Solution Set B-1 - Economic Allocation Mechanisms

Let a = (a (i))i=1;:::;n be the jungle equilibrium allocation achieved by using the algorithm in the notes (the agents choose the houses one by one, according to the order of power). Now assume by negation, that another equilibrium exists, with an allocation a0 = (a0 (i))i=1;:::;n 6= (a (i))i=1;:::;n. Let G be the set of agents such that a 0 (i) 6= a (i) 8i2G, and let j be the "most powerful" agent in G. Therefore, all agents who have more power than j own the same house as in a. By the algorithm used to construct a, we know that j cannot get any house better than a (j) when all agents who have more power than j own the same house as in a. This implies that a (j) j a0 (j) : In addition, given all agents who have more power than j own the same house as in a, and given allocation a0, j can move successfully to a (j) : So, in allocation a0, there exists an agent j who can deviate and strictly improve his situation, contradicting the assumption that a0 is an equilibrium.