Criteria to compare mechanisms that partially satisfy a property: an axiomatic study

Abstract

We study criteria that compare mechanisms according to a property (e.g., Pareto efficiency or stability) in the presence of multiple equilibria. The multiplicity of equilibria complicates such comparisons when some equilibria satisfy the property while others do not. We axiomatically characterize three criteria. The first criterion is intuitive and based on highly compelling axioms, but is also very incomplete and not very workable. The other two criteria extend the comparisons made by the first and are more workable. Our results reveal the additional robustness axiom characterizing each of these two criteria.

Appendix on school choice application

1.1 The school choice model

The model and notation are inspired from Haeringer and Klijn (2009). There are three students $i_1$, $i_2$ and $i_3$ and three schools $s_1$, $s_2$ and $s_3$, each endowed with one seat. Each student can be assigned to at most one school. Students have preferences over the schools they could be assigned to as well as the possibility of remaining unassigned (i.e., being self-matched). Each school has a strict priority ordering over the students. In this setting, a (school choice) problem is a pair $\pi = (R, \trianglerighteq )$ where

• 1. $R {:=}(R_{i_1}, R_{i_2}, R_{i_3})$ is the (strict) preference profile of students over the three schools, and

• 2. ${\trianglerighteq } {:=}(\trianglerighteq _{s_1}, \trianglerighteq _{s_2}, \trianglerighteq _{s_3})$ is the (strict) priority profile of schools over the three students.

The preference $R_i$ of student i is a linear order over $S \cup \{i\}$. If student i strictly prefers school s over school $s'$, we write $s~P_i~s'$. As usual, $s~R_i~s'$ denotes a weak preference, allowing for $s=s'$. We say that a school s is acceptable for a student i if $s~P_i~i$ and unacceptable if $i~P_i~s$. To avoid trivialities, we assume that all students find at least one school acceptable.

The priority $\trianglerighteq _s$ of school s is a linear order over the three students. If student i has a higher priority than student j at school s, then $i\vartriangleright _sj$ and we say that i is a competitor of j at school s. School s is a top-priority school for student i if i has no competitor at school s.

We denote by $\Pi$ the domain of problems satisfying these assumptions.

An assignment is a function $\mu : \{i_1,i_2,i_3\} \rightarrow \{s_1,s_2,s_3\} \cup \{i_1,i_2,i_3\}$ that matches every student with a school or with herself. We say that student i is assigned in the former case, and unassigned in the latter case.

An assignment is feasible if no two students are assigned to the same school.

Given any problem $\pi$, an assignment $\mu$ is stable if it satisfies each of the three following properties.

Individual rationality::

For any student i, we have $\mu (i) ~R_i~ i$.

Non-wastefulness::

For any student i and any school s, if $s~P_i~\mu (i)$, then $\# \{ j \in I~|~\mu (j) = s\} = 1$.

No justified-envy::

For any two students i and j, if $\mu (j)~P_i~\mu (i)$, then j is a competitor of i at school $\mu (j)$.

A (school choice) mechanism M is a function that associates every problem $\pi$ in some domain $\Pi ^M \subseteq \Pi$ of problems with a feasible assignment. We say that a mechanism is individually rational, non-wasteful or stable, if $M(\pi )$ is individually rational, non-wasteful or stable for all $\pi \in \Pi ^M$. As is common, when there is no ambiguity about $\trianglerighteq$, we often use M(R) to denote the assignment selected by mechanism M.

We assume that the three schools report their priority ordering truthfully to the mechanism. A type profile y is a school choice problem $\pi = (R, \trianglerighteq )$ (and thus $Y=\Pi$), and the players of mechanism M are the three students. For the two mechanisms that we consider, the strategy space $S_i$ of each student i consists in the set of reported preference $Q_i$ for which at least one school is unacceptable and at least one school is acceptable.

For any type profile y, the pair (M, y) defines a strategic form game for which students report a preference and the outcome is the assignment selected by M under the profile of reported preferences. Given (M, y), the strategy-space of student i is the set of all the preferences of i that are featured in at least one problem of $\Pi ^M$. We call these strategies reported preferences. A reported profile is a list $Q {:=}(Q_{i_1}, Q_{i_2}, Q_{i_3})$ of the reported preferences of all students.

The outcome of the game when students report Q is assignment M(Q). Student i evaluates this assignment according to her true preference $R_{i}$. In particular, strategy $Q_i$ is a (weakly) dominant strategy for student i if

$$\begin{aligned} M_i\left( Q_i,Q_{-i}\right) ~R_i~M_i\left( Q_i',Q_{-i}\right) , \qquad \text { for any } Q_{-i} \text { and any } Q_i'. \end{aligned}$$

In turn, strategy $Q_i$ is a dominated strategy for student i if

$$\begin{aligned} M_i\left( Q_i',Q_{-i}\right) ~R_i~M_i\left( Q_i,Q_{-i}\right) , \qquad \text { for any } Q_{-i} \text { and some } Q_i' \end{aligned}$$

and $M_i(Q_i',Q_{-i}')~P_i~M_i(Q_i,Q_{-i}')$ for some $Q_{-i}'$. A strategy is undominated if it is not dominated.

1.2 Two mechanisms

In this section we describe the two school choice mechanisms we compare, which are members of the class considered in Haeringer and Klijn (2009). We first describe $BOS^2$, a constrained version of the Boston mechanism for which students are allowed to report preferences on two schools only.

Input ::

A (reported) school choice profile.

Round 1::

Students apply to the school they reported as their favorite school. Every school that receives more applications than its capacity starts rejecting the lowest applicant in its priority ranking, up to the point where it meets its capacity. All other applicants are definitively accepted at the schools they applied to, and capacities are adjusted accordingly.

Round 2 ::

Students who are not yet assigned apply to the school they reported as their second favorite school. Every school that receives more new applications in round 2 than its remaining capacity starts rejecting the lowest new applicants in its priority ranking, up to the point where it meets its capacity. All other applicants are definitively accepted at the schools they applied to. The algorithm terminates and all students not yet assigned remain unassigned.

We now turn to $DA^2$, a constrained version of the Deferred Acceptance mechanism for which students are allowed to report preferences on two schools only.

Input ::

A (reported) school choice profile.

Round 1::

Students apply to the school they reported as their favorite school. Every school that receives more applications than its capacity definitively rejects the lowest applicant in its priority ranking, up to the point where it meets its capacity. All other applicants are temporarily accepted at the schools they applied to (this means they could be rejected at a later point).

Round 2: :

Students who were rejected in round 1 apply to the school they reported as their second favorite school. Every school considers the new applicants of round 2 together with the students it temporarily accepted. If needed, each school definitely rejects the lowest students in its priority ranking, up to the point where it meets its capacity. The algorithm terminates and all students not yet assigned remain unassigned.

1.3 Preliminary results on undominated strategies under $DA^2$ and $BOS^2$

Propositions 1 and 2 require identifying undominated strategies under $DA^2$ and $BOS^2$. The following lemmas provide the necessary results for such identification. They are direct implications of characterization results taken from Haeringer and Klijn (2009) and Decerf and Van der Linden (2018a).

Lemma 2

If student i finds only one school acceptable, then reporting only this school is a dominant strategy under both $BOS^2$ and $DA^2$.

Proof

This is a straightforward implication of the characterization of dominant strategies in constrained BOS and constrained DA in Decerf and Van der Linden (2018b). □

Lemma 3

Assume that the most-preferred school of student i is a top-priority school for i. Under both $BOS^2$ and $DA^2$, (1) i has a dominant strategy and (2) i is assigned to her most-preferred school when she plays her dominant strategy.

Proof

This is a straightforward implication of the characterization of undominated strategies in constrained BOS and dominant strategies in constrained DA in Decerf and Van der Linden (2018a) (Propositions 2 and 4). □

Lemma 4

Assume that the second most-preferred school of student i is a top-priority school for i. Student i has a dominant strategy under $DA^2$, which consists in reporting these two schools truthfully.

Proof

This is a straightforward implication of the characterization of dominant strategies in constrained DA in Decerf and Van der Linden (2018a) (Proposition 2). □

Let student i’s most-preferred top-priority school be the school that i prefers among the schools that are top-priority for i (if any).

Lemma 5

Assume that the most-preferred school of student i is not a top-priority school for i. Strategy $Q_i$ is undominated under $DA^2$ only if $Q_i$ reports two schools, $Q_i$ ranks these two schools according to i’s true preference and i weakly prefers these two schools over her most-preferred top-priority school.

Proof

Haeringer and Klijn (2009) (Proposition 4.2) show that a necessary condition for $Q_i$ to be undominated under $DA^2$ is that $Q_i$ reports two schools and $Q_i$ ranks these two schools according to i’s true preference. Decerf and Van der Linden (2018a) (Proposition 3) show that another necessary condition for $Q_i$ to be undominated under $DA^2$ is that i weakly prefers these two schools over her most-preferred top-priority school. □

Lemma 6

Strategy $Q_i$ is undominated under $BOS^2$ if and only if (i) the school reported first is i’s most-preferred top-priority school or (ii) the school reported first is not top-priority for i and $Q_i$ reports two schools, one of which is strictly preferred to i’s most-preferred top-priority school.

Proof

This is a straightforward implication of the characterization of undominated strategies in constrained BOS in Decerf and Van der Linden (2018a) (Proposition 4). □