A.L. Voynich¹
PhD, Associate Professor А.I. Chikurov¹
Dr. Hab., Professor V.G. Shilko_²
PhD, Associate Professor S.V. Radaeva²
M.V. Petrova^1
1Siberian Federal University, Krasnoyarsk
²Tomsk State University, Tomsk

Keywords: physical education, sports, mass sports, Nash equilibrium.

Background. Human resource is critical for the policies and practices geared to develop a dynamic innovative economy highly competitive on the global markets; and a special priority in the human resource development and employment efforts is commonly given to health as the key value largely determined by physical activity. It is common knowledge today that one of the key barriers for the popular physical education and sports promotion initiatives is the disharmony of the proclaimed wishful goals of the mass popular physical education and sports service and the actual service design and management practices [5].

Objective of the study was to provide theoretical and practical grounds for the model of competitions in mass children sports.

Methods and structure of the study. We applied the Nash equilibrium as a mathematical solution for the initiative to provide theoretical and practical grounds for the competitive model of mass children sports [7] striving, first of all, to remove any antagonistic attitude in competitions. This concept has proved its benefits for the modern game theory, management processes, economic theory and social sciences [1, 2] – and lately for the professional sport league design purposes [6, 8, 9].       

As provided by the modern game theory, a situation of the Nash equilibrium for non-cooperative game strategy (with the players’ actions being unpredictable and non-cooperative  at every stage of the game) may be described as х^* = (х₁^*, х₂^*, …, x_n^*), when all the players i and  and their strategies x_iX_i are determined by equality K_i(x^*_i, x^*_-i) ≥ K_i(x_i, x^*_-i), where K_i(x) means the win function. When situation х^* complies to the Nash equilibrium, it means that none of the players may break the equilibrium on his own at no loss; albeit some players may cooperate to shift the Nash equilibrium for the mutual benefit [1].

Based on this mathematical model, let us consider a hypothetical situation where eight players split up into two teams of four rationally thinking people are required to choose one of the following two models of competitions: elimination matches in couples with three prize winners (see Figure 1) and limited competition in couples with four prize winners taking the first, second and two third places (see Figure 2). When both of the teams independently and collectively take option 1, the competition will be run as required by the option. When both of the teams opt for the second model, the competitors will be split up into two categories, with every competitor guaranteed to win some prize. When the preferred options of the teams are different, the competition will be run in model 3 (see Figure 3) that means that those who opt for the elimination race are allowed to do so, and all the rest compete in the limited competitive format to get a guaranteed prize.

When every match win is scored by 1 point, and every loss by minus 1 point and the third, second and third places are scored by 1, 2 and 3 points, respectively, we can compute results of the matches (see circles on Figures 1-3) and benefit of every model for the competitors using the notion of gain common for the mathematical game theory and some other sciences [1]. We will interpret the gain as the ratio of the maximal possible points in every model (Figures 1-3) to the maximal cumulative loss. Thus gain of Model 1 is estimated at 6 – since the maximal possible win (6 points) to maximal loss (-1 point) ratio is 6. Such validation of gain for every model gives the means to find the Nash equilibrium situation: see Table 1, shaded cell.

Figure 1. Abstract model for a total-competition model

Figure 2. Abstract model for a limited-competition model

Figure 3. Abstract model for a combined-competition model

Having analyzed the above options, we find that neither of the teams, when the opponent’s power and game plan is unknown, may benefit from the total-competition model with its minimal gain. Both of the teams, when their decisions are rational and independent of one another, should rather opt for a limited loss under the limited-competition model with its modest albeit guaranteed win.

Table 1. Results of a bi-matrix game of teams 1 and 2

Benefits of the mathematical abstract models were tested at Magma Sport Club in Krasnoyarsk by a three-years-long (2014 through 2016) educational experiment. Sampled for the experiment were junior male competitors (n=30±5) of 10±3 and 13±2 years of age at the start and by the end of the experiment, respectively. The sample competed in 9 adapted Kyokushin karate tournaments, with 3 of them designed as total competitions (elimination model) with three prizes (Figure 1), and the rest as limited competitions (Figure 2). The limited-competition cases implied the Nash equilibrium being purposefully reached by splitting the competitors into categories so as to secure prize wins for each group. The groups were split up by age and weight classes plus extra nominations and categories. Benefits of the limited-competition model were tested by the players’ emotionality self-rating Wessman-Ricks test (in form of a questionnaire survey) a week after each competition [3].

Study findings and discussion. Given on Figure 4 hereunder are the comparative averaged emotionality self-rating test data arrays that demonstrate the significant difference of the post-competitive emotionality data arrays for the total- versus limited-competition models. The integral P5 rate produced by summation of P1, P2, P3 and P4 rates for the total-competition model was estimated at 26.28 points versus the limited-competition rate of 32.25 points – that means that the emotionality self-rate in the latter case was 23% higher. In addition, the dispersion of the emotionality rates in the total versus limited competition was 2.87 versus 1.52, respectively. This fact may be interpreted as indicative of the sample emotionality being more volatile (less homogenous) in the case of total competition; with the best emotional background tested in the athletes who made the highest competitive success.

Figure 4. Emotionality self-rating data: calm-anxiety rating P1 scale; energy-fatigue rating P2 scale; uplift-frustration rating P3 scale; and confidence-helplessness rating P4 scale [3]

Points Total competition Limited competition

The study data and analyses showed benefits of the proposed Nash-equilibrium-based limited-competition model for mass sports.

Conclusion. The study provided a sound theoretical and practical basis for the non-cooperative Nash-equilibrium-based limited-competition model recommendable for mass sports. It was found that the competitive model in application to the 7÷15 year-old male competitors in Kyokushin karate events secures a high positive emotional background in post-season associated with stable motivations for systemic trainings. The theoretical and practical grounds provided by the study may be applied for the education/ training process design purposes in popular mass sport disciplines.

References

1. Gubko M.V., Novikov D.A. Teoriya igr v upravlenii organizatsionnymi sistemami [Game theory in management of organizational systems]. RAS IPU. 2nd ed., Rev., Sup. Moscow, 2005, 138 p.
2. Zakharov A.V. Teoriya igr v obschestvennykh naukakh. Uchebnik dlya vuzov [Game theory in social sciences. Textbook for higher schools]. NRU of Higher School of Economics. Moscow: HSE publ., 2015, 304 p.
3. Karelin A.A. Bolshaya entsiklopediya psikhologicheskikh testov [Great encyclopedia of psychological tests]. Psikhodiagnostika: praktika. Metodika 'Samootsenka emotsionalnykh sostoyaniy [Psychodiagnostics: practice. Methodology 'Self-assessment of emotional states']. Moscow: Eksmo publ., 2007. Vol. 2, pp. 39 - 40.
4. Strategiya razvitiya fizicheskoy kultury i sporta v Rossiyskoy Federatsii na period do 2020 g. [Strategy for development of physical education and sports in the Russian Federation for the period until 2020] [Electronic resource]. Available at: http://www.minsport.gov.ru/activities/federal-programs/2/26363 (accessed: 25.03.2018)
5. Fiskalov V.D., Cherkashin V.P. Teoretiko-metodicheskie aspekty praktiki sporta [Theoretical and methodical aspects of sport practice]. Problemy razvitiya massovogo sporta v Rossii. Moscow, 2016. vol. 7, pp. 98 – 110.
6. Hirai S. Existence of Nash equilibria in sporting contests with capacity constraints. Journal title: 西南学院大学経済学論集, December 2014. Vol. 49, no.  2-3, pp. 147 – 168. Available at: http://repository.seinan-gu.ac.jp/handle/123456789/1155.
7. Nash J.F. Equilibrium points in N-person games. Proceedings National Academy of Sciences. 1950a, vol. 36, no. 2, pp. 48 – 49.
8. Palomino F., Rigotti, L. The Sport League's Dilemma: Competitive Balance versus Incentives to Win (November 2000). Tilburg University Center for Economic Research Working Paper No. 2000-109. Available at: https://ssrn.com/abstract=250793 or http://dx.doi.org/10.2139/ssrn.250793.
9. Szymanski S. The economic design of sporting contests. Journal of Economic Literature. 2003, vol. 41, pp. 1137 - 1187.

Corresponding author: Avoynich-fk17@stud.sfu-kras.ru

Abstract

The study gives theoretical and practical grounds for a mass sport competition model based on the Nash equilibrium. The model is essentially designed on the non-cooperative equilibrium concept that implies each player being prepared to sacrifice, to a reasonable degree, own gain for the mutual benefit in the competitive process. The Nash-equilibrium-based limited-competition model was tested by a 3-year-long educational experiment at Magma Sport Base in Krasnoyarsk city. Sampled for the study were 7-15 year-old males (n=30). As required by the Nash equilibrium concept, the children competitions were designed to offer a few competitive categories and formats to secure a win for each competitor. We found the proposed Nash-equilibrium-based limited-competition model for mass sports being beneficial as verified by the sample progress in the post-season emotionality self-rates that totaled 32.25 points on the Wessman-Ricks scale – versus the total-competition model that scored only 26.28 points (23% lower). The proposed Nash-equilibrium-based limited-competition model was tested beneficial and recommended for application.