## 1 Introduction

A fundamental problem in economics is the allocation of resources. Economists have long known that properly designed markets allocate resources efficiently. However, in many circumstances, markets are not feasible, leading economists to design a host of market-like mechanisms. Non-market settings abound, including political offices, grant allocations, spectrum, access to higher-education institutions, intra-company resources, and so on. Mechanisms for allocation in these settings include auctions, lotteries, contests, queues, tradable permits and various forms of rationing (for example, see Myerson 1981; Milgrom and Weber 1982; Sah 1987; Wijkander 1988; Dasgupta and Maskin 2000; Ergin 2002; Che, Gale, and Kim 2013; Ausubel 2006; Platt 2009). Some mechanisms allocate the resources efficiently. For private values, the Vickrey auction (for one good) or the Vickrey-Clarke-Groves auction (for multiple goods) is efficient. On the other hand, mechanisms such as lotteries or queues are not efficient.

In many circumstances, not only are markets absent, but even market-like alternatives such as auctions or tradable permits are simply not viable. Some examples where this may be the case include access to municipal parks, public libraries, street parking, and a variety of natural resources such as surface and ground water, atmospheric resources like air, and biological resources including fish stocks. Access to these resources may be constrained legally, politically, by custom or social norm, or physically. Yet even within the constraints that bind allocation mechanisms for these resources, there are management and policy differences. A public charter school may admit students with a pure lottery or using a part lottery part queuing system. Our goal is to construct models that facilitate valuation of resources under those institutional settings that do not yield efficient outcomes, as in the example of the school just mentioned.

In a second best setting, in which resources are allocated inefficiently, assessing the gains from changes in rules and regulations requires knowing the probabilities of access induced by those rules and regulations. This issue was first addressed by Seneca (1970) and Mumy and Hanke (1975), who studied the value of providing a local public good when access was rationed by arrival. They showed that equally likely access to the public good significantly reduced its expected benefits. The literature following Mumy and Hanke (1975) focused primarily on congestion (Harrington 1988) and allocation by lotteries and by hybrid mechanisms whereby a portion is allocated by price and the remainder through a non-price mechanism (Boyce 1994; Taylor, Tsui, Zhu 2003; Scrogin 2009; Evans, Vossler, and Flores 2009), although Che, Gale, and Kim (2013) have recently studied the assignment of initial ownership of a good when individuals differ in their wealth. They show that market mechanisms favor those able to pay, and may be less efficient than non-market assignments when reselling is allowed. While motivated by the insights in Mumy and Hanke (1975), we find it useful to develop a more general representation of conditions of access in terms of agents’ likelihood of obtaining units or services from the resource. Unlike recent work on mechanism design under incomplete information (see, for example, Yoon 2011; Condorelli 2012, 2013), focused primarily on constructing optimal allocation mechanisms when agents invest in costly signals and the designer trades off allocative and expenditure inefficiency, we take the many possible access scenarios –de facto mechanisms for sorting users’ valuations– as primitives. We then develop a typology of assignments that will enable us to rank them not only when access is dictated by a thoughtful design but also when it is determined by real-world political and similar constraints. The efficient version of access occurs when it is provided in decreasing order of marginal value, so that the agents with the highest willingness-to-pay gain access first. A simple alternative occurs when a priori all agents have equal probabilities of access, regardless of their marginal values (what we term “uniform random access”).

We develop an approach that has efficient access, uniform random access and a variety of other non-price sorting patterns as special cases. This framework will allow us to rank rationing and allocation policies in a second-best world where potential users have heterogeneous values and there is a motivation for rationing access. Throughout we assume that agents must gain access to the resource to receive value. This rules out any non-use values and means that agents that do not have access get no value. We apply our approach to settings (i) in which individuals’ valuations are independent of total consumption (non-rival), and (ii) to common-pool resources where external costs associated with intra-season depletion and congestion do not alter the ranking of valuations across individuals (e.g. congestion costs are equal across agents or proportional to the their valuations).
^{1}

Our results have implications for the management of quasi-public goods: knowledge of marginal value functions is insufficient to determine the aggregate value of the good. This conclusion holds across a variety of quasi-public goods. As an example, consider a commercial fishery, where access to fish stocks during a season may be unrelated to marginal values, but the length of the season is subject to policy and may depend on cumulative harvest.

We illustrate how, in the absence of markets for access rights, knowledge of marginal values is insufficient to determine the welfare effects of access. Consider the case of a local agency evaluating the construction of a public facility (e.g. a library). The available budget for constructing the facility is $7 per year, which provides total capacity for the new facility of 2 units. There are three potential users of this facility, with valuations *v*_{1}=$2, *v*_{2}=$4, and *v*_{3}=$6. Access by these different individuals will depend on traffic, which is unfortunately difficult to predict. If users 1 and 2 arrive systematically earlier than individual 3, the expected welfare associated with the use of the new facility is $6, which is less than the annual costs and the agency should drop the project. If, on the other hand, little information is available on the odds of any of these individuals arriving before the other two, the agency may assume equal probability of arrival. In these circumstances, the expected welfare associated with the use of the facility is $8; since this amount is higher than the $7 in annual costs, the project should be undertaken. Many alternative scenarios are possible. Hence, preferences tell us something, but in the absence of a pricing mechanism and the sorting of valuations it induces, they need to be combined with explicit information on the probabilities of access. A corollary is that when access is equally likely across individuals with different valuations, efficient access can be determined with knowledge of the mean values of access.

## 2 A Model of Access

In this section we build a general model of access. The purpose of this model is to value scarce resources that are rationed in the absence of markets, that is, are accessed under different patterns of non-price sorting of marginal values. We start by assuming that there is a collection of agents seeking access to a common-pool resource or a public facility with finite capacity. We assume there is a mass *m* of agents, each demanding a single unit, and characterized by a willingness-to-pay of *v*. The valuation (or type) *v* is distributed independently over [0,*F*(*v*) and density *f*(*v*). Capacity is costly to provide and insufficient to satisfy the entire demand, *q* < *m*, making the resource scarce and creating the need for the manager to ration access.
^{2} As indicated earlier, valuations *v* may be independent of total consumption *q*, or represent agents’ marginal willingness-to-pay net of the external costs associated with total consumption. All that is required for the model to accommodate this later case is that external costs preserve the ranking of agents’ willingness-to-pay (e.g. costs are identical across users, or proportional to users’ valuations).
^{3} We assume that the regulator’s objective is utilitarian efficiency.
^{4}

In the next sections we show that total welfare derived from consumption of the *q* units is contingent upon how the rules that govern access to the resource sort individuals with different willingness-to-pay.

### 2.1 When Exclusion is Feasible

When exclusion is feasible, *efficient* utilization of the resource maximizes total willingness-to-pay by granting access to the individuals with the highest valuations. Thus, the marginal valuation *z* defining efficient access is given by 1–*F*(*z*)=*q/*m. Only agents with willingness-to-pay *z* and above are granted access. Total willingness-to-pay corresponding to the efficient use is

*z*, that is, conditional on gaining access (Che, Gale, and Kim 2013).

Note that efficient access, denoted by

In the following sections, we pursue the idea of access to resources when exclusion is not viable, to help understand the interaction between access, sorting and valuation.

### 2.2 When Exclusion is Infeasible

Let the different sets of controls that the manager could use to ration the resource (e.g. lotteries for charter schools, queuing for driver’s licenses, gear restrictions and seasonal closures in commercial fishing) represent distinct *access scenarios*. Let *assignment*,

*v*, and

*k*all valuations have equal probability to access the resource. We would then have

For any access scenario

*q*denotes the available quantity or the resource capacity.

We now introduce the function *uniform random access*. In this form of assignment, each user has an equal probability of access, regardless of valuation, and the expected value of each accessed unit of the resource is simply *v*, thus resulting in the modified probability distribution *y*.

Given access scenario

*expected marginal value function*

*v*to access the first unit of the resource, how likely to access the second unit, and so on. In turn,

*y*and

Next, note that in Figure 1 the efficient allocation achieves the maximum value associated with the consumption of any number of units. Indeed, *m* can be satisfied, is the welfare associated with both scenarios equal. Graphically, this implies that the area under the curve

Next we introduce the function

*Let**denote the fraction of the q available units (or capacity) assigned by access scenario**to individuals with valuations equal or less than**is defined by*

^{5}

We say that access scenario *value-dominates**value-equivalent* if *uniform random* assignment, characterized by *q* and access scenario

we have the following result

*If access scenario**value-dominates**, then*

See Appendix

Figure 3 illustrates how a given number of available units (or capacity) *q* generates different levels of welfare depending on the sorting induced by the access scenario. Under the efficient allocation *q* is given by the *area**i* total value is given by the *area**area**m* are identical. For this to hold, however, it must be the case that *area**area*

We now look at a concrete example to illustrate how proposition 1 can be readily used to establish the conditions under which one mechanism is superior or equivalent to another. We focus on the case of hybrid assignment rules previously studied in the literature. As noted in Evans, Vossler, and Flores (2009), several U.S. states have recently used hybrid mechanisms to allocate big game permits, a setting in which there is often a small number of permits available relative to total demand. In one such hybrid mechanism, a few permits are auctioned off with the remaining permits allocated via lottery. An alternative hybrid mechanism rations permits using a lottery with one-part or two-part entry tariffs (Scrogin 2005). A question of interest for policy makers is the relationship between the fraction *value-equivalent*,

By the efficiency of the discriminatory and uniform auction formats (for single-unit demands and independent private values), we know that the

We finish this section by characterizing a class of well-behaved assignments, *monotone assignments*. This definition will later allow us to derive optimality conditions for allocation of access under these assignments, assignments that Mumy and Hanke (1975) depicted only in graphical form.

(**Monotone Assignments)***Assignment**is said to be*:

- (i)
*monotone decreasing if access to the*${y}^{{}^{\prime}}th$ *unit second-order stochastically dominates (SSD) access to the*${{y}^{\prime}}^{\prime}th$ *unit*,${a}_{i}(\cdot \mathrm{,}{y}^{\prime}){\ge}_{SSD}{a}_{i}(\cdot \mathrm{,}{{y}^{\prime}}^{\prime})$ *, for all* ,${y}^{\prime}<{{y}^{\prime}}^{\prime}$ *or*

- (ii)
*monotone increasing if access to the*${y}^{{}^{\prime}}th$ *unit is second-order stochastically dominated by access to the*${{y}^{\prime}}^{\prime}th$ *unit*,${a}_{i}(\cdot \mathrm{,}{y}^{\prime \prime}){\ge}_{SSD}{a}_{i}(\cdot \mathrm{,}{{y}^{\prime}}^{})$ *, for all*${y}^{\prime}<{{y}^{\prime}}^{\prime}$ *, or*

In words, *monotone decreasing assignments* grant higher willingness-to-pay units a larger probability of being satisfied before lower willingness-to-pay units. *Monotone increasing assignments*, on the other hand, grant lower willingness-to-pay units a larger probability of being satisfied first than higher willingness-to-pay units (see appendix for proof). The following lemma ranks monotone increasing and decreasing assignments

*For a given distribution of valuations**, monotone decreasing assignments always value-dominate monotone increasing assignments*.

See Appendix

Figure 4 depicts a marginal value function and the associated expected marginal value functions for access scenarios that induce different monotone assignments:

## 3 Access Across Multiple Sectors

To analyze the implications of different rules of access on optimal access allocation, we now introduce additional sectors whose agents seek to exploit the public resource. Our objective is to demonstrate that our framework can be extended to the case of competing sectors, typically the situation in real world settings such as fisheries and water resources. Let there be *n* sectors, with mass *j*, each characterized by willingness-to-pay of *X* among the different sectors to maximize welfare.

### 3.1 The Efficient Allocation

Let *j* such that (i) *j*. Denote *z* the value that simultaneously satisfies condition (i) for all sectors. Eliminating *z* solves *z* is the marginal valuation in each sector that defines the efficient allocation (i.e. as prescribed by the equi-marginal principle). In this efficient allocation only individuals with valuations *z* or higher gain access. To see that this is necessarily the case, note that if this condition did not hold, that is, if *j* receives

where

is the expectation of an individual’s valuation in sectors *j* conditional on receiving quota.

### 3.2 Allocation when Exclusion is Infeasible

We now derive the optimality condition for the allocation of capacity among multiple sectors for any access scenarios, when exclusion of low willingness-to-pay agents is infeasible. The optimal allocation maximizes total welfare, which is given by

where

*For any collection**of access scenarios, the (nx1) vector**of capacity shares allocated to different sectors maximizes total welfare if the following conditions hold*

See Appendix

According to this proposition, total capacity (or quota) should be allocated until the expected value of the last unit is equal across sectors (condition SC-1), with the expected value of that last unit in each sector given by the corresponding probability distribution of access to that unit across valuations.
^{6} Condition SC-2 is necessary to guarantee a local maximum. This proposition characterizes optimality over a wide range of access settings as it does not impose any restrictions on the shape of the expected value functions. Uniqueness is not guaranteed unless monotonicity assumptions are made on the functions

The following lemma characterizes the optimal allocation for the special case of monotone assignments. For both decreasing and increasing assignments uniqueness is guaranteed as monotonicity implies, respectively, global concavity and convexity.

*(i) Let the collection**of access scenarios induce monotone decreasing assignments: vector**maximizes total welfare if and only if*

*(ii) Let**denote the set of**, where**for**, and define**and**. Let the collection**of access scenarios induce monotone increasing assignments: vector**maximizes total welfare if and only if*

See Appendix

According to the lemma, the optimal allocation under monotone decreasing assignments is defined by the vector of quotas that equates marginal expected benefits across all sectors (condition (i)).
^{7}Figure 5 illustrates the case for the allocation between two sectors. The optimum is found at the point of intersection of the expected value functions. On the other hand, under monotone increasing assignments, that is, when high valuations are expected to access last, condition (ii) states that the optimal allocation is a corner solution that satisfies the entire demand of the sectors where access generates the largest welfare and excludes all other sectors. Figure 6 depicts this case for the allocation of access between two sectors. Total welfare in sector 1, when it receives the entire quota *X*, exceeds that in sector 2 when its entire demand is satisfied. Accordingly, the entire demand of sector 1 is satisfied, and sector 2 receives no quota. If, for example, Figure 6 depicted access to campgrounds along the Inca trail and sector 2 represented independent backpackers, then optimality would call for commercial, professionally guided trips only. The rationale for the result is straightforward. Backpackers would otherwise displace some high-value guided tourism by getting to campsites first.
^{8}

## 4 Conclusion

We have constructed a general model of access which has allowed us to value different assignments when resources are allocated in the absence of markets. These “assignments” appear in a variety of forms. They include lotteries, queues, priority lists, and restrictions based on personal characteristics uncorrelated with the privately held values. Our approach can be used to establish the design conditions under which one mechanism is superior to another. Because the assignment of transferable property rights tends to achieve the highest returns and most efficient use of a resource, economists have since long recommended its adoption. Those cases that resist tend do so for good reason –the high costs of imposing property rights, undesirable outcomes that emerge as a consequence, entrenched political interests, and the desire to allocate access for merit reasons can all contribute to the allocation of resources without property rights. We demonstrate that marginal value schedules are far less useful in allocating access when property rights are unfeasible. In these settings, the criterion for optimal allocation –a generalized version of the marginal principle– combines information on both the marginal value schedules and the assignments determining the probabilities of access to the resource. Our results highlight the need for practitioners handling scarce resources to think explicitly on how their rule making sorts heterogeneous users. The design of empirical strategies and survey instruments for assessing users’ probabilities of access under alternative scenarios should rank high in the agenda of managers limited in their ability to use price signals. Our framework has allowed us to rank rationing policies (and the access scenarios they induce) in a wide range of real-world, second-best settings. Alternatively, the approach could help inform managers’ optimal supply of (costly) capacity under various access scenarios. Information on who is likely to benefit from access is essential for the optimal provision of public resources.

We need to demonstrate that

Integrating by parts yields

The first term on the right-hand side is zero because

Monotone Assignments and Probability of Access

*Monotone decreasing assignments* grant higher willingness-to-pay units a larger probability of being satisfied before lower willingness-to-pay units. *Monotone increasing assignments*, on the other hand, grant lower willingness-to-pay units a larger probability of being satisfied first than higher willingness-to-pay units. To see that this is indeed the case, rewrite

*v*. Thus, we can rewrite the condition for monotone decreasing assignments as

The *k*th leading principal minor of expression [13] can be determined by Gaussian elimination. First, subtract the *k*th column from each of the first *i* by *k*th column, to obtain a lower triangular determinant whose diagonal components are

*k*th leading principal minor is given by

*k*possible

*k*th order principal submatrix of

*H*. Thus, it is sufficient that

*i*for the

*k*th leading principal minor to have the same sign as

*H*to be negative definite. Consequently, SC-2 is a sufficient condition for

*W*.

^{9}☐

By definition of monotone decreasing assignments (case (i)),

By definition of monotone increasing assignments (case (ii)),

Let

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## Footnotes

^{1}

In particular, the approach is directly applicable to the case of common pool resources where intra-season depletion is negligible, as in most recreational and commercial fisheries. Indeed, for the majority of fish stocks, managers set effort restrictions and removal targets for each entire season with the objective to ensure the sustainability of the stocks across seasons (rather than controlling the in-season evolution of the stock).

^{2}

We assume that the manager lacks the authority to charge users for access, that the willingness-to-pay of the lowest valuation types is insufficient to cover the costs of expanding capacity, or that capacity is otherwise exogenously fixed (e.g. national park, annual harvest quota set by the biologists, etc.).

^{3}

In the former case we write the marginal willingness-to-pay net of external costs as *u*(*v*, *q*)=*v*–*c*(*q*), where *c*′(*q*)≥0, in the later case as *u*(*v*, *q*)=*v*(1–

^{4}

While we acknowledge that the regulator may adopt alternative social welfare functions in order to advance objectives such as fairness and social inclusion, we focus on utilitarian welfare for being the most common and where our approach can inform policy.

^{5}

It is easy to show that in this case *y*, where

^{6}

For external costs associated with congestion or in-season depletion, SC-1 above, *i*, remains necessary for a local maximum.

^{7}

In the case of external costs due to congestion or in-season depletion, condition i) is modified as indicated in note 6.

^{8}

Indeed, since 2002 the permits required for access to the Inca Trial are sold exclusively through licensed tour operators. Independent, unguided visitors are no longer allowed. For details see http://www.mincetur.gob.pe/TURISMO/proyectos/regl_uso_turistico.htm.

^{9}

For the case of external costs,