ICML 2025

Expected Variational Inequalities
Brian Hu Zhang * 1 Ioannis Anagnostides * 1 Emanuel Tewolde 1 2 Ratip Emin Berker 1 2 Gabriele Farina 3
Vincent Conitzer 1 2 4 Tuomas Sandholm 1 5 6 7
Abstract
Variational inequalities (VIs) encompass many
fundamental problems in diverse areas ranging
from engineering to economics and machine learn-
ing.
However, their considerable expressivity
comes at the cost of computational intractabil-
ity. In this paper, we introduce and analyze a
natural relaxation—which we refer to as expected
variational inequalities (EVIs)—where the goal
is to find a distribution that satisfies the VI con-
straint in expectation. By adapting recent tech-
niques from game theory, we show that, unlike
VIs, EVIs can be solved in polynomial time under
general (nonmonotone) operators. EVIs capture
the seminal notion of correlated equilibria, but
enjoy a greater reach beyond games. We also
employ our framework to capture and generalize
several existing disparate results, including from
settings such as smooth games, and games with
coupled constraints or nonconcave utilities.
1. Introduction
Variational inequalities (VIs) provide a unifying framework
for analyzing a wide range of optimization and equilib-
rium problems. They have a host of important applica-
tions in engineering and economics (Facchinei & Pang,
2003), including identifying stationary points in constrained
optimization; computing Nash equilibria in multi-player
games (Nash, 1951), such as Cournot’s classical model of
oligopoly (Cournot, 1838); predicting economic activity—
commodity prices and consumer consumption—in a closed,
competitive economy (Arrow & Debreu, 1954), which is at
the heart of general equilibrium theory; traffic equilibrium
*Equal contribution 1Carnegie Mellon University 2Foundations
of Cooperative AI Lab (FOCAL) 3Massachusetts Institute of Tech-
nology 4University of Oxford 5Strategy Robot, Inc. 6Strategic Ma-
chine, Inc. 7Optimized Markets, Inc. Correspondence to: Brian Hu
Zha...