The Relationship Between the Core and the Modified Cores of A

mathematics Article The Relationship between the Core and the Modified Cores of a Dynamic Game Artem Sedakov 1,2,3,* and Hao Sun 1 1 Saint Petersburg State University, 7/9 Universitetskaya nab., 199034 Saint Petersburg, Russia; [email protected] 2 School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China 3 Institute of Applied Mathematics of Shandong, Qingdao 266071, China * Correspondence: [email protected] Received: 21 May 2020; Accepted: 22 June 2020; Published: 23 June 2020 Abstract: The core as a solution to a cooperative game has the advantage that any imputation from it is undominated. In cooperative dynamic games, there is a known transformation that demonstrates another advantage of the core—time consistency—keeping players adhering to it during the course of the game. Such a transformation may change the solution, so it is essential that the new core share common imputations with the original one. In this paper, we will establish the relationship between the original core of a dynamic game and the core after the transformation, and demonstrate that the latter can be a subset of the former. Keywords: discrete-time games; cooperation; the core; linear transformation MSC: 91A12; 91A50 1. Introduction The theory of cooperative dynamic games is useful for modeling and analyzing real world problems. Examples include advertising, public goods provision, resource extraction, environmental management, and others which are extensively discussed in [1,2]. The core as a solution to a cooperative game has the advantage that any imputation from it is undominated. This solution is quite popular in the literature on the application of dynamic game theory, not only because of the aforementioned property, but also because of its flexibility, allowing allocating the cooperative outcome in several ways, for instance, in lot sizing [3–5], pollution control [6–8], or non-renewable resource extraction [9]. In cooperative dynamic games, there is a known transformation of a characteristic function, which is a key component of any cooperative game measuring the claims of any group of players [10,11]. The core determined by the modified characteristic function possesses another advantage—time consistency—keeping players adhering to it and being non-negotiable during the course of the game [1,12]. Such a transformation, however, may change the core, so it is essential that the modified core share common imputations with the original one. This allows players to expect if not all of the imputations from the original one, but a part of them. For this reason, we will establish the relationship between the original core of a dynamic game and the core after the transformation, and demonstrate that in some instances the latter can be a subset of the former. It was proven in [11] that the proposed transformation rule applied an infinite number of times converges when the total players’ payoffs along the agreed upon behavior are positive. Here we will relax this assumption and refine the conditions that ensure the convergence of the transformation rule. The structure of the paper is as follows. Section2 introduces necessary definitions and concepts. The main results of the paper are formulated in Section3. We then study the relationship between Mathematics 2020, 8, 1023; doi:10.3390/math8061023 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 1023 2 of 13 the original core of a dynamic game and the modified ones and establish conditions for the limiting core to be a subset of the original core also when considering classes of linear symmetric games, linear-state games, and two-stage network games. Section4 concludes. 2. Background We consider a standard formulation of a dynamic game with complete information. Let N = f1, . , ng, jNj = n > 2, be a finite set of players. The set of game stages (periods) is described by a finite set T = f1, ... , Tg. We denote a state variable at stage t 2 T by x(t) which belongs to a state space X. Let x(1) = x1. Next, we denote an action of player i 2 N at stage t 2 T by ui(t) which belongs to her action space Ui(t). (Since the sets of actions U1(t), ... , Un(t), t 2 T , and the state space X have not been precisely defined, we suppose that they are not empty, and the values of all optimization problems below exist and are finite). We suppose that the state dynamics is governed by the difference equation x(t + 1) = ft(x(t), u(t)) 2 X (1) for any t 2 T from the initial state x1. It is supposed that x(t + 1) is uniquely defined. At each of T game stages, players simultaneously choose actions and thus form an action profile u = (u1, ... , un) with ui = (ui(1), ... , ui(T)) for i 2 N. The payoff to player i 2 N in the game is defined by the real-valued function T Ji(x1, u) = ∑ hit(x(t), u(t)), i 2 N, t=1 and amounts to the sum of her stage payoffs. (As with [11], players are not rewarded at a terminal state x(T + 1), yet the setting can easily be generalized to this case as well). A player chooses an action according to her strategy which accounts for the current information about the game available to her at the time of decision: this can be the information about the game stage, the value of the state variable, the actions that players have taken at previous stages, etc. We denote the information available to player i 2 N at stage t 2 T by hi(t). A strategy ui of player i is a rule that maps the player’s information space to her action space, i.e., at stage t player i with information hi(t) chooses the action ui(t) = ui(hi(t)) 2 Ui(t). (See [13] for more details). A collection u = (u1, ... , un) is a strategy profile. Each strategy profile generates a trajectory which is a profile x = (x(1), ... , x(T)) whose entries are determined by (1). One can introduce the payoff function Ji of player i 2 N defined on the set of players’ strategy profiles as follows: Ji(u) = Ji(x1, u) where u is an action profile corresponding to u. In the cooperative formulation of the game, players choose their strategies jointly to maximize ∗ ∗ ∗ the payoff they generate, that is to maximize the sum ∑i2N Ji(u). Let a strategy profile u = (u1, ... , un) maximize the latter sum. This profile is called a cooperative strategy profile and the associated trajectory ∗ ∗ ∗ ∗ x = (x (1),..., x (T)) with x (1) = x1 is called a cooperative trajectory. We now define a cooperative transferable utility game, or a TU game, (N, v) which is determined by the same player set N and a characteristic function v. This function is defined on 2N, that is the set of all subsets of set N, and for a subset S ⊆ N, called a coalition, its value (a real number) v(S) measures the worth, or claims, of this coalition in the game. Additionally, v(?) = 0. We will not specify how this function is determined as it is not relevant to the analysis we will perform; we only note that ∗ v(N) = ∑i2N Ji(u ), i.e., the grand coalition claims the maximum payoff it generates. (See different concepts for determining the characteristic function in dynamic games in [12]). Once the value of v(N) is obtained, players allocate it among them as an imputation which is a profile x(v) = (x1(v), ... , xn(v)) satisfying efficiency, i.e., ∑i2N xi(v) = v(N), and individual rationality, i.e., xi(v) > v(fig), i 2 N. The set of all imputations, or the imputation set, will be denoted by I(v).A cooperative solution, or simply a solution, to the cooperative dynamic game (N, v) is a rule that assigns a subset M(v) ⊆ I(v) to this game. In this paper, we suppose that the solution is the core, that is the subset of the imputation set given by C(v) = fx(v) 2 I(v) : ∑i2S xi(v) > v(S), S ⊂ Ng. Having chosen the agreed upon Mathematics 2020, 8, 1023 3 of 13 cooperative solution C(v), players jointly implement cooperative strategy profile u∗ moving along cooperative trajectory x∗, and after obtaining the value v(N) as their payoff, the players allocate it among them as an imputation from the chosen solution C(v). In cooperative dynamic games, it is important that players adhere to the same solution chosen at the initial stage as the game develops along the agreed upon cooperative trajectory x∗. A time-consistent solution is stable to its revision during the course of the game, and implementing certain mechanisms one can make cooperation sustainable. When the solution is time inconsistent, there are effective mechanisms of game regularization, that is a change in players’ stage payoffs along the cooperative trajectory, so that the solution becomes time consistent in the regularized game. (See [1,12] for a comprehensive analysis of sustainable cooperation and the associated time consistency property of a cooperative solution). In the vast majority of cases, such mechanisms are designed on a special redistribution of players’ stage payoffs determined by an imputation distribution procedure and they require consideration of proper subgames of the original game along the cooperative trajectory. Each subgame is a dynamic game of T − t + 1 stages starting from ∗ the initial state x (t), t 2 T n f1g. In a similar way, one can define a cooperative subgame (N, vt), the imputation set I(vt), and the cooperative solution (the core C(vt)) to each subgame, t 2 T n f1g.

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