Game Theory Market Design

Problem 1. Top-Trading Cycles (TTC) Consider the housing market (1, H,>, j) in which [1] = [H] = n, u denotes the agents' initial endowment, and > is a profile of strict preferences. (a) Let T be the number of rounds of the TTC algorithm. (i) Show that if T = 1, then the economy has a unique Pareto efficient allocation. (ii) Show that T = n if and only if the initial allocation is Pareto efficient. (b) Show that it is possible for a coalition of agents to exchange house prior to running TTC so that at least one agent in the coalition is better off strictly and all in the coalition are better off weakly compared to the result of running TTC from the initial allocation. Problem 1. Top-Trading Cycles (TTC) Consider the housing market (1, H,>, j) in which [1] = [H] = n, u denotes the agents' initial endowment, and > is a profile of strict preferences. (a) Let T be the number of rounds of the TTC algorithm. (i) Show that if T = 1, then the economy has a unique Pareto efficient allocation. (ii) Show that T = n if and only if the initial allocation is Pareto efficient. (b) Show that it is possible for a coalition of agents to exchange house prior to running TTC so that at least one agent in the coalition is better off strictly and all in the coalition are better off weakly compared to the result of running TTC from the initial allocation