Cooperative Behavior in Multicriteria Resource Sharing Problem

Abstract This paper considers a dynamic discrete-time game-theoretic model where players exploit a common resource and seek to optimize multiple objectives simultaneously. The center (referee) shares the resource available for exploitation among the participants and maintains the cooperative behavior. To construct multicriteria noncooperative and cooperative equilibria, modified bargaining schemes are applied. To maintain cooperative behavior, cooperative incentive equilibrium – a penalty mechanism adopted by the center – is presented. To define the cooperative sharing rule, the center is treated as a player with multiple objectives and the ability either to cooperate with other players or to act individually. A dynamic bi-criteria bioresource management problem is investigated to illustrate the proposed concepts. Similar content being viewed by others Data Availability No datasets were generated or analysed during the current study. 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Corresponding author Ethics declarations Competing interests The authors declare no competing interests. Additional information Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Appendices Proof of Proposition 1 First, consider the one-step game. Problem (20) takes the form As the resource sharing rule is fixed, we will use the designations \(\gamma _{1t}=u_{1t}(1-s_t)\) and \(\gamma _{2t}=u_{2t}s_t\). After an obvious simplification (\(B=1\) for the one-step game), problem (A1) becomes As the considered functions are concave, the first-order optimality conditions yield where \(W(\cdot )\) is the Lambert W function. Now consider problem (20) for the two-step game. The multicriteria payoff function for the first player takes the form With the simplification (\(B=1+\delta \) for the two-step game), we obtain for the first player, and similar for the second one. Hence, to construct the multicriteria Nash equilibrium, it is necessary to solve the problem Based on the first-order optimality conditions, we establish at the following relationship between the players’ strategies in the one- and two-step games: The players’ strategies at the last step again coincide: \(\gamma _{1m}=\gamma _{2m}=\gamma _m;\) they are found from the equation which yields where \(W(\cdot )\) is the Lambert W function. By repeating this procedure for the games with \(3,\ldots , m\) steps, we finally arrive at the multicriteria Nash equilibrium strategies (20). Proof of Proposition 3 Consider the second player’s deviation The center’s strategy is sought after in the form Hence, the first player’s strategy becomes The form of this strategy corresponds to the cooperative one, but under a different sharing rule. To determine the coefficient \(\eta _t\), it is necessary to maximize the individual payoff function of the second player provided that the center and the first player use the corresponding strategies (B3) and (B4). Therefore, we solve the problem where \(x_t\) evolves according to For \(\theta _{1t}(u_{2t})\) to be the incentive equilibrium (8), the solution of problem (B5), (B6) shall coincide with the cooperative one \(u_{2t}^c\). The individual payoff function of the second player becomes Note that here, the status quo points are the Nash equilibrium payoffs defined in subsection 5.1. First, consider the one-step game with the second player’s deviation \(u_{2m}=u_{2m}^c+\Delta _m\). Then the sharing rule is \(s_m^{d2}=s_m^c-\eta _m(u_{2m}-u_{2m}^c)\), and the first player’s strategy is given by \(\theta _{1m}(u_{2m})=\frac{\gamma ^c}{1-s_m^{2d}}\). To determine the second player’s strategy, we solve the problem Its solution shall coincide with the cooperative one \(u_{2m}^c\), which yields To proceed, consider problem (B5), (B6) for the two-step game. The individual payoff function becomes where The solution of the optimization problem (B7), shall coincide with the cooperative one, \(u_{2m-1}^c\) and \(u_{2m}^c\), which yields Following the above procedure for the games with \(3,\ldots , m\) steps, we finally arrive at the cooperative incentive equilibrium strategies (22), (23). Rights and permissions Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. About this article Cite this article Rettieva, A. Cooperative Behavior in Multicriteria Resource Sharing Problem. Dyn Games Appl (2026). https://doi.org/10.1007/s13235-026-00712-3 Received: Accepted: Published: Version of record: DOI: https://doi.org/10.1007/s13235-026-00712-3