Scoring a Goal Optimally in a Soccer Game Under Liouville-Like Quantum Gravity Action
Tóm tắt
In this paper, we present a new stochastic differential game-theoretic model of optimizing strategic behavior associated with a soccer player under the presence of stochastic goal dynamics by using a Feynman-type path integral approach, where the action of a player is on a
$$\sqrt{8/3}$$
-Liouville quantum gravity surface. Strategies to attack the oppositions have been used as control variables with extremes like excessive defensive and offensive strategies. Before determining the optimal strategy, we first establish an infinitary logic to deal with infinite variables on the strategy space, and then, a quantum formula of this logic is developed. As in a competitive tournament, all possible standard strategies to score goals are known to the opposition team, a player’s action is stochastic, and they would have some comparative advantages to score goals. Furthermore, conditions like uncertainties due to rain, dribbling and passing skills of a player, time of the game, home crowd advantages, and asymmetric information of action profiles are considered to determine the optimal strategy.
Tài liệu tham khảo
Santos RM (2014) Optimal soccer strategies. Econ Inq 52:183–200
Dobson S, Goddard J (2010) Optimizing strategic behaviour in a dynamic setting in professional team sports. Eur J Oper Res 205:661–669
Palomino F, Rigotti L, Rustichini A et al (1998) Skill, strategy and passion: an empirical analysis of soccer. Tilburg University Tilburg
Banerjee AN, Swinnen JF, Weersink A (2007) Skating on thin ice: rule changes and team strategies in the NHL. Canadian Journal of Economics/Revue canadienne d’économique 40:493–514
Feynman RP (1949) Space-time approach to quantum electrodynamics. Phys Rev 76:769
Fujiwara D (2017) Rigorous time slicing approach to Feynman path integrals. Springer
Pramanik P (2020) Optimization of market stochastic dynamics. In: SN Operations Research Forum, vol 1. Springer, pp 1–17
Pramanik P (2021) Effects of water currents on fish migration through a Feynman-type path integral approach under \(\sqrt{8/3}\) Liouville-like quantum gravity surfaces. Theory Biosci 140:205–223
Pramanik P (2023) Consensus as a Nash equilibrium of a stochastic differential game. European Journal of Statistics 3:10–10
Pramanik P, Polansky AM (2020a) Optimization of a dynamic profit function using Euclidean path integral. Preprint at arXiv:2002.09394
Hua L, Polansky A, Pramanik P (2019) Assessing bivariate tail non-exchangeable dependence. Statistics & Probability Letters 155:108556
Miller J, Sheffield S (2015) Liouville quantum gravity and the Brownian map I: The QLE (8/3, 0) metric. Preprint at arXiv:1507.00719
Miller J, Sheffield S (2016) Liouville quantum gravity and the Brownian map III: the conformal structure is determined. Preprint at arXiv:1608.05391
Polansky AM, Pramanik P (2021) A motif building process for simulating random networks. Computational Statistics & Data Analysis 162:107263
Pramanik P, Polansky AM (2020b) Motivation to run in one-day cricket. Preprint at arXiv:2001.11099
Pramanik P, Polansky AM (2022) Optimal estimation of Brownian penalized regression coefficients. International Journal of Mathematics, Statistics and Operations Research 2
Pramanik P, Polansky AM (2023) Semicooperation under curved strategy spacetime. Journal of Mathematical Socilogy 1–35
Sheffield S (2022) What is a random surface? Preprint at arXiv:2203.02470
Pitici M (2017) The best writing on mathematics. Princeton University Press
Pramanik P (2022a) On lock-down control of a pandemic model. Preprint at arXiv:2206.04248
Pramanik P (2023) Path integral control of a stochastic multi-risk SIR pandemic model. Theory Biosci 142:107–142
Pramanik P (2016) Tail non-exchangeability. Northern Illinois University
Kappen HJ (2007) An introduction to stochastic control theory, path integrals and reinforcement learning. In: AIP conference proceedings 887 149–181. AIP
Lasry J-M, Lions P-L (2007) Mean field games. Japan J Math 2:229–260
Sheffield S (2007) Gaussian free fields for mathematicians. Probab Theory Relat Fields 139:521–541
Brocas I, Carrillo JD (2004) Do the “three-point victory’’ and “golden goal’’ rules make soccer more exciting? J Sports Econ 5:169–185
Guedes JC, Machado FS (2002) Changing rewards in contests: has the three-point rule brought more offense to soccer? Empirical Economics 27:607–630
Dilger A, Geyer H (2009) Are three points for a win really better than two? A comparison of German soccer league and cup games. J Sports Econ 10:305–318
Garicano L, Palacios-Huerta II (2005) Sabotage in tournaments: making the beautiful game a bit less beautiful
Moschini G (2010) Incentives and outcomes in a strategic setting: the 3-points-for-a-win system in soccer. Econ Inq 48:65–79
Ross K (2008) Stochastic control in continuous time. Lecture Notes on Continuous Time Stochastic Control, Spring
Marcet A, Marimon R (2019) Recursive contracts. Econometrica 87:1589–1631
Frick M, Iijima R, Strzalecki T (2019) Dynamic random utility. Econometrica 87:1941–2002
Hellman Z, Levy YJ (2019) Measurable selection for purely atomic games. Econometrica 87:593–629
Gwynne E, Miller J (2016) Metric gluing of Brownian and \(\sqrt{8/3}\)-Liouville quantum gravity surfaces. Preprint at arXiv:1608.00955
Bettinelli J, Miermont G (2017) Compact Brownian surfaces I: Brownian disks. Probab Theory Relat Fields 167:555–614
Curien N, Le Gall J-F (2014) The Brownian plane. J Theor Probab 27:1249–1291
Granas A, Dugundji J (2003) Elementary fixed point theorems. In: Fixed Point Theory. Springer, pp 9–84
Falconer K (2004) Fractal geometry: mathematical foundations and applications. John Wiley & Sons
Kurtz DS, Swartz CW (2004) Theories of integration: the integrals of Riemann, Lebesgue, Henstock-Kurzweil, and Mcshane 9. World Scientific Publishing Company
Muldowney P (2012) A modern theory of random variation. Wiley Online Library
Pramanik P (2021b) Optimization of dynamic objective functions using path integrals, PhD thesis, Northern Illinois University
Pramanik P (2022b) Stochastic control of a SIR model with non-linear incidence rate through Euclidean path integral. Preprint at arXiv:2209.13733
Øksendal B (2003) Stochastic differential equations. In Stochastic differential equations 65–84. Springer
Parthasarathy TT, Theorems S, Applications T (1972) Lectures Notes in Math, nr. 263