Rewarding moderate behavior in a dynamic Nash Demand Game

I study the following repeated version of Nash’s Demand Game: whenever the demands are not jointly compatible, the player who stated the lower demand (the less greedy player) obtains the following advantage: his offer is the only one “on the table”, and the greedier player needs to respond to this offer by either accepting it (which terminates the game) or rejecting it (which triggers a one-period delay and a re-start of the game). If the feasible set is regular—meaning that the egalitarian point is also utilitarian—the game has a unique subgame perfect equilibrium. The equilibrium outcome is an immediate agreement on the egalitarian point. Regularity of a feasible set is a weakening of symmetry. Under some equilibrium refinement, regularity can be dispensed with.