We investigate first-order notions of correlated equilibria in smooth games, in which players do not incur any regret against small modifications of their actions prescribed by some vector field. We define two such notions, based on local deviations and on stationarity of the distribution, and identify the notion of coarseness as the setting where the strategy modifications are prescribed by gradient fields. For coarse equilibria, we prove that online (projected) gradient ascent has a universal approximation property for both variants of equilibrium; in the self-play setting, every differentiable function induces an equilibrium constraint, the approximation error of which depends on the modulus of continuity and magnitude of the gradient. In the adversarial setting, we instead obtain a characterisation of regret guarantees against continuous strategy modifications satisfied by projected gradient ascent; these are precisely deviations induced by gradient fields tangent to the action set. We also provide a generalisation of the Lagrangian Hedging framework, which identifies a novel refinement of correlated equilibrium which is tractable to approximate.

Projected gradient ascent is known to satisfy no-external regret as a learning algorithm. However, recent empirical work shows that projected gradient ascent often finds the Nash equilibrium in settings beyond two-player zero-sum interactions or potential games, including those where the set of coarse correlated equilibria is very large. We show that gradient ascent in fact satisfies a stronger class of linear Φ-regret in normal-form games; resulting in a refined solution concept which we dub semicoarse correlated equilibria. Our theoretical analysis of the discretised Bertrand competition mirrors those recently established for mean-based learning in first-price auctions. With at least two firms of lowest marginal cost, Nash equilibria emerge as the only semicoarse equilibria under concavity conditions on firm profits. In first-price auctions, the granularity of the bid space affects semicoarse equilibria, but finer granularity for lower bids also induces convergence to Nash equilibria. Unlike previous work that aims to prove convergence to a Nash equilibrium that often relies on epoch based analysis and probability theoretic machinery, our LP-based duality approach enables a simple and tractable analysis of equilibrium selection under gradient-based learning.

We study the Bayesian coarse correlated equilibrium (BCCE) of continuous and discretised first-price and all-pay auctions under the standard symmetric independent private-values model. Our goal is to determine how the canonical Bayes-Nash equilibrium (BNE) of the auction relates to the outcome when all buyers bid following no-regret algorithms. Numerical experiments show that in two buyer first-price auctions the Wasserstein-2 distance of buyers’ marginal bid distributions decline as O (1/n ) in the discretisation size in instances where the prior distribution is concave, whereas all-pay auctions exhibit similar behaviour without prior dependence. To explain this convergence to a near-equilibrium, we study uniqueness of the BCCE of the continuous auction, resulting in proofs of convergence of deterministic self-play to a near equilibrium outcome in these auctions. In the all-pay auction, we show that independent of the prior distribution there is a unique BCCE with symmetric, differentiable, and increasing bidding strategies, which is equivalent to the unique strict BNE. In the first-price auction, either the prior is strictly concave or the learning algorithm has to be restricted to strictly increasing strategies. Without such strong assumptions, no-regret algorithms can end up in low-price pooling strategies.

According to the fundamental theorems of welfare economics, any competitive equilibrium is Pareto efficient. Unfortunately, competitive equilibrium prices only exist under strong assumptions such as perfectly divisible goods and convex preferences. In many real-world markets, participants have non-convex preferences and the allocation problem needs to consider complex constraints. Electricity markets are a prime example, but similar problems appear in many real-world markets, which has led to a growing literature in market design.