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CoopQ: Cooperative Game Inspired Layerwise Mixed Precision Quantization for LLMs

Large Language Models (LLMs) promise impressive capabilities, yet their multi-billion-parameter scale makes on-device or low-resource deployment prohibitive. Mixed-precision quantization offers a compelling solution, but existing methods struggle when the average precision drops below four bits, as they rely on isolated, layer-specific metrics that overlook critical inter-layer interactions affecting overall performance. To address these limitations, we first frame the mixed-precision quantization problem as a cooperative game among layers and introduce Shapley-based Progressive Quantization Estimation (SPQE) to efficiently obtain accurate Shapley estimates of layer sensitivities and inter-layer interactions. Leveraging the SPQE estimates, we propose Cooperative Game Inspired Mixed-Precision Quantization (CoopQ) which translates these Shapley estimates into a binary quadratic optimization formulation, assigning either 2 or 4-bit precision to layers under strict memory constraints. Comprehensive experiments conducted on Llama-3, Gemma-2, and Qwen-3 models across three independent PTQ backends (Quanto, HQQ, GPTQ) demonstrate CoopQ's scalability and consistently superior performance compared to methods relying solely on isolated metrics. Across average precisions spanning 4 bit down to 2 bit, CoopQ cuts Perplexity by 20 - 80 % relative to the best baseline, with the margin growing as the bit-width tightens.

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