@inproceedings{cao-etal-2026-shapleylaw,
title = "{S}hapley{L}aw: A Game-Theoretic Approach to Multilingual Scaling Laws",
author = "Cao, Xuyang and
Liu, Qianying and
Xiao, Chuan and
Oda, Yusuke and
Wang, Jiayi and
Stenetorp, Pontus and
Kawahara, Daisuke and
Onizuka, Makoto and
Kurohashi, Sadao and
Zheng, Shuyuan",
editor = "Lee, Hung-Yi and
Dev, Sunipa and
Li, Jessy and
Elliott, Desmond",
booktitle = "Proceedings of the 2026 Conference on Empirical Methods in Natural Language Processing",
month = oct,
year = "2026",
address = "Budapest, Hungary",
publisher = "Association for Computational Linguistics",
url = "https://aclanthology.org/2026.emnlp-main.1080/",
pages = "24812--24833",
ISBN = "979-8-89176-459-0",
abstract = "In multilingual pretraining, the test loss of a pretrained model is heavily influenced by the proportion of each language in the pretraining data, namely the *language mixture ratios*. Multilingual scaling laws can predict the test loss under different language mixture ratios and can therefore be used to estimate the optimal ratios. However, the current approaches to multilingual scaling laws do not explicitly quantify *cross-lingual transfer* between individual languages, resulting in suboptimal mixture-ratio optimization. In this paper, we consider multilingual pretraining as a cooperative game in which each language acts as a player that jointly contributes to pretraining, gaining the resulting reduction in test loss as the payoff. Consequently, from the perspective of cooperative game theory, we quantify the cross-lingual transfer from each language by its contribution in the game, and propose a game-theoretic multilingual scaling law called *ShapleyLaw*. Our experiments show that ShapleyLaw outperforms baseline methods in model performance prediction and language mixture optimization. Our code and data are available at [https://github.com/YcTtXyLLM/ShapleyLaw](https://github.com/YcTtXyLLM/ShapleyLaw)."
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<abstract>In multilingual pretraining, the test loss of a pretrained model is heavily influenced by the proportion of each language in the pretraining data, namely the *language mixture ratios*. Multilingual scaling laws can predict the test loss under different language mixture ratios and can therefore be used to estimate the optimal ratios. However, the current approaches to multilingual scaling laws do not explicitly quantify *cross-lingual transfer* between individual languages, resulting in suboptimal mixture-ratio optimization. In this paper, we consider multilingual pretraining as a cooperative game in which each language acts as a player that jointly contributes to pretraining, gaining the resulting reduction in test loss as the payoff. Consequently, from the perspective of cooperative game theory, we quantify the cross-lingual transfer from each language by its contribution in the game, and propose a game-theoretic multilingual scaling law called *ShapleyLaw*. Our experiments show that ShapleyLaw outperforms baseline methods in model performance prediction and language mixture optimization. Our code and data are available at [https://github.com/YcTtXyLLM/ShapleyLaw](https://github.com/YcTtXyLLM/ShapleyLaw).</abstract>
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%0 Conference Proceedings
%T ShapleyLaw: A Game-Theoretic Approach to Multilingual Scaling Laws
%A Cao, Xuyang
%A Liu, Qianying
%A Xiao, Chuan
%A Oda, Yusuke
%A Wang, Jiayi
%A Stenetorp, Pontus
%A Kawahara, Daisuke
%A Onizuka, Makoto
%A Kurohashi, Sadao
%A Zheng, Shuyuan
%Y Lee, Hung-Yi
%Y Dev, Sunipa
%Y Li, Jessy
%Y Elliott, Desmond
%S Proceedings of the 2026 Conference on Empirical Methods in Natural Language Processing
%D 2026
%8 October
%I Association for Computational Linguistics
%C Budapest, Hungary
%@ 979-8-89176-459-0
%F cao-etal-2026-shapleylaw
%X In multilingual pretraining, the test loss of a pretrained model is heavily influenced by the proportion of each language in the pretraining data, namely the *language mixture ratios*. Multilingual scaling laws can predict the test loss under different language mixture ratios and can therefore be used to estimate the optimal ratios. However, the current approaches to multilingual scaling laws do not explicitly quantify *cross-lingual transfer* between individual languages, resulting in suboptimal mixture-ratio optimization. In this paper, we consider multilingual pretraining as a cooperative game in which each language acts as a player that jointly contributes to pretraining, gaining the resulting reduction in test loss as the payoff. Consequently, from the perspective of cooperative game theory, we quantify the cross-lingual transfer from each language by its contribution in the game, and propose a game-theoretic multilingual scaling law called *ShapleyLaw*. Our experiments show that ShapleyLaw outperforms baseline methods in model performance prediction and language mixture optimization. Our code and data are available at [https://github.com/YcTtXyLLM/ShapleyLaw](https://github.com/YcTtXyLLM/ShapleyLaw).
%U https://aclanthology.org/2026.emnlp-main.1080/
%P 24812-24833
Markdown (Informal)
[ShapleyLaw: A Game-Theoretic Approach to Multilingual Scaling Laws](https://aclanthology.org/2026.emnlp-main.1080/) (Cao et al., EMNLP 2026)
ACL
- Xuyang Cao, Qianying Liu, Chuan Xiao, Yusuke Oda, Jiayi Wang, Pontus Stenetorp, Daisuke Kawahara, Makoto Onizuka, Sadao Kurohashi, and Shuyuan Zheng. 2026. ShapleyLaw: A Game-Theoretic Approach to Multilingual Scaling Laws. In Proceedings of the 2026 Conference on Empirical Methods in Natural Language Processing, pages 24812–24833, Budapest, Hungary. Association for Computational Linguistics.