Probabilistic serial mechanism for multi-type resource allocation

In multi-type resource allocation (MTRA) problems, there are d≥ 2 types of items, and n agents who each demand one unit of items of each type and have strict linear preferences over bundles consisting of one item of each type. For MTRAs with indivisible items, our first result is an impossibility theorem that is in direct contrast to the single type (d= 1) setting: no mechanism, the output of which is always decomposable into a probability distribution over discrete assignments (where no item is split between agents), can satisfy both sd-efficiency and sd-envy-freeness. We show that this impossibility result is circumvented under the natural assumption of lexicographic preferences. We provide lexicographic probabilistic serial (LexiPS) as an extension of the probabilistic serial (PS) mechanism for MTRAs with lexicographic preferences, and prove that LexiPS satisfies sd-efficiency and sd-envy-freeness, retaining the desirable properties of PS. Moreover, LexiPS satisfies sd-weak-strategyproofness when agents are not allowed to misreport their importance orders. For MTRAs with divisible items, we show that the existing multi-type probabilistic serial (MPS) mechanism satisfies the stronger efficiency notion of lexi-efficiency, and is sd-envy-free under strict linear preferences and sd-weak-strategyproof under lexicographic preferences. We also prove that MPS can be characterized both by leximin-optimality and by item-wise ordinal fairness, and the family of eating algorithms which MPS belongs to can be characterized by lexi-efficiency.