Large-Scale Contextual Market Equilibrium Computation through Deep Learning

TL;DR

This work tackles large-scale contextual market equilibrium computation by introducing MarketFCNet, a neural network-based allocation function x_\theta(b_i,g_j) trained via Augmented Lagrangian methods to satisfy market constraints while decoupling complexity from the number of buyers. It provides unbiased gradient estimators to enable SGD/Adam optimization and defines Nash Gap (NG) as a principled measure of deviation from equilibrium, tying welfare gaps to a saddle-point interpretation. Empirical results on CES utilities show MarketFCNet achieves competitive equilibrium-like performance with dramatically reduced runtime compared to traditional solvers, and demonstrates strong scalability across market sizes and context distributions. The approach promises practical impact for accelerating large-scale contextual market analysis and opens avenues for online and stochastic-budget extensions.

Abstract

Market equilibrium is one of the most fundamental solution concepts in economics and social optimization analysis. Existing works on market equilibrium computation primarily focus on settings with relatively few buyers. Motivated by this, our paper investigates the computation of market equilibrium in scenarios with a large-scale buyer population, where buyers and goods are represented by their contexts. Building on this realistic and generalized contextual market model, we introduce MarketFCNet, a deep learning-based method for approximating market equilibrium. We start by parameterizing the allocation of each good to each buyer using a neural network, which depends solely on the context of the buyer and the good. Next, we propose an efficient method to unbiasedly estimate the loss function of the training algorithm, enabling us to optimize the network parameters through gradient. To evaluate the approximated solution, we propose a metric called Nash Gap, which quantifies the deviation of the given allocation and price pair from the market equilibrium. Experimental results indicate that MarketFCNet delivers competitive performance and significantly lower running times compared to existing methods as the market scale expands, demonstrating the potential of deep learning-based methods to accelerate the approximation of large-scale contextual market equilibrium.

Figures (4)

• Figure 1: Training process of MarketFCNet. On each iteration, the batch of $M$ independent buyers are drawn. each buyer and each good are represented as $k$-dimension context. The $(i,j)$'th element in the allocation matrix represents the allocation computed from $i$'th buyer and $j$'th good. MarketFCNet training process alternates between the training of allocation network and prices. The training of allocation network needs to achieve an unbiased estimator $\widehat{\mathcal{L}}_\rho(x_\theta;\lambda)$ of the loss function $\mathcal{L}_\rho(x_\theta;\lambda)$, followed by gradient descent. The training of prices need to get an unbiased estimator $\widehat{\Delta}\lambda_j$ of $\Delta \lambda_j$, followed by ALMM updating rule $\lambda_j \leftarrow \lambda_j + \beta_t \widehat{\Delta}\lambda_j$.
• Figure 2: The Nash Gap and GPU running time for different approaches: MarketFCNet, EG, and EG-m. Different colors represent different approaches. Market size is fixed with $n=4, 194, 304$ buyers and $m=10$ goods.
• Figure 3: The Nash Gap (left) and GPU time (right) for MarketFCNet, EG, and EG-m. Market size varies from $n=2^{18}, 2^{20}, 2^{22}$ buyers and $m=5, 10, 20$ goods.
• Figure 4: The curve of Nash Gap for different approaches: MarketFCNet, EG, and EG-m. Nash Gap is computed after each training epoch. Buyer size varies from $n= 2^{18},2^{20},2^{22}$, and the number of goods is fixed with $m=10$.

Theorems & Definitions (26)

• theorem thmcountertheorem: gao2020first
• proposition thmcounterproposition
• proposition thmcounterproposition: Unbiased estimator of $\mathcal{L}_p(x_\theta;\bm{\lambda})$
• definition thmcounterdefinition: Log Nash Welfare
• definition thmcounterdefinition: Fixed-price utility and Log Fixed-price Welfare
• definition thmcounterdefinition: Nash Gap
• proposition thmcounterproposition: Price constraints
• theorem thmcountertheorem
• proposition thmcounterproposition: Informal
• corollary thmcountercorollary