Two axiomatic approaches to the probabilistic serial mechanism

Abstract

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."

Efficient lottery design

“…Two by now well-known examples of this approach are the probabilistic serial (PS) mechanism by Bogomolnaia and Moulin (2001) (BM hereafter) and competitive equilibrium from equal incomes (CEEI) mechanism by Hylland and Zeckhauser (1979), 2 both of which have become the cornerstones of a rapidly growing body of literature concerning stochastic mechanisms (cf. Kojima and Manea (2010), Yilmaz (2010), Hashimoto et al (2014); Budish (2011); Kesten and Ünver (2011), He et al (2012)). BM have pointed out that the RSD outcome may suffer from unambiguous efficiency losses because a stochastic assignment needs to be decomposed into a feasible lottery before actual implementation, ex post considerations are comparably more difficult, if not impossible, to handle in the domain of stochastic assignments.…”

Size versus fairness in the assignment problem

“…Next [8] proposed the alternative Probabilistic Serial mechanism that fares better than RP in terms of e¢ ciency and fairness, but has worse incentive properties: PS is Envy-Free but RP is not, while RP is strategyproof but PS is not. Subsequent work considerably re…ned the comparison of RP and PS; for instance [14] discusses a di¤erent wasteful aspect of RP that PS does not share, while [7], and [15] characterize PS axiomatically. Particularly relevant here is the asymptotic equivalence of PS and RP along certain expansion paths of the economy with a …xed, …nite number of types of objects, while the number of copies of each object grows at roughly the same rate as the number of agents.…”

The Random Assignment Problem with Submodular Constraints on Goods

“…Define A ⊆ Z N ×E ≥0 to be the set of all functions φ : N × E → Z ≥0 such that vectors given by x i = (φ(i, e) | e ∈ E) for all i ∈ N satisfy (16) and (17). Every φ ∈ A determines a feasible allocation x i = (φ(i, e) | e ∈ E) for each agent i ∈ N .…”

“…Consider an independent-flow network N = (G = (S + , S − ; A), c, (S + , ρ + ), (S − , ρ − )), where S + = N , S − = E, G = (S + , S − ; A) is a complete bipartite graph with vertex bi-partition (S + , S − ) and arc set A = S + × S − , c (a) = +∞ (a sufficiently large positive integer) for all a ∈ A, (S − , ρ − ) is an integral polymatroid with rank function ρ − = ρ appearing in (17), and (S + , ρ + ) is a polymatroid with a (modular) rank function ρ + given by ρ + (X ) = d (X ) for all X ⊆ S + = N . 4 For simplicity we also denote the present independent-flow network by N = (N , E, d, (E, ρ)) (see Figure 11).…”