Nash Equilibria via Stochastic Eigendecomposition

Abstract

This work proposes a novel set of techniques for approximating a Nash equilibrium in a finite, normal-form game. It achieves this by constructing a new reformulation as solving a parameterized system of multivariate polynomials with tunable complexity. In doing so, it forges an itinerant loop from game theory to machine learning and back. We show a Nash equilibrium can be approximated with purely calls to stochastic, iterative variants of singular value decomposition and power iteration, with implications for biological plausibility. We provide pseudocode and experiments demonstrating solving for all equilibria of a general-sum game using only these readily available linear algebra tools.

Figures (6)

• Figure 1: (An Itinerant Loop) We identify a series of bridges— some new, some old, and some a mix— that connects the problem of approximating Nash equilibria in normal-form games to problems in algebraic geometry, linear algebra, machine learning, and back again to finite games (albeit not precisely where the loop began).
• Figure 2: (Projected Gradients) A player's gradient in a normal-form game $\nabla^i_{x_i} \in \mathbb{R}^{\vert \mathcal{A}_i \vert = 2}$ can point off the simplex and need not have zero length at an interior Nash equilibrium. In contrast, the gradient projected onto the tangent space of the simplex, $\Pi_{T\Delta}(\nabla^i_{x_i})$, does necessarily have zero length for all interior Nash equilibria and is the natural notion of "movement" on the simplex.
• Figure 3: (Least Squares vs Uniform) Violin plots of the exploitability attained by our proposed least-squares method (LstSq) versus a uniform strategy profile (Uni) on $10$ thousand games of each size with payoffs normalized to $[0.001, 1]$. Parameter $\tilde{\gamma} = \gamma / \vert \mathcal{A}_i \vert$. Percentages indicate the success rate of LstSq returning a valid output that lies on the product simplex.
• Figure 4: (Macaulay Matrix Growth) An upper bound on the number of rows R and columns C is shown for $2$-player games with various action spaces ($\vert \mathcal{A}_1 \vert, \vert \mathcal{A}_2 \vert$) as inverse temperature $\tau^{-1}$ is varied. The Macaulay matrix can quickly grow to sizes beyond typical memory limits (see equation (\ref{['eqn:macaulay_growth']}) for formulae), however, for fixed game sizes, this growth is actually polynomial for large $\tau^{-1}$, not exponential. The best fit dotted lines provide a visual aid— R and C behave as $\tilde{\mathcal{O}}(\tau^{-\sum_i \vert \mathcal{A}_i \vert})$.
• Figure 5: (EigenGame— Figure 2 reproduced from gempeigengame with permission) Players select unit vectors from a continuous set (the sphere). Trajectories show the dynamics of players updating their strategies simultaneously with gradient descent. A wireframe mesh can be overlayed on the sphere to convert it into a game with a finite strategy set assos2023online.