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1. United We Fall: On the Nash Equilibria of Multiplex Network Games

   https://doi.org/10.1109/Allerton58177.2023.10313426  

   Ebrahimi, Raman ; Naghizadeh, Parinaz ( September 2023 , 2023 59th Annual Allerton Conference on Communication, Control, and Computing (Allerton))

   Network games have provided a framework to study strategic decision making processes that are governed by an underlying network of interdependencies among agents. However, existing models do not account for environments in which agents simultaneously interact over multiple networks. In this paper, we propose a model of multiplex network games to capture the different modalities of interactions among strategic agents. We then explore how the properties of the constituent networks of a multiplex network can undermine or support the uniqueness of its Nash equilibria. We first show that in general, even if the constituent networks are guaranteed to have unique Nash equi- libria in isolation, the resulting multiplex need not have a unique equilibrium. We then identify certain subclasses of networks wherein guarantees on the uniqueness of Nash equilibria on the isolated networks lead to the same guarantees on the multiplex network game. We further highlight that both the largest and smallest eigenvalues of the constituent networks (reflecting their connectivity and two-sidedness, respectively) are instrumental in determining the uniqueness of the multiplex network equilibrium. Together, our findings shed light on the reasons for the fragility of the uniqueness of equilibria in multiplex networks, and potential interventions to alleviate them. 

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2. Multi-scale games: Representing and solving games on networks with group structure

   Jin, Kun ; Vorobeychik, Yevgeniy ; Liu, Mingyan ( January 2021 , Proceedings of the AAAI Conference on Artificial Intelligence)

   null (Ed.)

   Network games provide a natural machinery to compactly represent strategic interactions among agents whose payoffs exhibit sparsity in their dependence on the actions of others. Besides encoding interaction sparsity, however, real networks often exhibit a multi-scale structure, in which agents can be grouped into communities, those communities further grouped, and so on, and where interactions among such groups may also exhibit sparsity. We present a general model of multi-scale network games that encodes such multi-level structure. We then develop several algorithmic approaches that leverage this multi-scale structure, and derive sufficient conditions for convergence of these to a Nash equilibrium. Our numerical experiments demonstrate that the proposed approaches enable orders of magnitude improvements in scalability when computing Nash equilibria in such games. For example, we can solve previously intractable instances involving up to 1 million agents in under 15 minutes. 

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3. Interdependent Security Games in the Stackelberg Style: How First-Mover Advantage Impacts Free-Riding and Security (Under-)Investment

   Huang, Ziyuan ; Naghizadeh, Parinaz ; Liu, Mingyan ( June 2023 , The 22nd Workshop on the Economics of Information Security (WEIS'23))

   Network games are commonly used to capture the strategic interactions among interconnected agents in simultaneous moves. The agents’ actions in a Nash equilibrium must take into account the mutual dependencies connecting them, which is typically obtained by solving a set of fixed point equations. Stackelberg games, on the other hand, model the sequential moves between agents that are categorized as leaders and followers. The corresponding solution concept, the subgame perfect equilibrium, is typically obtained using backward induction. Both game forms enjoy very wide use in the (cyber)security literature, the network game often as a template to study security investment and externality – also referred to as the Interdependent Security (IDS) games – and the Stackelberg game as a formalism to model a variety of attacker-defender scenarios. In this study we examine a model that combines both types of strategic reasoning: the interdependency as well as sequential moves. Specifically, we consider a scenario with a network of interconnected first movers (firms or defenders, whose security efforts and practices collectively determine the security posture of the eco-system) and one or more second movers, the attacker(s), who determine how much effort to exert on attacking the many potential targets. This gives rise to an equilibrium concept that embodies both types of equilibria mentioned above. We will examine how its existence and uniqueness conditions differ from that for a standard network game. Of particular interest are comparisons between the two game forms in terms of effort exerted by the defender(s) and the attacker(s), respectively, and the free-riding behavior among the defenders. 

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4. On non-unique solutions in mean field games

   https://doi.org/10.1109/CDC40024.2019.9029906  

   Hajek, Bruce ; Livesay, Michael ( December 2019 , 2019 58th Conference on Decision and Control)

   The theory of mean field games is a tool to understand noncooperative dynamic stochastic games with a large number of players. Much of the theory has evolved under conditions ensuring uniqueness of the mean field game Nash equilibrium. However, in some situations, typically involving symmetry breaking, non-uniqueness of solutions is an essential feature. To investigate the nature of non-unique solutions, this paper focuses on the technically simple setting where players have one of two states, with continuous time dynamics, and the game is symmetric in the players, and players are restricted to using Markov strategies. All the mean field game Nash equilibria are identified for a symmetric follow the crowd game. Such equilibria correspond to symmetric $\epsilon$-Nash Markov equilibria for $N$ players with $\epsilon$ converging to zero as $N$ goes to infinity. In contrast to the mean field game, there is a unique Nash equilibrium for finite $N.$ It is shown that fluid limits arising from the Nash equilibria for finite $N$ as $N$ goes to infinity are mean field game Nash equilibria, and evidence is given supporting the conjecture that such limits, among all mean field game Nash equilibria, are the ones that are stable fixed points of the mean field best response mapping. 

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5. Subgame solving without common knowledge

   Zhang, B. H. ; Sandholm, T. ( January 2021 , ArXivorg)

   null (Ed.)

   In imperfect-information games, subgame solving is significantly more challenging than in perfect-information games, but in the last few years, such techniques have been developed. They were the key ingredient to the milestone of superhuman play in no-limit Texas hold'em poker. Current subgame-solving techniques analyze the entire common-knowledge closure of the player's current information set, that is, the smallest set of nodes within which it is common knowledge that the current node lies. However, this set is too large to handle in many games. We introduce an approach that overcomes this obstacle, by instead working with only low-order knowledge. Our approach allows an agent, upon arriving at an infoset, to basically prune any node that is no longer reachable, thereby massively reducing the game tree size relative to the common-knowledge subgame. We prove that, as is, our approach can increase exploitability compared to the blueprint strategy. However, we develop three avenues by which safety can be guaranteed. First, safety is guaranteed if the results of subgame solves are incorporated back into the blueprint. Second, we provide a method where safety is achieved by limiting the infosets at which subgame solving is performed. Third, we prove that our approach, when applied at every infoset reached during play, achieves a weaker notion of equilibrium, which we coin affine equilibrium, and which may be of independent interest. We show that affine equilibria cannot be exploited by any Nash strategy of the opponent, so an opponent who wishes to exploit must open herself to counter-exploitation. Even without the safety-guaranteeing additions, experiments on medium-sized games show that our approach always reduced exploitability even when applied at every infoset, and a depth-limited version of it led to--to our knowledge--the first strong AI for the massive challenge problem dark chess. 

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