A Jacobi-type Newton method for Nash equilibrium problems with descent guarantees

TL;DR

This work introduces a Jacobi-type Newton method for unconstrained two-player Nash equilibrium problems that uses a prediction of the opponent's action to form quadratic models and enforces descent via Armijo-type line search. The authors establish well-definiteness and convergence to stationarity under standard smoothness and boundedness assumptions, with safeguarding steps to handle stationary players. Numerical experiments on illustrative 1D problems and a 2D facility-location NEP demonstrate improved robustness and a higher likelihood of converging to true equilibria compared to traditional Jacobi or pure Newton approaches. The framework also opens avenues for extensions to multi-player, constrained, and learning-based dynamic settings, as well as links to game-theoretic concepts like potential and ratio-bounded games.

Abstract

A common strategy for solving an unconstrained two-player Nash equilibrium problem with continuous variables is applying Newton's method to the system of nonlinear equations obtained by the corresponding first-order necessary optimality conditions. However, when taking into account the game dynamics, it is not clear what is the goal of each player when considering that they are taking their current decision following Newton's iterates. In this paper we provide an interpretation for Newton's iterate in view of the game dynamics as follows: instead of minimizing the quadratic approximation of their objective function parameterized by the other player current decision (as a typical Jacobi-type strategy), we show that the Newton iterate follows this approach but with the objective function parameterized by a prediction of the other player action, considering that they are following the same Newtonian strategy. This interpretation allows us to present a new Newtonian algorithm where a backtracking procedure is introduced in order to guarantee that the computed Newtonian directions, for each player, are descent directions for their corresponding parameterized functions. Thus, besides favoring global convergence, our algorithm also favors true minimizers instead of maximizers or saddle points, differently from the standard Newton method, which does not consider the minimization structure of the problem in the non-convex case. Thus, our method is more robust in comparison with other Jacobi-type strategies or the pure Newtonian approach, which is corroborated by our illustrative numerical experiments. We also present a proof of the well-definiteness of the algorithm under some standard assumptions, together with a thorough analysis of its convergence properties taking into account the game dynamics.

Figures (2)

• Figure 1: Unidimensional objective function for the facility location problem.
• Figure 2: Numerical behavior in the bidimensional facility location problem.

Theorems & Definitions (18)

• Example 2.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Theorem 4.1