Time-Consistency of an Imputation in a Cooperative Hybrid Differential Game

This work is aimed at studying the problem of maintaining the sustainability of a cooperative solution in an n-person hybrid differential game. Specifically, we consider a differential game whose payoff function is discounted with a discounting function that changes its structure with time. We solve the problem of time-inconsistency of the cooperative solution using a so-called imputation distribution procedure, which was adjusted for this general class of differential games. The obtained results are illustrated with a specific example of a differential game with random duration and a hybrid cumulative distribution function (CDF). We completely solved the presented example to demonstrate the application of the developed scheme in detail. All results were obtained in analytical form and illustrated by numerical simulations.

FISH WARS WITH MANY PLAYERS

Discrete-time game-theoretic models related to a bioresource management problem (fish catching) are investigated. There are some players (countries or fishing firms) which harvest the fish stock. Power population's growth function and logarithmic players' profits are considered. We derive the Nash and cooperative equilibria. We construct the characteristic function for cooperative game in two unusual forms and determine the Shapley value and time-consistent imputation distribution procedure [Petrosjan (1977)]. We propose the condition which gives an incentive for the players to keep cooperation at each stage and compare it with the "irrational-behavior-profness" condition [Yeung (2006)]. We propose the linear programming method to find the time-consistent "rational" solution in C-core. The numerical modelling and the results' comparison were carried out.

STRATEGICALLY SUPPORTED COOPERATION

An n-person differential game Γ(x, T-t) with independent motions from the initial state x and with prescribed duration T - t is considered. Suppose that y(s) is the cooperative trajectory maximizing the sum of players' payoffs. Suppose also that before starting the game players agree to divide the joint maximal payoff V(x, T - t; N) according to the imputation α, which is considered as a solution of a cooperative version of the game Γ(x, T - t). Using individual rationality of the imputation α we prove that if in the game Γ(y(s),T - s) along the cooperative trajectory y(s), the solution will be derived from the imputation α with the use of the imputation distribution procedure (IDP), for each given ε > 0 there exists ε-Nash equilibrium in Γ(x, T - t) for which the payoffs of the players in the game will be equal exactly to the components of the imputation α (cooperative outcome). This means that the imputation α is strategically supported by some specially constructed ε-Nash equilibrium in Γ(x, T - t). A similar result is true for a discrete game with perfect information.