Tacit Coordination of Large Language Models

TL;DR

This work investigates tacit coordination among Large Language Models (LLMs) through Schelling-style focal points, aiming to understand when and why focal points emerge in cooperative and competitive settings. It introduces a formal framework linking Nash equilibria, salience-based focal points, and probabilistic coordination via a softmax mapping, alongside metrics like the Normalised Coordination Index (NCI) and Coordination Index (CI). The study conducts large-scale experiments across Amsterdam and Nottingham human datasets and a Bargaining Table to compare >20 open-source LLMs with humans, showing LLMs often coordinate as well as or better than humans, especially when culture-aware prompting is used, while struggles remain with numerically or culturally nuanced tasks. Key findings include the limited impact of deeper reasoning on coordination, the positive effect of model scale, and the practical importance of aligning LLM salience with human focal points for effective mixed human–AI coordination, with code and prompts made available for reproducibility.

Abstract

In tacit coordination games with multiple outcomes, purely rational solution concepts, such as Nash equilibria, provide no guidance for which equilibrium to choose. Shelling's theory explains how, in these settings, humans coordinate by relying on focal points: solutions or outcomes that naturally arise because they stand out in some way as salient or prominent to all players. This work studies Large Language Models (LLMs) as players in tacit coordination games, and addresses how, when, and why focal points emerge. We compare and quantify the coordination capabilities of LLMs in cooperative and competitive games for which human experiments are available. We also introduce several learning-free strategies to improve the coordination of LLMs, with themselves and with humans. On a selection of heterogeneous open-source models, including Llama, Qwen, and GPT-oss, we discover that LLMs have a remarkable capability to coordinate and often outperform humans, yet fail on common-sense coordination that involves numbers or nuanced cultural archetypes. This paper constitutes the first large-scale assessment of LLMs' tacit coordination within the theoretical and psychological framework of focal points.

Figures (27)

• Figure 1: Left: This paper uses the theoretical framework of Schelling/focal points to study how, why, and when tacit coordination emerges in heterogeneous LLMs. Right: Examples of focality principles that humans leverage when they have to tacitly coordinate include extremeness, centrality, as well as cultural factors kraus2000exploiting.
• Figure 2: Example of symmetry invariant (left) and common ordering (right) orbits that lead to a unique focal point equilibrium (in bold) on the question "Pick a number between $1$ and $100$".
• Figure 3: Left: Illustration of the rules and the pure Nash equilibria of the Amsterdam and Nottingham coordination games bardsley2010explaining. Right: Illustration of the Bargaining Table game mizrahi2020using, a semi-competitive game, and its pure Nash equilibria.
• Figure 4: Normalised Coordination Index (NCI) of humans and LLMs (Llama-3, 3.1, and 3.3 70B; Qwen 2 and 2.5 72B; GPT-oss 20B and 120B) on the Amsterdam and Nottingham datasets.
• Figure 5: The effect of reasoning (low, medium, and high: the darker the red, the higher the reasoning) on the coordination of GPT-oss-120B in Amsterdam and Nottingham. There is no clear evidence that reasoning improves the NCI of LLMs. Same results hold for other settings and prompting techniques (full results in Appendix \ref{['a:reasoning-AN']}).

Theorems & Definitions (1)

• Definition 1.1: Symmetry group and induced partition