A differentiable path-following method to compute Nash equilibria in robust normal-form games

The robust normal-form game formulated by Aghassi and Bertsimas [Robust game theory. Math Program. 2006;107(1-2):231-273. doi:10.1007/s10107-005-0686-0] makes an effective paradigm to deal with payoff uncertainty and has many applications in decision analysis. The computation of Nash equilibria plays an important role in these applications. Nonetheless, the existing method fails to converge globally and it remains a challenging problem to find such an equilibrium. To meet this challenge, this paper develops a differentiable path-following method to compute Nash equilibria in robust normal-form games when the payoff uncertainty can be characterized as a polytope. Incorporating logarithmic-barrier terms into payoff functions with an extra variable, we constitute a logarithmic-barrier robust normal-form game in which each player solves a strictly convex optimization problem. An application of the optimality conditions together with the equilibrium condition yields a polynomial equilibrium system. With this polynomial equilibrium system, we establish the existence of a smooth path that starts from an arbitrary totally mixed strategy profile and ends at a Nash equilibrium. For the purpose of numerical comparison, we secure one more smooth path from a convex-quadratic-penalty robust normal-form game. Numerical results show that the logarithmic-barrier path-following method significantly outperforms the convex-quadratic-penalty path-following method.