Conjecture on equitable equilibrium under a swap-symmetric, publicly known allocation rule

Prove the conjecture that if the time slot allocation rule A(y) is publicly known to all Railway Undertakings and possesses swap-symmetry—meaning that swapping the requests y_o of two Railway Undertakings causes the Infrastructure Manager to swap their time slot assignments—then the Nash equilibrium of the bidding game is equitable, with indistinguishable Railway Undertakings achieving the same revenue and the Infrastructure Manager playing a neutral role.

Background

The authors paper how different time slot allocation rules impact market equilibrium in a liberalized railway corridor. They compare a priority rule, which tends to induce dominance by the incumbent, with an equity rule aimed at balancing deviations and fairness.

While heuristic results suggest the equity rule can yield a fair equilibrium, the exact method shows smaller deviations but still exhibits revenue imbalances. Motivated by this, the authors formulate a conjecture about conditions on the allocation rule A(y) that would guarantee an equitable equilibrium.