Paraskevas V. Lekeas

Paraskevas V. Lekeas, Giorgos Stamatopoulos

We revisit games in partition function form, i.e. cooperative games where the payoff of a coalition depends on the partition of the entire set of players. We assume that each coalition computes its worth having probabilistic beliefs over the coalitional behavior of the outsiders, i.e., it assigns various probability distributions over the set of partitions that the outsiders can form. These beliefs are not necessarily consistent with respect to the actual choices of the outsiders. We apply this framework to symmetric partition function form games characterized by either positive or negative externalities and we derive conditions on coalitional beliefs that guarantee the non-emptiness of the core of the induced games.

Paraskevas V. Lekeas, Giorgos Stamatopoulos

LLM agents are known to deviate from Nash equilibria in strategic interactions, but nobody has looked inside the model to understand why, or asked whether the deviation can be reversed. We do both. Working with four open-source models (Llama-3 and Qwen2.5, 8B to 72B parameters) playing four canonical two-player games, we establish the behavioral picture through self-play and cross-play experiments, then open up the 32-layer Llama-3-8B model and examine what actually happens during a strategic decision. The mechanistic findings are clear. Opponent history is encoded with near-perfect fidelity at the first layer (96% probe accuracy) and consumed progressively by later ones, while Nash action encoding is weak throughout, never exceeding 56%. There is no dedicated Nash module. Instead, the model privately favors the Nash action through most of its forward pass, but a prosocial override concentrated in the final layers reverses this, reaching 84% probability of cooperation at layer 30. When we inject a learned Nash direction into the residual stream, the behavior shifts bidirectionally, confirmed through concept clamping. The behavioral experiments surface six scale- and architecture-dependent findings, the most notable being that chain-of-thought reasoning worsens Nash play in small models but achieves near-perfect Nash play above 70B parameters. The cross-play experiments reveal three phenomena invisible in self-play: a small model can unravel any partner's cooperation by defecting early; two large models reinforce each other's cooperative instincts indefinitely; and who moves first in a coordination game determines which Nash equilibrium the system reaches. LLMs do not lack Nash-playing competence. They compute it, then suppress it.