Continuous-Time Equilibrium Search Model

• Continuous-Time Equilibrium Search Models are dynamic frameworks that employ differential equations, variational inequalities, and control techniques to characterize equilibrium outcomes.
• They generalize classical tâtonnement by incorporating mirror extragradient methods and robust algorithms, ensuring convergence under monotonicity and Lipschitz conditions.
• These models are applied to matching markets, Nash games, and labor search, offering scalable computational schemes for large-scale and high-dimensional systems.

A continuous-time equilibrium search model is a class of dynamic frameworks for characterizing and computing equilibrium allocations, prices, or strategies in economic, game-theoretic, or network settings when agents interact in continuous time, and market clearing or equilibrium selection occurs as a result of endogenous adjustment processes. These models generalize discrete-time equilibrium approaches, leveraging the theory of differential equations, stochastic calculus, variational inequalities, and control to analyze convergence, stability, and optimality of equilibrium outcomes. Recent advances focus on rigorous convergence guarantees (e.g., via monotonicity or Minty-type conditions), tractable algorithms (such as mirror-extragradient or integral dynamics), high-dimensional and non-smooth environments, and practical computational schemes for large-scale systems (Goktas et al., 17 Feb 2025, Immorlica et al., 2021, Buhai, 7 Dec 2025, Persis et al., 2018, Bianchi et al., 2019).

1. Foundational Frameworks: Dynamic Price and Policy Adjustment in Continuous Time

Classical continuous-time equilibrium search models emerged in general equilibrium theory, notably in the Walras–Hurwicz–Uzawa tâtonnement dynamics, where a price vector $p(t) \in \mathbb{R}^n_+$ evolves as

$\frac{dp}{dt} = z(p),$

with $z(p)$ the excess-demand function. Each price rises proportionally to the excess demand for its corresponding good, encoding the principle of market adjustment toward a clearing equilibrium. Under regularity conditions—such as the Weak Axiom of Revealed Preferences (WARP)—these dynamics are globally convergent, as shown by the equivalence to monotone operator theory (Goktas et al., 17 Feb 2025).

However, in non-trivially constructed economies (such as the Scarf economy), classical tâtonnement cycles persistently, reflecting the failure of naive continuous-time adjustment in the presence of non-monotonicity or degeneracy. This motivates generalizing from classical price adjustment ODEs to operator-theoretic settings, such as variational inequalities (VIs) and fixed-point approaches, and extending to game-theoretic, network, or search/matching environments.

2. Variational Inequality Formulation and the Minty Condition

The modern approach reframes equilibrium as a variational inequality. For a closed convex price domain $\Xi \subset \mathbb{R}^n$, define $F(p) = -z(p)$. The strong (Stampacchia) VI $\operatorname{VI}(\Xi, F)$ seeks $p^* \in \Xi$ such that

$$\langle F(p^*), p - p^* \rangle \ge 0 \quad \forall\, p \in \Xi,$$

i.e., $\langle z(p^*), p^* - p \rangle \ge 0$ for all $p$. A solution to this VI is equivalent to a Walrasian equilibrium, satisfying feasibility and the budget-balancing property of Walras’s law (Goktas et al., 17 Feb 2025).

Critical for existence and algorithmic tractability is the Minty condition—guaranteeing the existence of a weak VI solution if there exists $\hat{p} \in \Xi$ with

$$\langle F(p), \hat{p} - p \rangle \ge 0 \ \forall\, p \in \Xi.$$

This holds in all balanced economies (homogeneity of $z$ and weak Walras’s law). Under mild Bregman-continuity or Lipschitz conditions on $z$, VI-based models sidestep failures of convergence that beset classical tâtonnement.

3. Algorithmic and Dynamic Models: Mirror Extragradient and Extensions

To achieve both theoretical convergence and computational practicality, recent work introduces the mirror extragradient ("mirror extratâtonnement") process. Let $h(\cdot)$ be a $1$-strongly convex, $\alpha$-smooth mirror kernel on $\Xi$. The discrete iterates are

$$\begin{aligned} p^{k+\frac{1}{2}} &= \arg\min_{p \in \Xi} \left\{ \langle z(p^k), p-p^k \rangle + \frac{1}{2\eta} D_h(p, p^k) \right\}, \ p^{k+1} &= \arg\min_{p \in \Xi} \left\{ \langle z(p^{k+\frac{1}{2}}), p-p^k \rangle + \frac{1}{2\eta} D_h(p, p^k) \right\}, \end{aligned}$$

where $$D_h(p, q) = h(p) - h(q) - \langle \nabla h(q), p - q \rangle$$ is the Bregman divergence. For $h(p) = \frac{1}{2}\|p\|^2$, this recovers the Korpelevich extragradient. In continuous time, the method yields a two-stage implicit dynamics that generalizes classical tâtonnement and is robust even in pathologically non-monotone economies (Goktas et al., 17 Feb 2025).

Extensions include projected-gradient flows, primal–dual integral dynamics for Nash/aggregative games (Bianchi et al., 2019, Persis et al., 2018), and FBSDE-based methods for dynamic games with asymmetric information (Qiao et al., 14 Jun 2025).

4. Rigorous Convergence Guarantees in Infinite Dimensions

For the mirror extragradient process, under Minty and Bregman/Lipschitz continuity conditions, and for step size $\eta \le 1/(\sqrt{2} L)$, a non-asymptotic convergence rate of order $O(1/\sqrt{k})$ is achieved: $$\max_{p \in \Xi} \langle z(p^{k+\frac{1}{2}}), p^{k+\frac{1}{2}} - p \rangle \le \frac{C}{\eta} \sqrt{\frac{D_h(p^*, p^0)}{k}},$$ for $p^*$ a solution and $C$ depending on $\alpha$ and the diameter of $\Xi$. This ensures $\epsilon$-feasibility and $\epsilon$-Walras’s law after $O(1/\epsilon^2)$ steps. Convergence is global and polynomial-time in economies with bounded elasticity, including those that defeat classical tâtonnement (e.g., Scarf) (Goktas et al., 17 Feb 2025).

In mean field game and matching settings, analogous results are established: e.g., the stationary mean field game for wage dispersion with on-the-job search admits a unique stationary equilibrium under Lasry–Lions monotonicity and general regularity conditions (Buhai, 7 Dec 2025).

5. Applications: Matching, Nash Network Games, and Search

Continuous-time equilibrium search frameworks are now standard across diverse domains:

• Matching Platforms: The platform optimizes directed meeting rates in a two-sided market subject to flow-balance and capacity constraints. Existence and uniqueness of stationary equilibrium in agent thresholds and steady-state distributions are analytically characterized. Approximation algorithms achieve at least 1/4 of the first-best social welfare; the problem is NP-hard to improve beyond a constant factor (Immorlica et al., 2021).
• Aggregative and Generalized Nash Games: Projected continuous-time integral dynamics ensure exponential convergence to aggregative equilibria when monotonicity conditions hold, with performance guarantees improved over prior work, especially as the number of agents grows large (Bianchi et al., 2019, Persis et al., 2018).
• Labor Markets with Search: The mean field game structure for wage dispersion and job-to-job mobility characterizes equilibrium distributions as a coupled system of coupled HJB and Kolmogorov–Fokker–Planck equations, with free-boundary rules for endogenous separation and policy feedback (Buhai, 7 Dec 2025).
• Dynamic Insider Trading: In the continuous-time Kyle-Back model, the equilibrium is characterized by a forward-backward SDE (FBSDE) system describing the insider and market maker’s joint strategies, allowing for non-Markovian, non-PDE-solvable equilibria (Qiao et al., 14 Jun 2025).

6. Computational Implementation and Empirical Calibration

Computationally, these models leverage finite-difference, finite-volume, and operator-splitting techniques:

• In general equilibrium, mirror extratâtonnement is demonstrated to converge robustly in large-scale Arrow–Debreu economies with Cobb-Douglas, CES, Leontief, and linear utilities, even in cases known to be PPAD-complete (Goktas et al., 17 Feb 2025).
• Matching and wage models employ monotone finite-difference discretizations and policy iteration for coupled HJB/Kolmogorov systems, compatible with large-scale empirical calibration against microdata (Buhai, 7 Dec 2025, Immorlica et al., 2021).
• Distributed Nash computation employs consensus-based and ISS-cascade robust integral controllers, scalable to large networks (Bianchi et al., 2019).

7. Broader Implications and Future Directions

Continuous-time equilibrium search models, grounded in variational inequality and operator-theoretic frameworks, systematically unify the analysis of equilibrium computation across economics, network games, and search-theoretic models. The mirror extragradient methodology resolves classical non-convergence pathologies, introduces robust global convergence theorems, and extends the tractable computation of equilibria to high-dimensional and empirical contexts. Recent advances demonstrate that the key mathematical barriers to practical computation are regularity and monotonicity/bounded elasticity, not inherent computational intractability. Future developments will likely focus on relaxing regularity conditions, exploiting sparsity/structure, fully stochastic (non-Markov) environments, and endogenizing network structures or agent heterogeneity (Goktas et al., 17 Feb 2025, Buhai, 7 Dec 2025, Qiao et al., 14 Jun 2025).