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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. 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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4. Distributed Storage Investment in Power Networks

   https://doi.org/10.23919/ACC.2019.8814777  

   Qin, Junjie; Li, Sen; Poolla, Kameshwar; Varaiya, Pravin (July 2019, 2019 American Control Conference (ACC))

   null (Ed.)

   The value created by aggregating behind-the-meter distributed energy storage devices for grid services depends on how much storage is in the system and the power network operation conditions. To understand whether market-driven distributed storage investment will result in a socially desirable outcome, we formulate and analyze a network storage investment game. By explicitly characterizing the set of Nash equilibria (NE) for two examples, we establish that the uniqueness and efficiency of NE depend critically on the power network conditions. Furthermore, we show it is guaranteed that NE support social welfare for general power networks, provided we include two modifications in our model. These modifications suggest potential directions for regulatory interventions. 

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5. Nash Equilibrium in Discontinuous Games

   https://doi.org/10.1146/annurev-economics-082019-111720  

   Reny, Philip J. (August 2020, Annual Review of Economics)

   We review the discontinuous games literature, with a sharp focus on conditions that ensure the existence of pure and mixed strategy Nash equilibria in strategic form games and of Bayes-Nash equilibria in Bayesian games. 

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