Large Language Models exhibit logical inconsistency across multi-turn inference processes, undermining correctness in complex inferential tasks. Challenges arise from ensuring that outputs align with both factual correctness and human intent. Approaches like single-agent reflection and multi-agent debate frequently prioritize consistency, but at the expense of accuracy. To address this problem, we propose a novel game-theoretic consensus mechanism that enables LLMs to self-check their outputs during the decoding stage of output generation. Our method models the decoding process as a multistage Bayesian Decoding Game, where strategic interactions dynamically converge to a consensus on the most reliable outputs without human feedback or additional training. Remarkably, our game design allows smaller models to outperform much larger models through game mechanisms (e.g., 78.1 LLaMA13B vs. 76.6 PaLM540B). As a model-agnostic method, our approach consistently improves even the latest models, enhancing DeepSeek-7B's performance on MMLU by 12.4%. Our framework effectively balances correctness and consistency, demonstrating that properly designed game-theoretic mechanisms can significantly enhance the self-verification capabilities of language models across various tasks and model architectures.

Inverse problems describe the process of estimating the causal factors from a set of measurements or data. Mapping of often incomplete or degraded data to parameters is ill-posed, thus data-driven iterative solutions are required, for example when reconstructing clean images from poor signals. Diffusion models have shown promise as potent generative tools for solving inverse problems due to their superior reconstruction quality and their compatibility with iterative solvers. However, most existing approaches are limited to linear inverse problems represented as Stochastic Differential Equations (SDEs). This simplification falls short of addressing the challenging nature of real-world problems, leading to amplified cumulative errors and biases. We provide an explanation for this gap through the lens of measure-preserving dynamics of Random Dynamical Systems (RDS) with which we analyse Temporal Distribution Discrepancy and thus introduce a theoretical framework based on RDS for SDE diffusion models. We uncover several strategies that inherently enhance the stability and generalizability of diffusion models for inverse problems and introduce a novel score-based diffusion framework, the Dynamics-aware SDE Diffusion Generative Model (D^3GM). The Measure-preserving property can return the degraded measurement to the original state despite complex degradation with the RDS concept of stability. Our extensive experimental results corroborate the effectiveness of D^3GM across multiple benchmarks including a prominent application for inverse problems, magnetic resonance imaging.