Existence and uniqueness of equilibrium in the base mean field game

Establish existence and uniqueness of equilibrium for the five-parameter mean field game of irreversible attack decisions with public-remedy coupling, optimal stopping, and no realized removal, beyond the numerical search evidence reported for the base model.

Background

The base model describes agents who decide when, if ever, to attack a shared record. Their payoffs combine a private remedy, a public remedy arriving at a rate proportional to the attacking mass, and a perceived detection hazard. Numerical fixed-point searches found no multiplicity over the explored parameter box for the base model, whereas the endogenous-capability extension produced multiple equilibria.

The paper explicitly distinguishes this computational observation from a mathematical theorem. It suggests that a monotonicity argument for the best-response map or a relaxed mean-field formulation might establish the result, but no proof is supplied. The related common-noise formulation introduces an additional unresolved equilibrium problem when the public discovery process remains coupled to the mean field.

References

That is a numerical statement over a box and not a uniqueness theorem; existence and uniqueness of equilibrium for the model as stated are open, and are plausibly within reach of a monotonicity argument on the best-response map under the optimal rule, or of the relaxed formulation of under either.

— Mean field games as a tool for AI safety: a worked example from the July 2026 Hugging Face incident  (2610.00902 - Graber, 1 Oct 2026) in Section 3.2, subsection “Strategic substitutes,” and Section 7, item 7 of “What is assumed and not derived”