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

In Gerber and Barberà (2016), the solution concept is “iterated elimination of weakly dominated strategies” and the correspondence is the possibility of agenda manipulation. In Dasgupta and Maskin (2008), the solution concept is “truthful revelation” and the correspondence is a collection of five voting properties.

The relevant primitives for our criteria are the numbers of equilibria that yield an outcome that is (resp. not) selected by the correspondence. In contrast, the relevant primitives for these rules include the “hit rate”, i.e. the fraction of observations predicted by the theory, and the “area”, i.e. the fraction of potential outcomes predicted by the theory. Neither the “hit rate” nor the “area” are relevant primitives for our criteria. We provide here the intuition why the “area” is not a relevant primitive for our criteria. The following two mechanisms have different area but should be considered equivalent by our criteria. The first mechanism has a unique equilibrium that yields an outcome that is selected by the correspondence. The second mechanism has multiple equilibria, all of which yield outcomes that are selected by the correspondence.

In particular, the weak relation is transitive.

This property assumes that all equilibria count the same. This is a natural assumption if one believes that all equilibria are equally likely to occur.

In particular, Parts 2 of Theorems 1, 2 and 3 require the construction of intermediate mechanisms that may have, for some type profiles, more numerous equilibria than the number of equilibria of the mechanisms being compared. However, Parts 1 of Theorems 1, 2 and 3 do not require Z to be unbounded.

Under our assumptions, the proportion is always well-defined. Indeed, we assume that solution concepts admit at least one equilibrium for each type profile. As a result, the denominateur of the proportion is never zero.

These three axioms are independent. Showing independence of Monotonicity is the most difficult part. We propose the criterion I2, which satisfies all these axioms except Monotonicity. Criterion I2 is based on the following function \(f:[0,1]\rightarrow [0,1]\) defined as \(f(x)=1-x\) for \(x \in \{0,1\}\) and \(f(x)=x\) for all \(x \in (0,1)\). That is, function f is strictly increasing for all \(x \in (0,1)\), but returns the smallest value for \(x=1\) and the greatest for \(x=0\). For any two \(F,F' \in {\mathcal {F}}\), we have \(F' \succeq _{I2} F\) whenever \(f\left( \frac{{F}'_1(y)}{{F}'_0(y)+{F}'_1(y)}\right) \ge f\left( \frac{{F}_1(y)}{{F}_0(y)+{F}_1(y)} \right) \) for all \(y \in Y\), and we have \(F' \succ _{I2} F\) if in addition the inequality is strict for some \(y^*\in Y\).

Consider the following complete order. For any two \(F,F' \in {\mathcal {F}}\), we have \(F' \succeq _{COMP} F\) wheneverObserve that these three axioms do not jointly imply this order. Additional properties would be required, typically imposing some form(s) of anonymity.

These type profiles are irrelevant in the sense that the exact fraction of outcomes in X of each mechanism does not matter.

An assignment has a blocking pair if a student is assigned to a school that another student prefers to her assignment and the other student has higher priority at this school than the first student.

See Appendix 7.2 for the description of both mechanisms.

Note that with \(F^{DA}\) and \(F^{BOS}\) the functions respectively associated to unrestricted DA and BOS by C and X, we also have \(F^{DA}\succ _{PHO}F^{BOS}\). Indeed, in DA all students have a single dominant strategy which consist in ranking all their acceptable schools without switches. There is therefore only one undominated strategy profile in DA, and this profile is always stable. It is then sufficient to show that some of the many undominated strategy profiles in BOS are not stable.