On the Characterization of Local Nash Equilibria in Continuous Games

By Lillian J. Ratliff, Samuel A. Burden, S. Sastry. 2014

157 citations recorded by the source corpus.

We present a unified framework for characterizing local Nash equilibria in continuous games on either infinite-dimensional or finite-dimensional non-convex strategy spaces. We provide intrinsic necessary and sufficient first- and second-order conditions ensuring strategies constitute local Nash equilibria. We term points satisfying the sufficient conditions differential Nash equilibria. Further, we provide a sufficient condition (non-degeneracy) guaranteeing differential Nash equilibria are isolated and show that such equilibria are structurally stable. We present tutorial examples to illustrate our results and highlight degeneracies that can arise in continuous games.

Subfields: Economics and Econometrics, Management Science and Operations Research, Public Health, Environmental and Occupational Health, Game & Decision Theory

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In Economics and Econometrics

1. Simon Lacoste-Julien
2. Pascal Vincent
3. Jimmy Ba
4. Hugo Berard
5. Guodong Zhang
6. Gauthier Gidel

In Management Science and Operations Research

1. Simon Lacoste-Julien
2. Gauthier Gidel
3. Guodong Zhang
4. Pascal Vincent
5. Hugo Berard
6. Amjad Almahairi
7. Guoqiang Hu
8. Jimmy Ba

In Public Health, Environmental and Occupational Health

1. Simon Lacoste-Julien
2. Guoqiang Hu
3. Gauthier Gidel

In Game & Decision Theory

1. Simon Lacoste-Julien
2. Pascal Vincent
3. Guoqiang Hu
4. Jimmy Ba
5. Maojiao Ye
6. Hugo Berard
7. Gauthier Gidel

Citations in the corpus

• A Closer Look at the Optimization Landscapes of Generative Adversarial Networks (2019, background): “Ratliff et al. (2016) derived some derivative-based necessary and sufficient conditions for being a LNE.”
• A Variational Inequality Perspective on Generative Adversarial Networks (2018, background): “tive and provide an interesting perspective on GANs, interpreting the training as the research of a two-players game equilibrium. A study of the continuous version of two player games has been done byRatliff et al. (2016) and interesting non-convex results have been proved in the context of GANs byG”
• Convergence of Learning Dynamics in Stackelberg Games (2019, unknown): “Definition 3 (Differential Nash Equilibrium [34]).”
• On Solving Minimax Optimization Locally: A Follow-the-Ridge Approach (2019, background): “As a special case of Stackelberg games (Ratliff et al., 2016) in the zero-sum setting, minimax optimization concerns the problem of solving min x ∈X max y ∈Y f ( x , y ) .”
• A Variational Inequality Perspective on Generative Adversarial Nets (2018, background): “A study of the continuous version of two player games has been done by Ratliff et al. (2016) and interesting non-convex results have been proved in the context of GANs by Grnarova et al. (2018).”
• Distributed Nash Equilibrium Seeking by a Consensus Based Approach (2016, background): “Noticing that k̄H is Hurwitz, the Nash equilibrium is exponentially stable under the gradient play in (19).”