Here is a fairly standard “minimax‐in‐infinite‐dimensions’’ argument which shows that nothing special about amenability is needed – as soon as you allow arbitrary finitely‐additive probabilities on your countable group, the multiplication‐game always has a value and a saddle‐point (hence a Nash equilibrium). In particular one does not get a new characterization of amenability by asking “solvable for every W ⊆ G.”

1. Let G=F₂ be the free group, and let
   M := {μ a finitely additive proba...

Here is a “cook‐book’’ for getting the equilibrium temperature that a little planet in circular orbit around a Kerr (or Schwarzschild) black hole will settle to, when the only “heater’’ is the cosmic microwave background at 2.7 K as seen from infinity. In outline you do precisely the same thing you would do in special relativity (blue‐shift plus Doppler beaming of an otherwise isotropic bath) except that now you must combine

• the gravitational red–(or blue–)shift from infinity into your or...