Effective Allocation of Affordable Housing

In cities across the world, public agencies oversee the allocation of affordable housing. In the United States, 5 million households rely on federal rental assistance. In Amsterdam, 40% of the population lives in public housing, and in Singapore, 80% does. Despite this scale, demand for affordable housing greatly exceeds supply, forcing agencies to clear the market using lotteries or wait-lists. The rules governing this allocation vary widely. Many public housing authorities in the United States use a wait-list, and force applicants who reject an offer to re-enter at the bottom of the list. In Amsterdam, by contrast, applicants maintain their position on the list until they are successfully matched. Several cities in North America use lotteries to allocate newly-built affordable units, and allow applicants to enter as many lotteries as they wish. Singapore also uses lotteries, but permits applicants to enter only one lottery in each quarterly launch. To ensure that the limited supply of affordable housing is put to the best possible use, it is important to understand how these alternatives compare.

The paper “Design of Lotteries and Wait-lists for Affordable Housing Allocation” (Nick Arnosti and Peng Shi, Management Science, 2020) evaluates several simple allocation mechanisms using two objectives: targeting apartments to applicants with the greatest need, and matching applicants to suitable apartments. It analyzes a stylized model in which apartments arrive at a steady rate and are assigned dynamically to a pool of waiting agents. Applicants have different outside options, and idiosyncratic values for each apartment. The paper assumes that applicants respond optimally to the steady-state Markov Decision Process induced by the allocation mechanism and others’ choices. An outcome succeeds at targeting if most apartments go to agents with poor outside options, and succeeds at matching if most matched agents receive apartments that they value highly.

One finding is that two widely-used allocation systems do not succeed at matching. In the “wait-list without choice,” applicants lose their position in the wait-list after rejecting an offer. As a result, they frequently choose to instead accept a poor match. An alternative is to use “independent lotteries,” in which each apartment building holds a lottery, and applicants may enter as many lotteries as they wish. While this seems to offer choice, in equilibrium applicants enter lotteries for many buildings, and few receive their most preferred options. Despite the apparent differences between the wait-list without choice and independent lotteries, within the model they result in equivalent equilibrium outcomes.

Several alternatives achieve better matching. The authors show that a wait-list in which applicants may reject offers without punishment is equivalent to a virtual currency system that gives applicants a steady income of tickets to enter into lotteries of their choosing.  When the cost of participation is low, these approaches are equivalent to two others: a variant of independent lotteries in which each applicant may enter at most one lottery, and a common lottery that simultaneously determines priority for all buildings. All of these mechanisms achieve better matching than the wait-list without choice or independent lotteries. However, this comes at a cost: these systems increase the number of apartments assigned to agents with good outside options, implying that fewer are allocated to those with the greatest need.

The mechanisms described above illustrate a trade-off between matching and targeting. Using a mechanism design approach, the authors show that if there are many applicants with good outside options, this trade-off cannot be avoided by any anonymous mechanism. In practice, this tension could be resolved by using observable characteristics to prioritize high-need applicants, and then granting these applicants a choice of apartments.

For more perspective on this research, the E-i-C received comments and reflections from a few experts including Professor Paul Milgrom and Mr. Mitchell Watt from Stanford University and Professor Martin A. Lariviere of Northwestern University. Their comments are given below:

Paul Milgrom and Mitchell Watt Stanford University

The Design of Lotteries and Waitlists for Affordable Housing Allocation by Nick Arnosti and Peng Shi illustrates how mathematical analysis in the hands of skilled researchers can lend deep new insights into important practical problems. This paper is distinguished by both its practical relevance and its theoretical elegance. The authors show an equivalence in terms of agent outcomes among very different looking matching rules. At first sight, the equivalence is quite surprising. They also identify a trade-off between allocating the right houses to the right agents and ensuring that housing goes to those most in need. Their findings suggest that certain commonly used allocation rules for public housing may successfully target lower-need individuals, but in so doing reduce the benefit of the allocation to successful applicants. Understanding trade-offs like this one will be especially valuable to policymakers in cities that use public subsidies to encourage or create affordable housing.

Equivalence Results

Arnosti and Shi present a dynamic model of the allocation of subsidized housing. Their analysis is aided by three clever formulation decisions:

  1. Agents are modelled as a continuum of types defined by the value of their outside option, a measure of their need for affordable housing. Agents’ strategies are type-dependent and Markovian. As a consequence, the authors can study the distribution of outcomes experienced by types rather than particular individuals.
  2. Houses and applicants arrive at a constant rate, and the analysis takes place in the ‘steady state’ or stationary equilibrium of the model. This suppresses a lot of minor details and focuses analytical attention squarely on the long-run average performance of a matching rule for each of the types.
  3. Time preferences arise from a per-period participation cost and an exogenous probability of departure from the system, which is common across the agents. This excludes behavioural preferences over the time of assignment but ensures that the dynamic system remains stable.

These are simplifying assumptions that are not a perfect match with reality, but they capture much of what matters for welfare analysis and, at the same time, lead to mathematical tractability.

The surprising and insightful finding that independent lotteries and the waitlist without choice are equivalent flow from the simplifying assumptions. In fact, the equivalence result applies more generally to any ‘periodic-offer’ matching rule in which the waiting time between offers is some random variable such that the expected waiting time to the next offer is decreasing in the number of periods since the last offer. A key insight is that the analysis for such mechanisms hinges on a single critical number: the fraction of surviving agents that receive an offer in each period. The argument is as follows:

  1. For each type of participant and for any periodic-offer matching rule, the decisions about whether to participate and which houses to accept are both determined by this critical fraction. Combining these two decisions, the total outflow of agents accepting offers is found to be an increasing function of the critical fraction.
  2. For equilibrium of periodic offer mechanisms, there is a unique critical fraction that balances the outflows from participants accepting offers with the inflows of housing. It follows that this fraction is the same for all such mechanisms and hence that each type adopts the same participation and acceptance strategies in all such mechanisms.
  3. Consequently, the type-dependent payoffs and expected costs of waiting must be the same in all periodic-offer mechanisms as well. 

The authors show that there is also a Myersonian interpretation of this argument. Adopting that perspective, agent utilities are determined via an envelope theorem by the probability of assignment, which in turn fixes the price in terms of waiting times that must be paid by participating agents in order to achieve incentive-compatibility. The idea that a market without financial transfers may equilibrate via waiting times is not new[1], but through their model, the authors remind us that the prices paid by agents may depend on their need, with consequent implications for welfare.

The authors do not explicitly include time preferences in their model, instead capturing temporal trade-offs via the departure probability. In many cases, the two modelling choices lead to similar conclusions, but this decision is consequential here. For example, in a model with classical time discounting, if we compare an independent lotteries rule to a waitlist without choice where agents face the same expected waiting times, the realized waiting times would be less random under the waitlist rule. By Jensen’s inequality, this would imply that the waitlist rule has a lower expected cost of participation and so agents will be more discerning as to which houses they accept, leading the outflow of agents to differ between the two mechanisms. But since this outflow is pinned down by the supply of housing, the expected waiting times must differ in the equilibria of the two matching rules, even if the expected utility is the same. This may explain why in practice some people may prefer the certain timing of a waitlist over a lottery-based system. However, this also highlights an advantage of the common lottery proposed by the authors – this scheme resolves uncertainty up front and reduces inefficiency in the sense that unmatched agents do not pay large participation costs under the scheme.

Comparing Mechanisms

This paper introduces two meaningful standards by which to evaluate a matching rule: ‘matching’, the extent to which applicants are matched to the right houses, and ‘targeting’, the extent to which houses are allocated to those most in need. These two standards appear, prima facie, to both be desirable goals for a matching mechanism, but as Arnosti and Shi demonstrate, if there are enough low-need candidates, there is a tension between them. The intuitive reason is simple when viewed through the lens of market-clearing wait-times for housing. Other things equal, if an allocation is expected to match successful agents better, then agents with better outside options will enter the mechanism, with the result that fewer houses remain for the targeted group with poorer outside options.

This finding can be important for policy design in affordable housing markets. Many cities around the world assign public and affordable housing using waitlists and lotteries. When the odds of winning a lottery are low (or waitlists are very long), the cost of turning down an offer for housing becomes large, and as a consequence people are more likely to accept housing that may be less suitable to them. Given increasing evidence about the importance of proximity to work for economic and other outcomes[2], improving the match quality of affordable housing may be a worthwhile goal. This paper shows that a simple change that empowers participants to be more selective over their eventual housing allocation, such as assigning each participant an annual budget of lottery tickets, may substantially improve the quality of matches.

References

Andersson, F., Haltiwanger, J. C., Kutzbach, M. J., Pollakowski, H. O., & Weinberg, D. H. (2018). Job displacement and the duration of joblessness: The role of spatial mismatch. Review of Economics and Statistics, 100(2), 203-218.

Barzel, Y. (1974). A Theory of Rationing by Waiting. The Journal of Law & Economics, 17(1), 73-95.

Martin A. Lariviere of Northwestern University

Many American cities have become expensive places to live. Rents are high and buying even a small home is beyond the reach of many. Residents consequently are left vulnerable – settling for dilapidated housing, dangerous neighborhoods, or long commutes. Even those who find acceptable housing are at risk. With rent taking a large chunk of monthly budgets, there is no way to build a financial reserve to buffer a lost job or unexpected bill. Homelessness inevitably looms.

Having a large swath of a city’s population so exposed is clearly problematic. Beyond the obvious humanitarian and moral concerns, civic leaders must also consider issues tied to economic growth and business development. How can an entrepreneur launch a new venture – particularly one dependent on a ready supply of workers – if no one earning what she is able to pay can afford to live in town? Cities themselves face challenges. Working as a fireman or schoolteacher is steady employment but is certainly not a quick way to get rich. When real estate costs rise enough, municipal workers may be forced out of the city.

There are, of course, ways for government to intervene to allow low-income renters a chance to compete in a market in which they are uncompetitive. A municipality could offer rent subsidies for those meeting certain criteria. Section 8 housing vouchers – a federally created program that is administered locally – would be one example. That gives support to the demand side of the market. Alternatively, an intervention can target the supply-side. In New York City and other locations, developers must often provide affordable apartment units to win approval for new projects. (“Affordable” here should not be confused with cheap – at least by national standards. An apartment that is inexpensive by Manhattan standards can still cost over two grand a month a require a household income above the national median.) But these steps create new problems of their own. They are not enough to bring housing security to every – or even most – low-income renters. The demand for housing subsidies or discounted apartments far exceeds the available supply.

There is consequently the need for a mechanism to award the limited supply to recipients. Some aspects of the process can be tied to verifiable information (e.g., earnings or number of dependents) but there will still be a large number of possible recipients. Any mechanism then needs to work at scale while promising some measure of fairness. In Washington State, King County runs a lottery for the privilege of joining a waiting list to receive a Section 8 voucher – a process compared to winning a golden ticket to Willy Wonka’s factory (Patrick, 2020). A rent voucher is generic and transportable; the difference to those who receive them at different points in time is at some level just time. The sooner one receives a voucher, the sooner one can move to better housing. The situation in New York City is somewhat more complicated. Here, the mechanism doles out apartments in specific buildings in specific neighborhoods and not all possible recipients value each of the options in the same way. How to think about matching heterogeneous recipients to heterogeneous properties is then important.

This gets us to “Design of Lotteries and Waitlists for Affordable Housing Allocation”

by Arnosti and Shi. At a high level, there are two ways one could approach the question of thinking about these allocation problems in an economic framework. One is to set up the machinery of mechanism design and consider what best scheme maximizes some objective. This approach has been taken by, for example, Chakravarty and Kaplan (2013) and Condorelli (2012). From these works we learn that fully differentiating between applicants or fully pooling them may both be optimal. Alternatively, one might pool some types while differentiating between others. That is, there are settings in which allocating goods efficiently (i.e., making the best possible match between items and applicants) is so critical that it is worth forcing some agents to incur unproductive costs (e.g., waiting). In other settings, the heterogeneity between agents is so small that they should all be treated the same (e.g., the planner could just run a lottery). From these insights one can argue that various mechanisms used in practice mimic the optimal mechanism – or at least a form of mechanism that is optimal in some settings.

Arnosti and Shi take a second approach – essentially running the mechanism design approach in reverse. As opposed to starting with the optimal scheme and seeing how real world mechanisms stack up, they start with mechanisms that are used in practice and ask how agents behave and what kind of performance different schemes deliver. While there are virtues to focusing on optimal design, there are also important benefits from evaluating existing mechanisms. First, Arnosti and Shi are able handle more general preferences – agents differ in both how they evaluate a particular opportunity as well as how dire their current situation. Second, they are able to show that the universe of real world mechanism may not be as big as one might think; that is, there are mechanisms whose descriptions differ but that induce the same behavior and ultimate allocation. Third, this is probably the right way to frame results for policy makers. While an optimal mechanism design approach may allow, for example, one to characterize settings where a simple lottery is appropriate, that does not help a policy maker unless one can assure her that those conditions are or are not met.

Furthermore, the analysis highlights two issues that determine the efficiency of the resulting allocation. If agents have different values for the properties being offered, then efficiency calls for matching well. A given apartment should go to someone who values it highly. But agents also differ in how acceptable their current living arrangements are. Ideally, a mechanism should target those whose outside options are most grim. A policy maker should care about both matching and targeting. To the extent that there is a tradeoff (and the paper offers some insight on this), different mechanisms can be ranked on these measures. Alternatively, a policy maker can defend their choice of mechanism based on which measure it favors. To my mind, this shows the strength of this paper. It elucidates an important problem, highlighting a clear tradeoff and offering decision makers insights on how that tradeoff can be managed using existing mechanisms.

 

References

Chakravarty S, Kaplan TR (2013) Optimal allocation without transfer payments. Games and Economic Behavior 77(1):1–20.

Condorelli D (2012) What money can’t buy: Efficient mechanism design with costly signals. Games and Economic Behavior 75(2):613–624.

Patrick, A. (2020), Vying for a golden ticket: King County Housing Authority reopens subsidized housing lottery, Seattle Times, Feb 12, https://www.seattletimes.com/seattle-news/homeless/vying-for-a-golden-ticket-king-county-housing-authority-reopens-subsidized-housing-lottery/

[1] Barzel (1974) provides an analysis of rationing via wait-times in the Chicago tradition.

[2] See, for example, (Andersson et al., 2018).

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