Existence of cycling equilibria with far-sighted sharing outside the combinatorial criterion

Determine whether a cycle that violates the stated individual-harmonicity connectivity criterion can be an equilibrium under far-sighted sharing.

Background

Under far-sighted sharing, the paper gives a combinatorial criterion ensuring that deterministic recurrent cycles disappear as the discount factor approaches one. A cycle escaping the criterion must keep at least one player inside an agreement that the player does not dissolve at multiple states.

The paper does not know whether such a cycle can satisfy the equilibrium conditions. Resolving this would identify whether the combinatorial criterion is close to necessary for ruling out patient cycling or whether additional structural arguments exclude all such cycles.