In this section, we present our first main result that ABM has intermediate incentive properties: while it is not strategyproof, it satisfies the intermediate incentive requirement of partial strategyproofness (Mennle and Seuken, 2015b). This establishes a hierarchy of manipulability between DA, ABM, and NBM. Our finding is in contrast with the rather surprising fact that a comparison by vulnerability to manipulation (Pathak and S¨onmez,
2013) fails to differentiate between NBM and ABM, except in very special cases.
3.5.1 Failure of Comparison by Vulnerability to Manipulation
It is well-known that DA is strategyproof (Roth, 1982) while NBM is not even weakly strategyproof (Proposition 11 in Appendix 3.B.1). Even though ABM is not fully strategyproof, intuitively, it should have better incentive properties than NBM: under ABM, students automatically skip exhausted schools, which removes some obvious opportunities for manipulation. The motivating example in the introduction further
3.5 Incentives for Truth-telling supports this intuition. However, a formal justification for this intuition has remained elusive so far.
One may hope to obtain a distinction via the vulnerability to manipulation concept (Pathak and S¨onmez, 2013). There are three ways to conduct this comparison. Unfor- tunately, neither of them delivers satisfactory results: first, for fixed priority profiles π P ΠM, NBMπ
is indeed as manipulable as ABMπ (Dur, 2015). However, the strict comparison that NBMπ is more manipulable than ABMπ only holds if some students are unacceptable at some schools. Second, the comparison cannot be strengthened to the statement that NBMπ is strongly as manipulable as ABMπ (see Examples 8& 9in
Appendix 3.A.3). Third, when ties are broken randomly, as is common in school choice, then the mechanisms are not even comparable by the weaker as manipulable as-relation (see Examples 6 & 7in Appendix 3.A.2).
This highlights that a different approach must be taken to obtain a conclusive compar- ison of ABM and NBM by their incentive properties. To this end, we employ the partial strategyproofness concept, which we review in the next section.
3.5.2 Review of Partial Strategyproofness
In (Mennle and Seuken,2015b), we have shown that strategyproofness can be decomposed into three simple axioms. These axioms restrict the way in which a mechanism may change the assignment of some student when that student changes her report by swapping two consecutive schools in her reported preference order, e.g., from P : a→ b to P✶ : b → a. ϕ is called upper invariant if this swap leaves the student’s assignment unchanged for any school that she strictly prefers to a, and ϕ is called lower invariant if it leaves her assignment unchanged for any school that she likes strictly less than b. Finally, ϕ is called swap monotonic if the swap either does not lead to a change of the student’s assignment at all, or if it induces any change, then her probability for a must decrease strictly, and her probability for b must increase strictly.
Fact 3 (Mennle and Seuken, 2015b). A mechanism is strategyproof if and only if it is
upper invariant, swap monotonic, and lower invariant.
Now suppose that a student i has a vNM utility function ui that is consistent with her
preference order Pi. We say that ui satisfies uniformly relatively bounded indifference
Pi : a→ b we have that r ✂ ui♣aq ✁ min jPM ui♣jq ✡ ➙ ui♣bq ✁ min jPM ui♣jq, (202)
This implies that the factor difference between i’s (normalized) preference intensity for a over b is a least 1④r. Lower r means that the student differentiates more strongly, while higher r allows her to be closer to indifferent between the two schools a and b.
Definition 18 (Partially Strategyproof). For a given setting (i.e., set of students, set of schools, and school capacities), a mechanism ϕ is r-partially strategyproof if truthful reporting is a dominant strategy for any student whose vNM utility ui satisfies URBI(r).
ϕ is partially strategyproof if it is r-partially strategyproof for some r → 0.
Fact 4 (Mennle and Seuken, 2015b). For a given setting, a mechanism is partially
strategyproof if and only if it is swap monotonic and upper invariant.
Partial strategyproofness is implied by strategyproofness, and it implies weak strate- gyproofness (Bogomolnaia and Moulin, 2001), convex strategyproofness (Balbuzanov,
2015), approximate strategyproofness (Carroll, 2013), strategyproofness in the large (for rÑ 1) (Azevedo and Budish, 2015), and lexicographic strategyproofness (Cho, 2012). Thus, partial strategyproofness can be understood as an intermediate incentive require- ment. We further discuss its implications in the context of our partial strategyproofness result for ABM in Section 3.5.3.
3.5.3 Partial Strategyproofness of ABM
Our first main result formally establishes that the incentive properties of ABM are in fact intermediate between those of DA and NBM.
Theorem 9. ABMU
is partially strategyproof but not strategyproof.
Proof Outline (formal proof in Appendix 3.B.2). We prove partial strategyproofness of
ABMU
by showing upper invariance and swap monotonicity and using Fact4. For upper invariance, we first show that ABMπ is upper invariant for any priority profile π, and then we show that this property is inherited by any mechanism that randomly selects the priority profile π according to some priority distribution P. The more challenging proof is swap monotonicity: we first observe that ABMπ is always monotonic (i.e., bringing a school up in one’s ranking never decreases the chances of obtaining that
3.5 Incentives for Truth-telling school). Next, given any priority profile π such that ABMπ changes the manipulating student’s assignment under a swap of some schools (a and b, say), we construct a single priority profile π✝ such that under ABMπ✝, the manipulating student receives either a or b, depending on the relative ranking of a and b in her report. Thus, the change in probability for a and b is strict because π✝ is chosen with positive probability.
Theorem 9has a number of interesting consequences. First, partial strategyproofness is the strongest incentive requirement (for finite markets) that has been shown to hold for the celebrated Probabilistic Serial (PS) mechanism (Mennle and Seuken, 2015b). Thus, from an axiomatic perspective, Theorem 9means that the incentive properties of ABMU
are in the same class as those of the PS mechanism.
Second, partial strategyproofness implies weak strategyproofness: a student cannot obtain a stochastically dominant assignment by misreporting her preferences. Put differently, any manipulation will necessarily involve a trade-off on the part of the student between probabilities for different schools. This is illustrated by the example in the introduction: recall that student 1 could obtain schools a, b, d with probability 1④3 each under truthful reporting, or she could obtain her third choice school c with certainty by ranking c in first position. By misreporting, the student had to “sacrifices” all probability for her first choice a in order to convert chances to obtain her last choice d into chances to obtain c. Whether or not she would prefer this manipulation to reporting truthfully depends on her relative preference intensities for the different schools. Theorem 9teaches us that any manipulation will take such a form, and no student can gain unambiguously (in a first order-stochastic dominance sense) from misreporting.
Third, partial strategyproofness by Theorem 9 implies that ABMU
makes truthful reporting a dominant strategy for all students who differentiate sufficiently between different schools. Formally, for any setting, there exists r → 0 such that any student whose vNM utility satisfies URBI(r) will have a dominant strategy to be truthful. Thus, even though ABMU is not strategyproof, we can give honest and useful strategic advice to the students: they are best off reporting their preferences truthfully as long as they are not too close to indifferent between any two schools.
Remark 10. It is worth noting that NBM satisfies the upper invariance axiom, which is
essentially equivalent to truncation robustness: students cannot improve their chances of obtaining a better school by “truncating” their preference reports and falsely claiming that some lower ranking schools are unacceptable (Hashimoto et al., 2014). However, NBM violates swap monotonicity (Proposition 11in Appendix 3.B.1), and therefore it
Property DAU ABMU NBMU Upper invariant ✓ ✓ ✓ Swap monotonic ✓ ✓ ✗ Lower Invariant ✓ ✗ ✗ Partially strategyproof ✓ ✓ ✗ Strategyproof ✓ ✗ ✗
Table 3.1: Incentive properties of mechanisms
cannot be partially strategyproof. Table 3.1provides and overview of the properties that each of the mechanisms violate or satisfy.
Generality of Theorem 9: We have proven that Theorem 9 continues to hold for a larger class of priority distributions. We say that a priority distribution P supports all
single priority profiles if any single priority profile is selected with positive probability.
This means that Pr♣π, . . . , πqs → 0 for all π P Π, but multiple priority profiles may also be selected. Our proof of Theorem 9 covers the more general statement:
Theorem 10. For any priority distribution P that supports all single priority profiles,
ABMP
is partially strategyproof.
3.5.4 Partial Strategyproofness for Arbitrary Priority Distribution
Partial strategyproofness of ABMP
by Theorem 10 hinges on the randomness induced by the random choice of the priority profile. However, if the priority distribution is not sufficiently random (or if the priority profile is fixed), then ABMP may no longer be partially strategyproof. To obtain a meaningful comparison of ABMP an NBMP even in this case, we can consider a second source of randomness: if students are uncertain about the preference reports from the other students, then they face a random mechanism because the outcome depends on the unknown reports P✁i. This random mechanism is guaranteed to be partially strategyproof if the original mechanism is upper invariant, monotonic, and sensitive4 (see Theorem 8 of (Mennle and Seuken,2015b)). The following Proposition8 shows that NBM and ABM both satisfy these conditions.
4
A mechanism is sensitive if for any swap that changes student i’s assignment for some P✁i, there also exist Pi✶, P✁i✷ such that the swap changes that i’s assignment of each of the two objects that she swaps, respectively.
3.6 Efficiency Comparison