Convex strategyproofness with an application to the probabilistic serial mechanism

Abstract

We consider two natural notions of strategyproofness in random objectassignment mechanisms based on ordinal preferences. The two notions are stronger than weak strategyproofness but weaker than strategyproofness. We demonstrate that the two notions are equivalent, provide a geometric characterization of the new intermediate property which we call convex strategyproofness, and then show that the (generalized) probabilistic serial mechanism is, in fact, convexly strategyproof. We finish by showing that the property of weak envy-freeness of the random serial dictatorship can be strengthened in an analogous manner.

Strategy-proofness of the randomized Condorcet voting system

“…Thus, if we relax this condition, we may be able to prove the (thus weaker) strategy-proofness of some voting systems that are not random dictatorships. This is what has been proposed by Bogomolnaia and Moulin (2001) and Balbuzanov (2016), who respectively introduced ordinal efficiency and convex undomination. In this paper, we shall take another path, by studying a very distinct (but still fairly natural) class of preferences based on pairwise comparisons of drawn alternatives, which was previously introduced by Fishburn (1982) and Aziz et al (2015).…”

Partial Strategyproofness: Relaxing Strategyproofness for the Random Assignment Problem

“…Proposition 2 already shows that it implies LD-strategyproofness. Expanding on this insight, we consider other relaxed incentive requirements that have previously been discussed in the context of the assignment problem, namely weak SD-strategyproofness (Bogomolnaia and Moulin, 2001), convex strategyproofness (Balbuzanov, 2015), approximate strategyproofness (Carroll, 2013), and strategyproofness in the large (Azevedo and Budish, 2015). We formalize these requirements and we show in what sense they are implied by partial strategyproofness.…”

“…Various relaxed notions of strategyproofness have been employed in prior work to describe the incentive properties of non-strategyproof assignment mechanisms. Partial strategyproofness provides a unified perspective on these because it implies all of them in meaningful ways (Theorem 3): r-partial strategyproofness for some r ą 0 implies weak SD-strategyproofness (Bogomolnaia and Moulin, 2001) and convex strategyproofness (Balbuzanov, 2015); a positive degree of strategyproofness r ą 0 implies ε-approximate strategyproofness (Carroll, 2013) (where ε is guaranteed to be close to 0 for r sufficiently close to 1); if the degree of strategyproofness of a mechanism converges to 1 in large markets, then this mechanism is strategyproof in the large (Azevedo and Budish, 2015); and the converse holds for neither of these implications. The relation to approximate strategyproofness is especially important because it allows us to further refine the strategic advice we can give: By definition of r-partial strategyproofness, agents are best off reporting their preferences truthfully if their preference intensities for any two objects differ sufficiently (by the factor r); otherwise, if they are close to being indifferent between some objects, then we now know that their potential gain from misreporting may be positive but it is bounded (by the corresponding value of ε).…”

“…We conclude that the axiomatic motivation, the ability to parametrize a broad spectrum of incentive properties, and the unified perspective on incentives make the partial strategyproofness concept an important addition to the mechanism designer's toolbox. To illustrate its usefulness in mechanism design, we provide two applications: First, we show that it yields the most demanding description of the incentive properties of the Probabilistic Serial mechanism in finite settings (refining (Bogomolnaia and Moulin, 2001;Balbuzanov, 2015)) as well as in the limit as markets get large (refining (Kojima and Manea, 2010;Azevedo and Budish, 2015)). Second, we show that the concept can be used to distinguish between two common variants of the Boston mechanism in terms of their incentive properties: Under the classical Boston mechanism (BM) (Abdulkadiroglu and Sönmez, 2003), all applications in round k go to the students' k th choices, but under the adaptive Boston mechanism (ABM) (Alcalde, 1996;Miralles, 2008;Harless, 2015;Dur, 2015), students apply to their respective best available school in each round.…”

Ordinal Bayesian incentive compatibility in random assignment model

“…Thus, in large economies, the PS mechanism is strategy-proof. Balbuzanov (2016) introduce a notion of strategy-proofness which is stronger than weak strategy-proofness and show that the PS mechanism satisfies it. His notion of strategy-proofness is based on the "convex" domination of lotteries, and hence, called convex strategy-proofness.…”