A complex system exhibiting uncertainty as well as randomness may be portrayed by an uncertain random singular difference equation. This paper investigates two-person nonzero-sum and zero-sum games based on uncertain random singular difference equations. First, an approach is proposed to translate the two-person nonzero-sum game into an equivalent game for a standard uncertain random dynamic system. The relevant recursive equations are developed to search the Nash equilibrium for the converted game. Solving the related recursive equations yields the solution to such a game. Following that, a Max-Min Theorem is provided for finding the saddle-point equilibrium of an uncertain random two-person zero-sum game. Finally, a numerical example is offered to demonstrate the validity of the findings. 

https://onlinelibrary.wiley.com/doi/pdfdirect/10.1049/cth2.12400

Two‐person games for uncertain random singular dynamic systems

In deterministic team problems every closed-loop representation of an optimal open-loop solution is also optimal. This property, however, no longer holds true when the optimization problem is a zero-sum or a nonzero-sum game. In zero-sum games, two weaker (but still general enough) versions of this statement are valid, which still fail to hold in the case of nonzero-sum games. In this correspondence we state and prove these two general properties of the saddle-point solution in dynamic games.

Two general properties of the saddle-point solutions of dynamic games

In this article, we study the feedback Nash strategy of the model-free nonzero-sum difference game. The main contribution is to present the Q -learning algorithm for the linear quadratic game without prior knowledge of the system model. It is noted that the studied game is in finite horizon which is novel to the learning algorithms in the literature which are mostly for the infinite-horizon Nash strategy. The key is to characterize the Q -factors in terms of the arbitrary control input and state information. A numerical example is given to verify the effectiveness of the proposed algorithm. 

Q-Learning for Feedback Nash Strategy of Finite-Horizon Nonzero-Sum Difference Games

In stochastic optimal control, change of measure arguments have been crucial for stochastic analysis. Such an approach is often called static reduction in dynamic team theory and has been an effective method for establishing existence and approximation results for optimal policies. These arguments have also been applied to the study of dynamic games. In this paper, we demonstrate the limitations of such an approach for a wide class of stochastic dynamic games, where additionally, unlike the team setting considered in Part I, informational dependence of equilibrium behavior is significantly more complicated. We identify three types of static reductions: (i) those that are policy-independent (as those introduced by Witsenhausen for teams), (ii) those that are policy-dependent (as those introduced by Ho and Chu for partially nested dynamic teams), and (iii) a third type that we will refer to as static measurements with control-sharing reduction as in Part I (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. However, for the second type, we show that there is generally no isomorphism between Nash equilibrium solutions under the original information structure and their policy-dependent static reductions. Sufficient conditions are presented 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, these sufficient conditions are relaxed and stronger results are established. We also study three classes of multi-stage games, where we establish connections between closed-loop, open-loop, and control-sharing Nash and saddle point equilibria.

Isomorphism Properties of Optimality and Equilibrium Solutions under Equivalent Information Structure Transformations II: Stochastic Dynamic Games.

A salient feature of many optimal decision-making policies in adversarial environments is a level of unpredictability, or randomness, which keeps opponents uncertain about the system’s strategies. These considerations, along with feedback from adversarial behaviors, are crucial in ensuring the security of modern infrastructures and complex systems. This paper considers policies that do just the opposite, namely ones that reveal strategic intentions to an opponent before engaging in competition. We consider such scenarios in the context of General Lotto games, which models the competitive allocation of resources between opposing players. Here, we consider a dynamic extension where one of the players has the option to publicly pre-commit assets to a battlefield in the first stage. In response, the opponent decides whether to secure the battlefield by matching the pre-commitment with its own resources, or to withdraw from it entirely. They then engage over the remaining set of battlefields in the second stage. We show that the weaker-resource player can have incentives to pre-commit when the battlefield values are asymmetric across players. Previous work asserts this never holds when the values are symmetric across players. Our analysis demonstrates the viability of alternate strategic mechanisms that a competitor may be able to employ.

Analyzing pre-commitment strategies in General Lotto games

This paper contains exact expressions for the complete class of uncountably many globally optimal affine Nasb equilibrium strategies for a two-stage two-person nonzero-sum game problem with quadratic objective functionals and with dynamic information for beth players. Existence conditions for each of these Nash equilibrium solutions are derived and it is shown that a recursive Nash solution is not necessarily globally optimal. Cost-uniqueness property of the derived Nash strategies is investigated and it is proven that the game problem under consideration admits a unique Nash cost pair if and only if it can be made equivalent to either a team problem or a zero-sum game. It is also shown that existence conditions of a globally optimal Nash solution will be independent of the parameters characterizing the nonuniques of the Nash strategies only if the game problem can be made equivalent to a team problem.

Properties of Nash solutions of a two-stage nonzero-sum game

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Properties of Nash solutions of a two-stage nonzero-sum game