An ideal market mechanism allocates resources efficiently such that welfare is maximized and sets prices in a way so that the outcome is in a competitive equilibrium and no participant wants to deviate. An important part of the literature discusses Walrasian equilibria and conditions for their existence. We use duality theory to investigate existence of Walrasian equilibria and optimization algorithms to describe auction designs for different market environments in a consistent mathematical framework that allows us to classify the key contributions in the literature and open problems. We focus on auctions with indivisible goods and prove that the relaxed dual winner determination problem is equivalent to the minimization of the Lyapunov function. This allows us to describe central auction designs from the literature in the framework of primal-dual algorithms. We cover important properties for existence of Walrasian equilibria derived from discrete convex analysis, and provide open research questions.

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Applications of combinatorial auctions (CA) as market mechanisms are prevalent in practice, yet their Bayesian Nash equilibria (BNE) remain poorly understood. Analytical solutions are known only for a few cases where the problem can be reformulated as a tractable partial differential equation (PDE). In the general case, finding BNE is known to be computationally hard. Previous work on numerical computation of BNE in auctions has relied either on solving such PDEs explicitly, calculating pointwise best-responses in strategy space, or iteratively solving restricted subgames. In this study, we present a generic yet scalable alternative multi-agent equilibrium learning method that represents strategies as neural networks and applies policy iteration based on gradient dynamics in self-play. Most auctions are ex-post nondifferentiable, so gradients may be unavailable or misleading, and we rely on suitable pseudogradient estimates instead. Although it is well-known that gradient dynamics cannot guarantee convergence to NE in general, we observe fast and robust convergence to approximate BNE in a wide variety of auctions and present a sufficient condition for convergence.