Construction of Strongly Time-Consistent Subcores in Differential Games with Prescribed Duration

Abstract

A new strongly time-consistent (dynamically stable) optimality principle is proposed in a cooperative differential game. This is done by constructing a special subset of the core of the game. It is proposed to consider this subset as a new optimality principle. The construction is based on the introduction of a function \(\hat V\) that dominates the values of the classical characteristic function in coalitions. Suppose that V (S, \(\bar x\) (τ), T −τ) is the value of the classical characteristic function computed in the subgame with initial conditions \(\bar x\) (τ), T −τ on the cooperative trajectory. Define

\(\hat V\left( {S;{X_0},T - {t_0}} \right) = \mathop {\max }\limits_{{t_0} \leqslant \tau \leqslant T} \frac{{V\left( {S;{x^ * }\left( \tau \right),T - \tau } \right)}}{{V\left( {N;{X^ * }\left( \tau \right),T - \tau } \right)}}V\left( {N;{x_0},T - {t_0}} \right)\)

Using this function, we construct an analog of the classical core. It is proved that the constructed core is a subset of the classical core; thus, we can consider it as a new optimality principle. It is also proved that the newly constructed optimality principle is strongly time-consistent.