Measuring the Non-Transitivity in Chess

It has long been believed that Chess is the Drosophila of Artificial Intelligence (AI). Studying Chess can productively provide valid knowledge about complex systems. Although remarkable progress has beenmade on solving Chess, the geometrical landscape of Chess in the strategy space is still mysterious. Judging on AI-generated strategies, researchers hypothesised that the strategy space of Chess possesses a spinning top geometry, with the upright axis representing the transitive dimension (e.g., A beats B, B beats C, A beats C), and the radial axis representing the non-transitive dimension (e.g., A beats B, B beats C, C beats A). However, it is unclear whether such a hypothesis holds for real-world strategies. In this paper, we quantify the non-transitivity in Chess through real-world data from human players. Specifically, we performed two ways of non-transitivity ∗Corresponding author quantifications—Nash Clustering and counting the number of RockPaper-Scissor cycles—on over one billion match data from Lichess and FICS. Our findings positively indicate that the strategy space occupied by real-world Chess strategies demonstrates a spinning top geometry, and more importantly, there exists a strong connection between the degree of non-transitivity and the progression of a Chess player’s rating. In particular, high degrees of non-transitivity tend to prevent human players from making progress on their Elo rating, whereas progressions are easier to make at the level of ratings where the degree of non-transitivity is lower. Additionally, we also investigate the implication of the degree of non-transitivity for population-based training methods. By considering fixed-memory Fictitious Play as a proxy, we reach the conclusion that maintaining large-size and diverse populations of strategies is imperative to training effective AI agents in solving Chess types of games. ar X iv :2 11 0. 11 73 7v 1 [ cs .A I] 2 2 O ct 2 02 1 Trovato and Tobin, et al.

[1]  Murray Campbell,et al.  Deep Blue , 2002, Artif. Intell..

[2]  Yujing Hu,et al.  Unifying Behavioral and Response Diversity for Open-ended Learning in Zero-sum Games , 2021, ArXiv.

[3]  Siddhartha Sen,et al.  Aligning Superhuman AI with Human Behavior: Chess as a Model System , 2020, KDD.

[4]  H. Young,et al.  Handbook of Game Theory with Economic Applications , 2015 .

[5]  O. Arenz.  Extensive Games , 2012 .

[6]  Nathan Ensmenger,et al.  Is chess the drosophila of artificial intelligence? A social history of an algorithm , 2012, Social studies of science.

[7]  Thore Graepel,et al.  Re-evaluating evaluation , 2018, NeurIPS.

[8]  Eric van Damme,et al.  Non-Cooperative Games , 2000 .

[9]  Peter Norvig,et al.  Artificial Intelligence: A Modern Approach , 1995 .

[10]  Matthew E. Taylor,et al.  Diverse Auto-Curriculum is Critical for Successful Real-World Multiagent Learning Systems , 2021, AAMAS.

[11]  Ying Wen,et al.  MALib: A Parallel Framework for Population-based Multi-agent Reinforcement Learning , 2021, J. Mach. Learn. Res..

[12]  A. Elo.  The rating of chessplayers, past and present , 1978 .

[13]  Lisa Dresner.  Lectures On The Theory Of Games Am 37 , 2016 .

[14]  Max Jaderberg,et al.  Open-ended Learning in Symmetric Zero-sum Games , 2019, ICML.

[15]  Sanjeev Khanna,et al.  Approximating Longest Directed Paths and Cycles , 2004, ICALP.

[16]  Demis Hassabis,et al.  Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm , 2017, ArXiv.

[17]  Yaodong Yang,et al.  Modelling Behavioural Diversity for Learning in Open-Ended Games , 2021, ICML.

[18]  Yuhao Li,et al.  On the Complexity of Computing Markov Perfect Equilibrium in General-Sum Stochastic Games , 2021, Electron. Colloquium Comput. Complex..

[19]  M. Glickman.  The Glicko system , 2011 .

[20]  Wojciech M. Czarnecki,et al.  Grandmaster level in StarCraft II using multi-agent reinforcement learning , 2019, Nature.

[21]  Claude E. Shannon,et al.  XXII. Programming a Computer for Playing Chess 1 , 1950 .

[22]  Sriram Srinivasan,et al.  OpenSpiel: A Framework for Reinforcement Learning in Games , 2019, ArXiv.

[23]  Max Jaderberg,et al.  Real World Games Look Like Spinning Tops , 2020, NeurIPS.

[24]  Yaodong Yang,et al.  An Overview of Multi-Agent Reinforcement Learning from Game Theoretical Perspective , 2020, ArXiv.

[25]  Pieter Spronck,et al.  Monte-Carlo Tree Search: A New Framework for Game AI , 2008, AIIDE.