RecurBid: A Multi-Scale Recursive Agent with Differentiable Dual Reflection for Real-Time Bidding

Real-Time Bidding (RTB) serves as the economic backbone of digital advertising, demanding optimal constrained decision-making across millions of high-frequency second-price and first-price auctions under volatile market dynamics and rigid budget constraints. While recent pioneering attempts to incorporate Large Language Model (LLM) agents into RTB (such as RTBAgent) have demonstrated the promise of cognitive reasoning, existing agent frameworks suffer from four fundamental structural pathologies: (1) $\text{temporal horizon misalignment}$, where flat single-pass prompting fails to bridge multi-day macro budget planning, intra-day meso pacing, and sub-millisecond micro-auction execution; (2) $\text{the stationary market fallacy}$, ignoring competitive opponent dynamics and falling prey to the winner's curse; (3) $\text{absence of theoretical stability}$, lacking convergence guarantees for budget pacing; and (4) $\text{prohibitive inference latency}$, rendering hundred-millisecond autoregressive token generation unusable within standard ad exchange Service Level Agreements ($<20\text{–}30\text{ms}$). To overcome these foundational limitations, we introduce RecurBid, a principled multi-scale recursive agent framework that integrates cognitive reasoning with differentiable dual control for real-time bidding. RecurBid is founded upon four synergistic pillars: (1) $\text{Hierarchical Temporal Recursion (HTR)}$, which formally decomposes the constrained POMDP across nested Macro-Meso-Micro hierarchies through a recursive Bellman formulation; (2) $\text{Recursive Theory-of-Mind (R-ToM)}$, executing bounded Level-$k$ opponent belief filtering to anticipate competitor pacing adaptations and eliminate adverse selection; (3) $\text{Differentiable Recursive Dual Engine (R-Dual)}$, a micro-level state-space controller with provable Lyapunov contraction mapping guarantees, ensuring exponential pacing convergence and zero asymptotic budget overspend; and (4) $\text{Recursive Tree-of-Thought Reflection (R-Tree)}$, an asynchronous background reasoning engine that simulates counterfactual pacing trajectories and distills strategic guidance into closed-form, sub-millisecond analytical micro-bidding ($<0.42\text{ms}$ per request). We provide rigorous theoretical proofs for asymptotic budget stability ($\gamma$-contraction), bounded Level-$k$ competitive regret $\mathcal{O}(K^{-1}\sqrt{T\log |\mathcal{A}|})$, and Pareto-optimality over flat agents. Extensive empirical evaluations across three benchmark datasets (iPinYou multi-campaigns, Criteo attribution, and Taobao ad exchange) against 12 competitive baselines demonstrate that RecurBid achieves +26.4% higher conversions, reduces Effective Cost Per Click (eCPC) by 31.8%, reaches 99.4% budget utilization without morning starvation, and establishes state-of-the-art bidding performance under non-stationary market conditions.