This paper studies the problem of assigning a set of indivisible objects to a set of agents when monetary transfers are not allowed and agents reveal only ordinal preferences, but random assignments are possible. We offer two characterizations of the probabilistic serial mechanism, which assigns lotteries over objects. We show that it is the only mechanism that satisfies non-wastefulness and ordinal fairness, and the only mechanism that satisfies sd-efficiency, sd-envy-freeness, and weak invariance or weak truncation robustness (where "sd" stands for first-order stochastic dominance). 6 We thank an anonymous referee for suggesting that we weaken HH's definition of truncation robustness to the current definition (Definition 3). Upon showing that this new definition is strong enough to characterize PS, we observed that the proof also extends to the general case where the null object may not exist. This motivated us to obtain our second characterization result using the current definition of weak invariance (Definition 2), which is the counterpart of Definition 3 in environments without the null object. 7 Also, our proof immediately implies that we can weaken sd-efficiency in Theorem 2 and Corollary 2 as in HH and BH. 8 A more recent paper by Heo and Yılmaz (2012) extends the results of BH to the case with weak preferences for Katta and Sethuraman's (2006) extended probabilistic serial correspondence. 9 This axiom was previously introduced by Heo (2013) as one of her auxiliary axioms. She referred to it as "limited invariance."

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Many real-life applications of house allocation problems are dynamic. For example, each year college freshmen move in and seniors move out of on-campus housing. Each student stays on campus for only a few years. A student is a “newcomer” in the beginning and then becomes an “existing tenant.” Motivated by this observation, we introduce a model of house allocation with overlapping generations. In terms of a dynamic rule without monetary transfers, we examine two static rules of serial dictatorship and top trading cycles. We support these seniority-based rules in terms of their dynamic Pareto efficiency and incentive compatibility. (JEL D13, D61, D82)

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