Isomorphism properties of optimality and equilibrium solutions under equivalent information structure transformations: stochastic dynamic games and teams

Static reduction of information structures is a method that is commonly adopted in stochastic control, team theory, and game theory. One approach entails change of measure arguments, which has been crucial for stochastic analysis and has been an effective method for establishing existence and approximation results for optimal policies. Another approach entails utilization of invertibility properties of measurements, with further generalizations of equivalent information structure reductions being possible. In this paper, we demonstrate the limitations of such approaches for a wide class of stochastic dynamic games and teams, and present a systematic classification of static reductions for which both positive and negative results on equivalence properties of equilibrium solutions can be obtained: (i) those that are policy-independent, (ii) those that are policy-dependent, and (iii) a third type that we will refer to as static measurements with control-sharing reduction (where the measurements are static although control actions are shared according to the partially nested information structure). For the first type, we show that there is a bijection between Nash equilibrium policies under the original information structure and their policy-independent static reductions, and establish sufficient conditions under which stationary solutions are also isomorphic between these information structures. For the second type, however, we show that there is generally no isomorphism between Nash equilibrium (or stationary) solutions under the original information structure and their policy-dependent static reductions. Sufficient conditions (on the cost functions and policies) are obtained to establish such an isomorphism relationship between Nash equilibria of dynamic non-zero-sum games and their policy-dependent static reductions. For zero-sum games and teams, these sufficient conditions can be further relaxed. In view of the equivalence between policies for dynamic games and their static reductions, and closed-loop and open-loop policies, we also present three classes of multistage games and teams with partially nested information structures, where we establish connections between closed-loop, open-loop, and control-sharing Nash and saddle point equilibria. By taking into account a playerwise concept of equilibrium, we introduce two further classes of ``playerwise"" static reductions: (i) independent data reduction under which the policy-independent reduction holds through players and time, and (ii) playerwise (partially) nested independent reduction under which measurements are independent through players but (partially) nested through time for each player.