FINITE APPROXIMATION OF NONCOOPERATIVE 2-PERSON GAMES PLAYED IN STAIRCASE-FUNCTION CONTINUOUS SPACES

16 ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ, СИСТЕМНИЙ АНАЛІЗ ТА КЕРУВАННЯ DOI: 10.20535/kpisn.2023.1-2.270281 UDC 519.833+519.833.3 V. V. Romanuke Polish Naval Academy, Gdynia, Poland FINITE APPROXIMATION OF NON-COOPERATIVE 2-PERSON GAMES PLAYED IN STAIRCASE-FUNCTION CONTINUOUS SPACES Background. There is a known method of approximating continuous non-cooperative 2-person games, wherein an approximate solution (an equilibrium situation) is considered acceptable if it changes minimally by changing the sampling step minimally. However, the method cannot be applied straightforwardly to a 2-person game played with staircase-function strategies. Besides, the independence of the player’s sampling step selection should be taken into account. Objective. The objective is to develop a method of finite approximation of 2-person games played in staircase-function continuous spaces by taking into account that the players are likely to independently sample their pure strategy sets. Methods. To achieve the said objective, a 2-person game, in which the players’ strategies are staircase functions of time, is formalized. In such a game, the set of the player’s pure strategies is a continuum of staircase functions of time, and the time is thought of as it is discrete. The conditions of sampling the set of possible values of the player’s pure strategy are stated so that the game becomes defined on a product of staircase-function finite spaces. In general, the sampling step is different at each player and the distribution of the sampled points (function-strategy values) is non-uniform. Results. A method of finite approximation of 2-person games played in staircase-function continuous spaces is presented. The method consists in irregularly sampling the player’s pure strategy value set, finding the best equilibria in “smaller” bimatrix games, each defined on a subinterval where the pure strategy value is constant, and stacking the equilibrium situations if they are consistent. The stack of the “smaller” bimatrix game equilibria is an approximate equilibrium in the initial staircase game. The (weak) consistency of the approximate equilibrium is studied by how much the payoff and equilibrium situation change as the sampling density minimally increases by the three ways of the sampling increment: only the first player’s increment, only the second player’s increment, both the players’ increment. The consistency is decomposed into the payoff, equilibrium strategy support cardinality, equilibrium strategy sampling density, and support probability consistency. It is practically reasonable to consider a relaxed payoff consistency. Conclusions. The suggested method of finite approximation of staircase 2-person games consists in the independent samplings, solving “smaller” bimatrix games in a reasonable time span, and stacking their solutions if they are consistent. The finite approximation is regarded appropriate if at least the respective approximate (stacked) equilibrium is e-payoff consistent. Keywords: game theory; payoff functional; staircase-function strategy; bimatrix game; irregular sampling; approximate equilibrium consistency. Introduction Non-cooperative 2-person games model processes where two sides referred to as persons or players struggle for optimizing the limited resources distribution implying as real-world resources, facilities, tools, funds, energy, etc., as well as more abstract objects whose utility is assessed as the player’s payoff [1, 2]. A possible action of the player is called its (pure) strategy used to receive closely the best possible payoff under conditions of uncertainty generated by actions of the other player [3, 4]. The strategy can be as a simple (point) action, as well as a process consisting of an order of simple actions [1, 5, 6]. In the simplest case, the player’s pure strategy is a short action whose duration is negligible. This negligible-duration action is represented as just a time point. In a more complicated case, the player’s pure strategy is a function of time [4, 7, 8], so the player’s action is a complex process [6, 9]. Пропозиція для цитування цієї статті: В. В. Романюк, “Скінченна апроксимація безкоаліційних ігор двох осіб, що розігруються у неперервних просторах сходинкових функцій”, Наукові вісті КПІ, 1–4, с. 16–44, 2023. doi: 10.20535/kpisn.2023.1-2.270281 Offer a citation for this article: Vadim V. Romanuke, “Finite approximation of non-cooperative 2-person games played in staircase-function continuous spaces”, KPI Science News, no. 1–4, pp. 16–44, 2023. doi: 10.20535/kpisn.2023.1-2.270281 © The Autor(s). The article is distributed under the terms of the license CC BY 4.0 ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ, СИСТЕМНИЙ АНАЛІЗ ТА КЕРУВАННЯ Such strategies are used in multistage optimization [10], planning and control processes [11], scheduling [12], multistage corrective action processes [13], etc., modelled under uncertainties and influence of other competitive factors [5, 6, 9]. Whichever the pure strategy form is, the simplest 2-person game is a bimatrix game. Any bimatrix game has an equilibrium – a finite number or continuum of equilibria, either in pure or mixed strategies [1, 2]. Infinite or continuous 2-person games, where the players’ payoff functions are meshes or surfaces of two variables defined on finite-dimensional compact Euclidean subspaces, are far more complicated [1, 2, 7, 14]. A simple example of the subspace is a unit square [2, 15]. Even if the surfaces do not have a discontinuity, the equilibrium is not always determinable as opposed to bimatrix games [2]. Moreover, 2-person games defined on open (or half-open) subspaces (e. g., open square) may not have an equilibrium at all [2, 16, 17]. Therefore, rendering a 2-person game to a bimatrix one is a crucial task in game modelling as it allows assuredly having a game solution (equilibrium point) as a pair of the players’ best strategies. Without rendering, a 2-person game may have an intractable equilibrium (if any), when the equilibrium strategy support is infinite or continuous (e. g., see the examples in [1, 7, 16, 17]). A 2-person game, in which the player’s strategy is a function (e. g., of time), is a far more complicated case. In such games, the payoff kernel must be a functional mapping every pair of functions (pure strategies of the players) into a real value [7, 8, 18, 19]. A game played with such function-strategies is rendered down to a bimatrix game only when each of the players possesses a finite set of one’s function-strategies. Obviously, the rendering is theoretically impossible if the set of the player’s strategies is infinite. The question of rendering an infinite game to a finite one was studied in [14, 20]. Regardless of antagonism of the players’ interests, it consists in approximating the infinite game so that the approximated game would not lose the properties of the initial game. There are two fundamental conditions in the game approximation core that allow rendering a (...truncated)