2.1 Preliminaries

There are n agents (refugee families/school children) randomly labeled $1,...,n$ , each of which has to be assigned to a location (host country city or town/school). Let L denote the finite set of locations. Each location $l \in L$ has a capacity $q_l \geq 1$ as to how many agents it can accommodate. We assume that $n \leq \sum _l q_l$ so that it is feasible to assign all agents. For each agent i, let $g_i(l)$ be a measure of success at location l (employment probability/test scores) when assigned to that location. In practice, this measure may need to be estimated, in which case it represents an agent’s success at location l in expectation. This measure may be accounted for in the agent’s preferences but is the key consideration for a social planner. We refer to $g_i(l)$ as the planner’s outcome score for agent i at location l.

Each agent i has a complete and transitive preference ordering $\succsim _i$ over the set of locations.Footnote ⁷ Indifference and the strict preference relations are denoted by $\sim _i$ and $\succ _i$ , respectively, and $\succsim = {(\succsim _1,...,\succsim _n)}$ denotes the vector of preferences.

We make the assumption on agents’ preferences that the only indifferences are over the worst-ranked locations. That is, apart from possibly having ties among a set of locations that an agent deems to be the worst, each agent has a strict preference over all of the other locations. Formally, for all agents i, if $l \sim _i l'$ for some $l' \neq l$ , there is no $l''$ such that $l \succ _i l''$ . This still allows for an agent to be indifferent over all locations. This assumption is motivated by our application to refugee assignment: refugee families often do not have full information on all possible locations, but they may have (strict) preferences over a limited set of top choices.Footnote ⁸

Define the set $S_i = L \backslash \{l \in L : \exists l' \sim _i l\}$ , which are all of the locations except any that agent i is indifferent over. Agent i has a strict preference across all locations in $S_i$ , and if any location is left out of $S_i$ , then it must have been ranked worst.

A matching $\mu $ maps the set of agents to locations. A matching $\mu $ is

1. 1. feasible if it satisfies the capacity constraints: $|\mu ^{-1} (l)| \leq q_l, \forall l$ ;

2. 2. $\bar g$ -acceptable if the average outcome score is not lower than $\bar g$ : $\frac {1}{n} \sum _i g_{i} (\mu (i)) \geq \bar g$ .

$\bar g$ -acceptability reflects the idea that the planner wants the average outcome score not to fall below a specified threshold $\bar g$ . The planner wants to ensure that the allocation is such that agents have some minimum level of expected outcomes (e.g., a minimum expected employment rate/GPA or test score).

Note that not all values of $\bar g$ can produce a feasible matching. Let $\bar g^{\text {max}}$ denote the highest possible average outcome score that can be generated by a feasible matching

(1) $$ \begin{align} \bar g^{\text{max}} := \max_{\mu} \frac{1}{n} \sum_i g_i ( \mu(i)) \text{ subject to } |\mu^{-1} (l)| \leq q_l, \forall l. \end{align} $$

Feasible $\bar g$ -acceptable matchings exist only for $\bar g \leq \bar g^{\text {max}}$ .

2.2 The Assignment Procedure

Given a value of $\bar g \leq \bar g^{\max }$ , the algorithm starts with agent 1 and works down to agent n in a sequence of n steps before completing in either the nth or an additional $(n+1)$ th step. At Step $i \leq n$ , agent i is either assigned to a location or put on hold by being added to a set of temporarily unassigned agents that will all get assigned simultaneously at Step $n+1$ . At each Step i, let $N_i$ denote the set of agents $j < i$ that have been put on hold. $N_1 = \emptyset $ , since at the start of the algorithm, no agent is on hold.

If agent $j < i$ was assigned a location prior to Step i, then let $\alpha _i (j)$ denote the location and $(j, \alpha _i(j))$ the assignment, viewing $\alpha _i$ as a function. Refer to this function as the completed assignment at Step i. Note that $\alpha _1 = \emptyset $ , so the completed assignment at Step 1 is trivial. A remaining assignment $\beta _i$ at Step i is a mapping of the unassigned agents $\{i,...,n\} \cup N_i$ to locations such that

$$ \begin{align*} \mu_{(\alpha_i, \beta_i)} (j) := \left\{\begin{array}{ll} \alpha_i(j) &\quad \text{if } j < i\\ \beta_i(j) &\quad \text{if } j \in \{i,...,n\} \cup N_i \end{array}\right. \end{align*} $$

is a matching. We refer to $\mu _{(\alpha _i, \beta _i)}$ as the matching associated with the pair of completed and remaining assignments $(\alpha _i, \beta _i)$ . The existence of these matchings will be guaranteed recursively by the algorithm.

At each Step $i \leq n$ , given $\alpha _i$ , define the set

$$ \begin{align*} L^{\bar g}_i (\alpha_i) = \{ l \in L : \exists \beta_i \text{ s.t. } l = \beta_i (i) \text{ and } \mu_{(\alpha_i, \beta_i)} \text{ is a feasible } \bar g\text{-acceptable matching}\}. \end{align*} $$

This is the set of locations that are not at full capacity and for which there is a way to finish assigning all unassigned agents so as to create a feasible $\bar g$ -acceptable matching.

Let $q^i_l$ be the remaining capacity of location l after any agents ahead of i (i.e., $j < i$ ) have been assigned in the previous $i-1$ steps. At the start, we have $q^1_l = q_l$ for all l. It will also be convenient to define the following problems: for all Steps $i=1,...,n+1$ , and given a vector $q^i := (q^i_l)_{l \in L}$ ,

(2) $$ \begin{align} G_i (q^i) := \max_{\beta_i} \sum_{j \in \{i,...,n\} \cup N_i} g_j (\beta_i (j)) \text{ subject to } |\beta_i^{-1} (l) | \leq q^i_l, \forall l, \end{align} $$

with the convention that $\{i,...,n\} := \emptyset $ if $i = n+1$ . At each Step i, the problem in (2) finds the remaining assignment that maximizes the total outcome score subject to the updated capacity constraints at Step i. The solution to this problem at each step determines whether the associated matching is $\bar g$ -acceptable. In fact, to verify whether or not a location l belongs in $L^{\bar g}_i(\alpha _i)$ , we must first check whether the highest possible value of the average outcome score that can be achieved under the remaining assignment is at least $\bar g$ ; i.e., whether

$$ \begin{align*} \bar g_i(l) := \frac{1}{n} \left( G_{i+1}(q^{i+1}) + g_i (l) + \sum_{j < i \text{ s.t.~}j \notin N_i} g_j (\alpha_i(j)) \right) \geq \bar g, \end{align*} $$

where $q^{i+1}_{l'} = q^i_{l'}$ for all $l' \neq l$ and $q^{i+1}_{l}= q^i_{l} -1$ . If indeed $\bar g_i(l) \geq \bar g$ and $q^i_{l}> 0$ , then l belongs to $L^{\bar g}_i (\alpha _i)$ ; otherwise, it does not. Constructing $L^{\bar g}_i(\alpha _i)$ at each Step $i=1,...,n+1$ , therefore, requires solving the problems given in (2). In addition, to verify whether $\bar g \leq \bar g^{\max }$ also requires solving one of these problems, since the problem in (1) equals $G_1 (q^1)/n$ .

The steps of the algorithm are as follows.

1. Step 0. Verify that $\bar g \leq \bar g^{\max }$ and proceed only if it holds.

2. Step i ≤ n. If $S_i \cap L^{\bar g}_i (\alpha _i)$ is empty (meaning that there is no location that agent i ranked strictly to which it could be assigned that would allow us also to find a remaining assignment that generates a feasible $\bar g$ -acceptable matching), then place agent i on hold. In this case, set

   $$ \begin{align*} N_{i+1} = N_i \cup \{i\}, \alpha_{i+1}= \alpha_{i}, q^{i+1}_l = q^i_l~\forall l \end{align*} $$
   and move on to Step $i+1$ . Otherwise, if $S_i \cap L^{\bar g}_i (\alpha _i)$ is nonempty, then it contains a unique best location from the perspective of agent i—i.e., a location $l_i^*$ such that $l_i^* \succ _i l$ for all $l \in S_i \cap L^{\bar g}_i(\alpha _i)$ . This follows from the fact that i ranks the elements of $S_i$ strictly. Assign agent i to $l_i^*$ , and set
   $$ \begin{align*} N_{i+1} = N_i, \alpha_{i+1}= \alpha_{i} \cup \{(i,l_i^*)\}, q^{i+1}_{l_i^*} = q^i_{l_i^*}-1, \text{ and } q^{i+1}_l = q^i_l \forall l \neq l_i^*. \end{align*} $$
   If $i<n$ , then move to Step $i+1$ . If $i = n$ , then move to Step $n+1$ only if an agent was ever put on hold (i.e., $N_{n+1} \neq \emptyset $ ); otherwise, stop.

3. Step n + 1. At this stage, the only unassigned agents are those that were put on hold in $N_{n+1}$ . Here, choose any remaining assignment that maximizes the average outcome score given the completed assignment and the capacity constraints; that is, solve (2) for $i = n+1$ and stop.

For any preference vector $\succsim $ satisfying our assumptions, our algorithm produces a matching, namely $\mu _{(\alpha _s, \beta _s)}$ , where $s \in \{n, n+1\}$ was the step at which the algorithm stopped. The algorithm defines a mechanism $\varphi $ , which, given the other parameters of the model, is a mapping from preference vectors to feasible matchings. We refer to the mechanism as $\bar g$ -constrained priority, since it is a modification of the usual priority mechanism (Satterthwaite and Sonnenschein Reference Satterthwaite and Sonnenschein1981).

At each Step i, implementation of the mechanism involves iteratively solving the maximization problem in Equation 2 to verify that $\bar g$ -acceptability can still be met if agent i were assigned to each available location in order of preference, until such a location is found. This amounts to iteratively solving a standard linear sum assignment problem, for which various polynomial-time algorithms exist.Footnote ⁹ Under a worst-case scenario where every agent is put on hold after unsuccessfully considering all of its strictly ranked locations, this would require solving an equally sized maximization problem in Equation 2 a total of $n (|L|-2)$ times.Footnote ¹⁰

2.3 Properties of the Mechanism

Let $\varphi (\succsim )$ denote the matching produced by the $\bar g$ -constrained priority mechanism for any preference vector $\succsim $ that satisfies our assumptions, and $\varphi (\succsim )(i)$ the location assignment of agent i under this matching. By construction, the matching produced by this mechanism is feasible and $\bar g$ -acceptable. In addition, the mechanism satisfies two key properties. It is:

1. 1. constrained efficient in the sense that for all preference vectors $\succsim $ that satisfy our assumptions, $\varphi (\succsim )$ is not Pareto dominated by another feasible $\bar g$ -acceptable matching $\mu $ . That is, it is not the case that $\mu (i) \succsim _i \varphi (\succsim )(i)$ for all agents i, and $\mu (i) \succ _i \varphi (\succsim )(i)$ for some agent i.

2. 2. strategy-proof in the sense that truthful reporting is a dominant strategy of the induced preference reporting game. That is, for every preference vector $\succsim $ satisfying our assumptions, every agent i, and every alternative preference $\succsim _i'$ that i could report that also satisfies our assumptions, $\varphi (\succsim )(i) \succsim _i \varphi (\succsim ^{\prime }_i, \succsim _{-i})(i)$ .

The proof that the mechanism is constrained efficient and strategy-proof is straightforward, but for completeness, we include it in the SI.

One important property of the canonical priority mechanism that does not carry over to our $\bar g$ -constrained priority mechanism is the property that the mechanism characterizes the full set of Pareto-efficient assignments. Abdulkadiroglu and Sonmez (Reference Abdulkadiroglu and Sonmez1998) showed that for any Pareto-efficient assignment, there exists an ordering of agents under which implementing the priority mechanism for that ordering generates that assignment. Given this, one could ask whether for every $\bar g$ -constrained efficient assignment, there exists an ordering of the agents for which the $\bar g$ -constrained priority mechanism generates that assignment. The answer to this question turns out to be no, as demonstrated by the following example with two agents $1$ and $2$ and three locations $A, B$ , and C. The table gives the ranking of the three locations for each agent and in parentheses the outcome score $g_i(l)$ for each agent–location pair.

Suppose that each location has a capacity of 1 seat. If the planner’s threshold $\bar g$ is set to $0.45$ and agent 1 goes first, then he will be assigned to location A, and agent 2 will be assigned to B. If agent 2 goes first, then she will be assigned to A, and agent $1$ will be sent to C. But the possibility of sending 1 to B and 2 to C also meets the planner’s constraint and is not Pareto dominated by any other assignment that is acceptable to the planner.

3.1 Simulation Data

We apply the mechanism to simulated data to show these properties. For simplicity, our simulations involve assigning 100 agents to 100 locations with one seat each. For each agent, we randomly generate a preference rank vector (with $1$ indicating the most desired location and $100$ the worst) and an outcome score vector (with values in $[0,1]$ ). The simulations vary both the correlation between preference and outcome vectors ( $-0.5$ , $0$ , and $0.5$ ) and the correlation between preference vectors across agents ( $0$ , $0.5$ , and $0.8$ ).Footnote ¹² This yields nine different scenarios, and in each, we apply our mechanism to make the assignment for various values of $\bar g$ . See the SI for details.

3.2 Refugee Data

As a simulated illustration of how the mechanism could perform in a real-world scenario, we apply it to data from refugees in the United States, where refugee families are the agents and resettlement cities are the locations. Early employment is a core goal of the U.S. resettlement program, which strives to quickly transition refugees into self-sufficiency after arrival. This application illustrates how our mechanism could hypothetically be employed in the United States to achieve a desired level of early employment while geographically assigning refugees based on their location preferences.

Our real-world refugee data include deidentified information on working-age refugees (ages 18–64; N = 33,782) who have been resettled to the United States during the 2011–2016 period by one of the largest U.S. refugee resettlement agencies. Over this time period, the agencies’ placement officers centrally assigned refugees to one of approximately 40 resettlement locations in the agency’s network. The data contain details on the refugee characteristics such as age, gender, origin, and education. The data also include the assigned resettlement location, whether the refugee was employed at 90 days after arrival, and whether the refugee migrated from the initial location within 90 days.

We applied our mechanism to data on the refugee families who arrived in the third quarter (Q3) of 2016, specifically focusing on refugees who were free to be assigned to different resettlement locations (561 families), in contrast to refugees who were predestined to specific locations on the basis of existing family or other ties. To generate each family’s outcome score vector across each of the locations, we employed the same methodology in Bansak et al. (Reference Bansak2018), using the data for the refugees who arrived from 2011 up to (but not including) 2016 Q3 to generate models that predict the expected employment success of a family (i.e., the mean probability of finding employment among working-age members of the family) at any of the locations, as a function of their background characteristics. These models were then applied to the families who arrived in 2016 Q3 to generate their predicted employment success at each location, which comprise their outcome score vectors. See the SI and Bansak et al. (Reference Bansak2018) for details.

Our mechanism also requires data on location preferences of refugees. To the best of our knowledge, such data do not currently exist in the United States, where refugees are assigned to locations by the resettlement agencies. We therefore infer revealed location preferences from secondary migration behavior. Specifically, we use the same modeling procedures used in the outcome score estimation, simply swapping in outmigration in place of employment as the response variable. This allows us to predict for each refugee family that arrived in 2016 Q3 the probability of outmigration at each location as a function of their background characteristics. For each family, we then rank locations such that the location with the lowest (highest) probability of outmigration is ranked first (last). Details about the data and sample are provided in the SI.

We acknowledge that inferred location preferences from secondary migration behavior are not necessarily the same as the stated location preferences that refugees would express in an application form if given the opportunity to do so by host country governments. That said, there are reasons to believe that the inferred location preferences provide a useful proxy. Outmigration is a costly signal indicating that a refugee prefers to move rather than stay in the originally assigned location. Mossad et al. (Reference Mossad, Ferwerda, Lawrence, Weinstein and Hainmueller2020) provide a comprehensive analysis of the secondary migration patterns of refugees in the United States and find that secondary migration is mostly driven by refugees relocating in search of employment opportunities and coethnic communities. One of the main channels through which these effects operate is the refugee’s nationality, which is also an important predictor in the model that we use to infer revealed location preferences from secondary migration.

3.3 Education Data

As a second simulated illustration of how the mechanism could perform in a real-world scenario, we apply it to education data from the United States, where the agents are individual students and the locations are schools. In particular, we consider data from the Tennessee’s Student Teacher Achievement Ratio (STAR) project conducted by the Tennessee State Department of Education (for details, see Achilles et al. Reference Achilles2008). These data contain information on the choice of elementary schools for a large sample of students as well as information on the test score performance of these students. We focus on the Kindergarten grade level and apply our mechanism to generate new assignments of students to schools with the goal of improving the outcomes of students as measured by standardized tests administered at the end of Kindergarten. One could imagine a school district setting a minimum test score that should be achieved on average.

To generate each student’s outcome score vector across each of the schools, we employ the same methods as in the previous application to predict the expected test scores of a student at any of the schools, as a function of their background characteristics. The background characteristics included the students’ age, gender, and race, as well as information on whether they are eligible for free school lunches (a proxy for socioeconomic status) or special education. The test score outcome was defined as the sum of reading, math, and listening scaled scores on the SAT for the Kindergarten level.

We inferred revealed school preferences of students from the observed transfers out of the schools. Specifically, we used the same modeling procedure as for the test scores but instead used a response variable that measured whether a student had transferred to another school by the first, second, or third grade. Based on these models, we can then predict for each student the propensity for leaving each school as a function of their background characteristics. For each student, we then rank schools such that the school for which they have the lowest (highest) propensity for transferring out is ranked first (last).

We generate these outcome score and preference vectors and apply our mechanism to a random sample of 1,000 students from 33 schools that are observed for all grades from Kindergarten through third grade and have nonmissing data for tests scores and background characteristics. Details about the data and sample are provided in the SI.

5.1 Using Predicted Preferences

An alternative approach to capturing part of the gains that we are seeing from reordering the agents that does not sacrifice either strategy-proofness or constrained efficiency is to use historical (or other) data to predict the preferences of the agents. For example, the planner could use historical data to predict preferences based on the demographic similarities of past agents to current ones and fix the ordering of agents to be the one that maximizes the top-3 metric according to the predicted preferences.Footnote ¹⁹ In particular, note that if the planner can perfectly predict the preferences of agents, then strategy-proofness is not a concern, since the planner already has the agents’ preferences. In this case, the planner can recover the full gain from rerandomizing the order of agents. If the planner cannot perfectly predict the preferences of the agents, but can come close, then the planner should, in expectation, be able to recover some of this gain. Note that because we are using historical data from past agents to set the order of the current agents, the agents cannot use their reported preferences to manipulate the mechanism.

In order to evaluate this approach, we employed the data from our refugee application described earlier. Specifically, we began by randomly drawing 100 families from the full set of data used in the application. We then randomly generated 100 different orderings of those families, and for each ordering, we implemented the constrained priority mechanism along a sequence of values of $\bar g$ . We used the families’ outcome score vectors and “actual” preference vectors (i.e., the preference vectors employed in the application presented earlier). This allows us to asses the extent to which different orderings can result in varying levels of the top-3 metric. The components of Figure 6 labeled “Many Orders” display these results, with the black dots corresponding to the average across the 100 orderings and the intervals denoting the maximum and minimum results obtained across the 100 orderings.

Figure 6 Results from applying our $\bar g$ -constrained priority mechanism to 100 random orderings of a random sample of 100 families from the 2016 Q3 refugee data, along with “best guess” ordering based on pseudo preferences. The black dots correspond to the average results across the 100 orderings, and the intervals denote the maximum and minimum results obtained across the 100 orderings. The triangles (labeled “Pseudo-Inferred Order”) denote the actual results when employing the ordering that yielded the best pseudo top-3 metric according to the pseudo preferences. The three scenarios successively increase the amount of perturbation applied to the actual preference vectors to generate the pseudo preferences. Upper panel shows the average probability that an agent was assigned to one of its top-three locations. Lower panel shows the realized average outcome score. $N=100$ .

Furthermore, for each of the 100 orderings, we also evaluated the results of applying the constrained priority mechanism using pseudo-preference vectors in place of the actual preference vectors. These pseudo-preference vectors are intended to represent the imperfectly predicted preferences of the agents. At each level of $\bar g$ , we identified the random ordering that resulted in the best pseudo aggregate welfare as measured by the fraction of families receiving one of their top-three locations according to these pseudo preferences. We were then able to assess the actual welfare results (based on the families’ actual preferences) of applying these “best guess” orderings. In order words, we ran through the process by which a researcher could (i) in advance/independent of actual preference reporting employ simulations to identify orderings likely to lead to higher levels of aggregate welfare based on predicted preferences and then (ii) use those as the final orderings by which to actually apply the $\bar g$ -constrained priority mechanism to assign the families.

To simulate the process of imperfectly predicting the families’ pseudo preferences, we constructed their pseudo preference vectors by randomly perturbing their actual preference vectors. In order to investigate the performance of this approach across different levels of effectiveness in predicting preferences (i.e., the extent to which it is possible to construct pseudo preference vectors that are similar to the actual preference vectors), we imposed varying degrees of perturbation and evaluated the results across those different specifications. The results can be seen in the components labeled “Pseudo-Inferred Order” in Figure 6, where the separate panels correspond to scenarios with increasing amounts of perturbation.Footnote ²⁰ For each scenario, the triangles labeled “Pseudo-Inferred Order” denote the actual results when applying the ordering deemed best according to the pseudo preference results, as described above. The figure shows that a large portion of the gain from carefully fixing the order of agents could be recovered if the planner is able to very accurately predict preferences, but how much can be gained could be very sensitive how good a prediction the planner makes.Footnote ²¹

5.2 Ordering Agents by Outcome Score Variance

We also explored an alternative strategy for identifying a priori (and hence without sacrificing strategy-proofness) an agent ordering that is likely to result in a favorable level of the top-3 metric. Rather than attempting to predict pseudo preferences, this strategy instead utilizes the families’ outcome scores. Specifically, for each family the variance of outcome scores across locations can be computed, and then agents can be ordered according to their variances. We propose ordering the agents in increasing variance (from lowest variance to highest variance). The intuition for this proposal is the following. In making assignments, the $\bar g$ -constrained priority mechanism is faced with the trade-off between sending agents to their preferred location and sending them to a location that will enable the $\bar g$ constraint to be met. For each agent, the extent to which this particular trade-off can bite depends to some degree on the variance of their outcome scores across locations. In the extreme case, there is no trade-off for an agent whose outcome score is identical across locations: no matter where they are sent, their assignment will have an equivalent implication for the $\bar g$ constraint. However, for agents whose outcome score variance is very high, the extent to which their assignment can work in favor of or against the $\bar g$ constraint varies greatly across locations. In other words, the high-variance agents’ assignments offer opportunity to create buffer for the $\bar g$ constraint, whereas the low-variance agents’ assignments do not. Therefore, assigning low-variance agents earlier ensures that such units are more likely to be assigned to a highly preferred location without occurring at the expense of excessively cutting into the $\bar g$ constraint.Footnote ²²

Figure 7 shows the results of applying this increasing-variance ordering strategy (left panels) as well as the opposite decreasing-variance ordering for illustrative purposes (right panels). The components labeled “Many Orders” display the results from the same 100 random orderings as in Figure 6, with the black dots corresponding to the average across the 100 orderings and the intervals denoting the maximum and minimum results obtained across the 100 orderings. The components labeled “Variance-Based Order” display the results when applying the mechanism to the families put in the proposed increasing-variance order (left panels), or in the decreasing-variance order (right panels). The figures show that when agents are ordered by increasing outcome score variance, a substantial share of the gain from fixing the order of agents can be recovered. It also depicts what we expect would happen when agents are ordered by decreasing outcome score variance, which is that welfare measured by the top-3 metric is generally worse than under the typical random ordering.

Figure 7 Results from applying our $\bar g$ -constrained priority mechanism to 100 random orderings of a random sample of 100 families from the 2016 Q3 refugee data, along with outcome score variance-based orderings. The black dots correspond to the average results across the 100 orderings, and the intervals denote the maximum and minimum results obtained across the 100 orderings. The triangles (labeled “Variance-Based Order”) denote the results when employing orderings based on the families’ outcome score variances across locations, with families ordered by increasing variance on the left and decreasing variance on the right. Upper panel shows the average probability that an agent was assigned to one of its top-three locations. Lower panel shows the realized average outcome score. $N=100$ .

5.3 Overview

One property of the increasing outcome variance ordering approach is that it does not rely on historical (or any other) data to predict the preferences of agents. Another desirable property of this approach relates to fairness/distributive considerations: the agents that cannot gain much by way of their outcome score through different assignments can be prioritized to achieve gains in terms of their preferences, and those agents that have a lot to gain by way of outcome scores could enjoy such gains even if they do not get one of their top preferences. A concern with fixing the order of agents using either of these techniques, however, is that it may unintentionally though systematically put agents of particular backgrounds higher or lower in the priority order, a form of potential disparate impact that may not be desirable to the planner. In choosing the order of agents, one might want to incorporate additional constraints to overcome part of the bias that may be implicit in these techniques. More theoretical work will be necessary to evaluate how workable these and other approaches of using data and machine-learning techniques to determine the order of agents will be in various applications.

6.1 Top Trading Cycles

One candidate for an alternative mechanism, among matching mechanisms with one-sided preferences, is Gale’s TTC mechanism (Shapley and Scarf Reference Shapley and Scarf1974; Roth Reference Roth1982), which has already been employed in previous proposals for refugee matching (Delacrétaz et al. Reference Delacrétaz, Kominers and Teytelboym2016). However, adding a planner’s $\bar g$ constraint to this mechanism while retaining the feature that it is strategy-proof and constrained efficient is not straightforward.

Consider, for example, the simple adjustment of this mechanism that begins by provisionally assigning agents to locations to maximize the planner’s objective and removes cycles until the planner’s welfare measure falls below the threshold, at which point it stops and everyone that is unassigned receives the assignment that they currently provisionally have. The following three person-three location example depicted in Figure 8 shows that this mechanism is not strategy-proof. The agents are $1, 2,$ and $3$ , and the locations are A, B, and C.

Figure 8 Three person-three location example showing violation of strategy-proofness when adding a planner’s $\bar g$ constraint to TTC mechanism.

To maximize the average outcomes score, agent 1 is provisionally assigned to A, 2 to B, and 3 to C. Preferences over locations are given on the left. In the middle, we have the values of $g_i(l)$ . Under truthful reporting, agent 1 points to B, and 2 and 3 point to A. The only cycle is between $1$ and $2$ . However, if the planner’s threshold $\bar g$ is set to 0.5, then swapping 1 and 2’s locations guarantees an average outcome score below this threshold. The algorithm would terminate with the outcome maximizing assignment being assigned. However, agent 1 could do better by misreporting and pointing to C instead. In this case, the assignment would be 1 to C, 2 to B, and 3 to A, which gives an average outcome score above the threshold.

Thus, while it may be possible to incorporate an outcome constraint into the TTC mechanism that preserves strategy-proofness and constrained efficiency, it appears that there is no straightforward way to do so. For the priority mechanism, however, incorporating this constraint is both straightforward and computationally tractable.

6.2 Two-Sided Mechanisms

Finally, we could also consider matching mechanisms with two-sided preferences, such as the deferred acceptance mechanism (Gale and Shapley Reference Gale and Shapley1962), where we incorporate the planner’s welfare objective into the preferences for the locations. Here, there are at least two possibilities. First, we could assume that locations care about maximizing the planner’s welfare score, along with other considerations; that is, we allow the locations to express their genuine preferences. At least for the refugee assignment application, this appears to be politically challenging, partly because policymakers are concerned that this could result in political problems, where some locations might discriminate against refugees from certain groups/nationalities. The second possibility is we assume that each location simply wants to maximize the average outcome score among agents assigned to it. This creates competition among locations. Again, at least in the refugee assignment application, it is not clear why the planner (national government) would want to allow this—i.e., it is not clear what this assignment mechanism would achieve that serial priority does not, given the objectives of the planner.