial population is generated starting from ${\beta }^{\ast }$ by adding deviations following a normal distribution with standard deviation $\sigma$. Parameter $\sigma$ controls how much the initial population is spread in the search space around ${\beta }^{\ast }$. A very small value would lead to premature convergence to ${\beta }^{\ast }$, while a higher value would slow the search as the initial solutions may need a lot of improvement before satisfying restrictions in the payoff functions.

Variation operators A DE/rand/1/exp scheme, presented in Algorithm 1, is used to create offspring (Thomsen, 2004). With probability CR, some of the components of the offspring are modified based on the values of three distinct parents from the current population by adding the difference of two of them multiplied by a scaling factor F to the third. This is a standard DE scheme that has been proven efficient in optimization problems.

Nash ascendancy In order to guide the search towards the equilibrium of game $\mathrm{\Gamma }$ the Nash ascendancy relation is used (Lung & Dumitrescu, 2008). Two strategy profiles of the game—here represented by the offspring $o$ and parent $\beta$—are compared by counting how many players can improve their payoffs by unilaterally changing their strategy from one to the other. The strategy profile having less number of such players is considered to be better with respect to NE than the other. If the number of players that can unilaterally improve their payoffs is the same, they are considered indifferent (Algorithm 2). In the context of game $\mathrm{\Gamma }$ an extra step is added to take into account conditions in the payoff functions (Eqs. (4) and (6)). Thus, the Nash ascendancy relation is tested only if both individuals fulfill conditions in Eq. (5), otherwise, if only one of them fulfills them, it will be considered better, and if none of them fulfills them, the one having a better probit likelihood value (Eq. 2) is considered better.

Fitness function While the Nash ascendancy relation is used for evolution purposes, in order to identify the best individual in the population, a specific classification-based fitness is used: the Area under the Curve (AUC) indicator (Fawcett, 2006) computed based on the prediction made using the cumulative normal distribution function for individuals in the population-based on training data. The AUC metric indicates the probability that a classifier will rank a positive instance higher than a negative instance. A maximum value of 1 indicates a correct classification of the tested data. A higher value indicates a better classification from the positive label point of view.

Termination condition PrEDE repeats iterations until a maximum number MaxGen of iterations is reached, or if the best AUC in the population computed on the training data supersedes the AUC of the probit estimator ${\beta }^{\ast }$ for a successive number of $\eta$ iterations. The motivation behind stopping the search is double-fold: to reduce the computational complexity of a run and to avoid overfitting, as better AUC values on the training data may indicate both a better classification and the danger of overfitting. If a better solution is identified, and it is held for $\eta$ iterations, then it might represent a genuine improvement to probit, and it should be enough; continuing the search may indeed improve the AUC value but only for the training data, and even the probit log-likelihood function, but may not lead to an actual improvement in results.

PrEDE parameters PrEDE uses three types of parameters:

• parameters that are specific to any evolutionary algorithm: population size ( $popsize$), and the maximum number of iterations ( $MaxGen$);

• two parameters that are specific to the differential evolution algorithm: the scaling factor F, and crossover probability CR;

• two parameters specific to the classification problem: number of iterations the AUC fitness of the best individual exceeds the AUC of the probit estimator before the search is stopped $\eta$, and the standard deviation used to generate the initial population around probit parameters, $\sigma$.

Synthetic data and parameter testing

A set of synthetically generated data is used to preliminary assess the performance of PrEDE in various settings and provide an overview of the effect of parameter settings on results. The datasets are generated by using the make_classification function in the scikit-learn Python (https://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_classification.html, last accessed Feb. 2022). To simulate an environment that is similar to the one created by the oil price data, we generated datasets with 100 and 200 instances, with three and six attributes, and with different degrees of separation between classes by setting the class separator parameter to 0.1, 0.5 and 1. A higher value creates instances with better-separated classes. As a performance indicator, the AUC (Fawcett, 2006) is used. AUC takes values between 0 and 1, 1 indicates a correct classification, and it can be used to compare results. A total of 10-folds cross-validation is used and the average of the 10 AUC values reported on the tested folds are presented (Hastie, Tibshiran & Friedman, 2016; Stone, 1974). Figure 1 presents the tested values and corresponding AUC values.

The differential evolution parameters, F, and CR are set to take values from 0.1 to 1 with a step of 0.1, Fig. 1. We find, as expected, that there is no ‘gold’ setting for these parameters. However, smaller CR and F values seem to provide a good trade-off on datasets with smaller class separator values which are more difficult to solve.

The parameter $\eta$ is used to stop the search if for a successive number of $\eta$ iterations the fitness of the best individual in the population is better than the fitness of the probit parameter. This ensures that the solution provided by PrEDE is indeed better than that provided by probit while avoiding overfitting (Fig. 1). The value of $\sigma$ least influences results (Fig. 1), indicating that the choice of initial population setting does not influence the search.

Application: oil data

Data used in the study consists of the closing price for the crude oil WTI and the Brent oil collected by the authors from the Bloomberg database for almost two decades, starting from 2000 until 2019. From the same database and for the same period, the closing prices for three stocks: Exxon Mobil (XOM, Irving, TX, USA) from the US, OMV AG Austria (OMV, Vienna, Austria), and OMV Petrom SA Romania (SNP, Bucharest, Romania) were also collected. To better compare and avoid any distortions, all the data were converted into US dollars. We selected these three oil companies because they are from different geographical areas, and they are also different in size, but they have similar activities. Yet, what is more important, these companies have some close connections: OMV AG owns 51 percent of Petrom, while Petrom and Exxon have made together a joint company to prospect and exploit natural gas from the Black Sea. Other reasons for this selection were the fact that, on the one hand, the oil companies from the area of Central and Eastern Europe (in our case, OMV Petrom, Bucharest, Romania) were not researched enough yet and, on the other hand, Exxon Mobil, besides its links with OMV Petrom, is probably the most representative oil company in the world. Figure 2 illustrates the collected data used for the analysis.

One of the issues of interest when looking at oil price data is to predict if an increase/decrease in price is expected for these three stocks. In this approach, we test the following model: a binary variable taking values of 0 and 1 is created for the stock prices of each oil company, with value 1 representing an increase in the stock price from the previous day and 0 a price decrease. Figure 3 presents the overlapping degree of the two obtained classes for all pairs of considered data. Probit models changes in the oil price and PrEDE, and results are compared and discussed.

Ten intervals of 50 days were selected for training, and the subsequent 10 days were used in the testing phase. This is a reasonable setting as it is common to look for the trends of the last 50 days to make predictions related to the near future. AUC values were reported for each time slot. Various combinations of data were tested in this manner. Our focus was mainly on the Romanian oil company OMV Petrom (SNP, Bucharest, Romania); thus, we tested the changes in eight different combinations with Brent and Crude oil and with the other two oil companies, combinations that we considered that are the most useful ones, as we can see from in Fig. 4. We were primarily interested in SNP because it is the least researched of the three companies and has been less studied. In order to do this analysis, we tested SNP in combination with the oil prices and the other oil companies in pairs of two, three, or four data sets (Fig. 4). Firstly, we tested SNP stock price variation based on Brent oil, because this is the European benchmark, and together with OMV, because it is the majority shareholder of SNP, and then based on two more combinations with Brent. Secondly, we continued by testing SNP stock price variation with the crude oil, which is the US benchmark, in four ways, including Exxon, because it is a US company, then adding OMV, and then Brent. Thirdly, we tested SNP stock price variation based on the stock prices of the other two oil companies, OMV and XOM. In the end, mainly for confirmation purposes, we tested XOM and OMV stock price variation, in relation with each-others, with the two oil benchmarks, and with SNP.

Results are presented as error bars of AUC values in Fig. 5 and corresponding numerical values in Table 1. While values are comparable, we find the PrEDE reports better average AUC values than probit in eight out of the 12 scenarios tested. Figure 5 illustrate AUC values for each set of days tested. A paired t-test performed overall AUC values for each data set, and each starting date shows a significant difference between AUC values reported by PrEDE and probit ( $t=5.86$, and $p<0.0001$). For seven out of the 12 combinations tested, the differences between results are significant, while there is no situation in which probit results can be considered better. A detailed representation of differences is illustrated in Fig. 5. Considering the high overlapping degree of the data (Fig. 3), it is to be expected that average AUC values are around 0.5; however, the fact that PrEDE has been able to improve results reported by probit indicates the potential of exploring such an approach. Results reported by other standard classification methods are presented in Table 2. Logit (Seabold & Perktold, 2010), k-nearest neighbor (kNN), and a random forest (RF) (Pedregosa et al., 2011) were used on the same data, and the mean and standard deviation of AUC values are reported. Only in one instance (SNP $\leftrightarrow$ Brent, Crude) kNN and RF results were significantly better than those reported by PrEDE.