An agency holds Q^* units of an indivisible good to be distributed over N > Q^* individuals, where N is assumed large and Q^* may be small or large relative to N. Each individual possesses unit demand and a private valuation v for the good that is independently drawn from the interval [0, ∞) according to the distribution function F(v) with density f ( v ) = F ′ ( v ) . Q^*, N, and F(v) are assumed commonly known, and each individual also knows his/her valuation but not the valuation of any other individual. Individual expected demand is then represented by the survival function F⁺(v) = 1 − F(v). That is, F⁺(v) is the probability an individual’s valuation is at least v or the probability the individual would be willing to purchase a unit if the price were v. Aggregate expected demand is NF⁺(v), and the expected market clearing price v^* satisfies NF⁺(v) = Q^*.

The agency’s objective in allocating Q^* is to maximize aggregate expected consumer surplus by posting a uniform price p Î [0, v^*] and randomly assigning units if excess demand exists. Individual i’s expected probability of acquiring a unit conditional upon a valuation v_i Î [p, ∞) is π ( p ) = Q + / N F + ( p ) and i’s expected consumer surplus is written:

E C S ( p ) i = ∫ p ∞ π ( p ) ( v − p ) f ( v ) d v = Q * N F + ( p ) ( 1 F + ( p ) ∫ p ∞ v f ( v ) d v − p ) ∀ i ∈ N (1)

Aggregate expected consumer surplus the agency seeks to maximize is obtained by multiply (1) by expected demand, and the pricing problem is written:

max p E C S ( p ) = Q * ( 1 F + ( p ) ∫ p ∞ v f ( v ) d v − p ) (2)

Given the value distribution and clearing price, the solution to (2) may be described by one of three mechanisms for allocating Q^*. These will be referred to as market assignment at the clearing price p = v^*, pure lottery assignment at price p = 0, and hybrid lottery assignment at a price p Î (0, v^*).

To proceed, note the term in brackets in (1) and (2) is the individual expected consumer surplus if a unit could be purchased outright for p. That is, it is the difference between a price p and the individual mean private valuation conditional upon it being at least as large as p. The expression is the value distribution equivalent of a lifetime distribution’s mean residual life function, defined M R L ( t ) = E [ X − t | X ≥ t ] . For a lifetime random variable X, MRL(t) is the expected remaining life of the respective entity (e.g., a machine or individual) conditional upon its ‘surviving’ to time t. MRL(t) is a close counterpart of the hazard rate function H R ( t ) = f ( t ) / F + ( t ) and [9] ’s eta function η ( t ) = − f ′ ( t ) / f ( t ) , and all are long-standing workhorses in the reliability engineering and stochastic ageing literature (see e.g., [11] and [12] ). Denoting the value distribution’s mean residual value function by MRL(v) and its hazard rate and eta functions by HR(v) and ƞ(v), ECS(p) in (2) is then the product of Q^* and MRL(v) evaluated at price p = v for v Î [0, v^*]. That is, ECS(p) = Q^*MRL(p).

The relationship between ECS(p) and MRL(v) is appealing for characterizing the solution to (2) because the shape of ECS(p) (monotonic or non-monotonic) is the same as that of MRL(v) for vÎ [0, v^*] and because the shape of MRL(v) is implied directly by HR(v) and indirectly by ƞ(v) as shown by [9] [13] and [14] ³. Evaluation of HR(v) or ƞ(v) may then be sufficient for determining the optimal allocation mechanism. The results are formalized for monotonic and non-mo- notonic classes of MRL(v), HR(v), and ƞ(v) in the remainder of this section and selected cases are evaluated numerically in Section 3.

In order to distinguish the monotonic and non-monotonic classes of MRL(v) from one another, let IMRL (DMRL) denote MRL(v) that is monotonic increasing (decreasing). That is, MRL(v) is IMRL (DMRL) if MRL'(v) > 0 (<0) for v Î [0, ∞). For the non-monotonic classes, let DIMRL (IDMRL) denote the case in which MRL(v) initially decreases (increases) to a minimum (maximum) at v₁ Î (0, ∞) and then increases (decreases). That is, MRL(v) is DIMRL (IDMRL) if MRL(v₁) = 0, MRL'(v) < 0 (MRL'(v) > 0) for v < v₁, and MRL'(v) > 0 (MRL'(v) < 0) for v > v₁. Further, as MRL'(v) = HR(v)MRL(v) − 1, it follows that at MRL(v)’s interior minimum or maximum, MRL(v₁) = 1/HR(v₁). The monotonic and non- monotonic classes of HR(v) and ƞ(v) will be similarly referenced below (e.g., IHR and DIHR).

Given ECS(p) = Q^*MRL(p) if p = v, it is clear that ECS(p) is maximized by market (pure lottery) assignment of Q^* if MRL(v) is IMRL (DMRL). If instead MRL(v) is non-monotonic, then ECS(p) may or may not be monotonic for p Î [0, v^*) and information on the relative scarcity or abundance of the good is needed to determine the optimal mechanism directly from MRL(v) or indirectly from HR(v) and ƞ(v). To link HR(v) and ECS(p) through MRL(v) the following sufficient conditions from [13] and [14] are used:⁴

i) If HR(v) is IHR (DHR), then MRL(v) is DMRL (IMRL).

ii) If HR(v) is IDHR and f(0)μ< 1 (f(0)μ ≥ 1), then MRL(v) is DIMRL (IMRL).

iii) If HR(v) is DIHR and f(0)μ> 1 (f(0)μ ≤ 1), then MRL(v) is IDMRL (DMRL).

Monotonicity of HR(v) therefore implies monotonicity of MRL(v), whereas non-monotonicity HR(v) may or may not imply non-monotonicity of MRL(v). For linking ƞ(v) and ECS(p) through HR(v), sufficient conditions from [9] are used. Defining ε and δ, respectively, as lim v → ∞ f ( v ) = ε and lim v → ∞ η ( v ) / H R ( v ) = δ , the conditions are:

i) If ƞ(v) is I (D), then HR(v) is IHR (DHR).

ii) If ƞ(v) is DI and ε = 0 or δ < 1 (ε = ∞ or δ > 1), then HR(v) is IHR (DIHR).

iii) If ƞ(v) is ID and ε = 0 or δ < 1 (ε = ∞ or δ > 1), then HR(v) is IDHR (DHR).

Monotonicity of ƞ(v) therefore implies monotonicity of HR(v) and of MRL(v) by transitivity, whereas if ƞ(v) is non-monotonic, then HR(v) and MRL(v) may or may not be non-monotonic. While the above conditions are used here for characterizing optimal allocation rules, they have proven invaluable for determining the shape of MRL(t) in reliability engineering and stochastic ageing studies because MRL(t) typically does not appear in closed form (see e.g., [11] ).

The solution to (2) is given first in Proposition 1 and Corollary 1 for cases of HR(v) and ƞ(v) that imply MRL(v) that is monotonic, followed by the cases of non-monotonic MRL(v) in Propositions 2 and 3.

Proposition 1: Given Q^*, N, and F(v):

i) If HR(v) is DHR (IHR), then ECS(p) is maximized by market (pure lottery) assignment of Q^*.

ii) If HR(v) is IDHR and μf(0) ≥ 1 (DIHR and μf(0) ≤ 1), then ECS(p) is maximized by market (pure lottery) assignment of Q^*.

Proof: i) If HR(v) is DHR (IHR), then MRL(v) is IMRL (DMRL). Therefore, ECS(v^*) > ECS(p) for p Î [0, v^*) (ECS(0) > ECS(p) for p Î (0, v^*]) and ECS(p) is maximized by market (pure lottery) assignment of Q^*.

ii) If HR(v) is IDHR and μf(0) ≥ 1 (DIHR and μf(0) ≤ 1), then MRL(v) is IMRL (DMRL). Therefore, ECS(v^*) >ECS(p) for pÎ [0, v^*) (ECS(0) >ECS(p) for p Î (0, v^*]) and ECS(p) is maximized by market (pure lottery) assignment of Q^*.

Distributions for which Proposition 1 applies (and Propositions 2 and 3 below) are reported in Table 1. If private values are Pareto distributed (see [10] for a detailed example) or exponential-geometric distributed, then MRL(v) is IMRL and ECS(p) is maximized by market assignment of Q^*, and if private values are uniform, truncated normal, or Gompertz-Makeham distributed, then MRL(v) is DMRL and ECS(p) is maximized by allocating Q^* by pure lottery. If instead values are Weibull or gamma distributed, then market or pure lottery assignment could be optimal as MRL(v) is either IMRL or DMRL.

Corollary 1 follows immediately from Proposition 1 and [9] ’s sufficient condi-

^aThe distributions and their properties are discussed in [11] [12] or [17] .

tions on ƞ(v) discussed above.

Corollary 1: Given Q^*, N, and F(v):

i) If ƞ(v) is D(I), then ECS(p) is maximized by market (pure lottery) assignment of Q^*.

ii) If η(v) is DI (ID) and ε = 0 or δ < 1 (ε = ∞ or δ > 1), then ECS(p) is maximized by pure lottery (market) assignment of Q^*.

Proof: The results follow by transitivity from Proposition 1 and [9] ’s sufficient conditions on η(v) for HR(v) to be DHR or IHR, implying MRL(v) is IMRL or DMRL, respectively.

In the remaining cases, HR(v) implies MRL(v) is non-monotonic. A graphical depiction of HR(v) and MRL(v) that are i)IDHR and DIMRL and ii)DIHR and IDMRL appear in Figure 1 and Figure 2, respectively. Included are MRL(v) and 1/HR(v). From Figure 1, if the clearing rate v^* is such that MRL(v^*) > MRL(v_μ) = μ, then F⁺(v^*) < F⁺(v_μ) and ECS(p) is maximized by market assignment, and if MRL(v^*) < MRL(v_μ)= μ, then F⁺(v^*) >F⁺(v_μ) and ECS(p) is maximized by pure lottery assignment. From Figure 2, if v^* > v₁ then MRL(v^*) < MRL(v₁), F⁺(v^*) < F⁺(v₁), and ECS(p) is maximized by allocating Q^* by hybrid lottery at price p = v₁; otherwise, market assignment of Q^* is optimal.

Figure 1 and Figure 2 also indicate that the optimal mechanism can be determined from evaluation of HR(v) and μ if v^* is sufficiently scarce or abundant. The results are formalized in Proposition 2 for HR(v) implying DIMRL and in Proposition 3 for HR(v) implying DIMRL.

Proposition 2: Given HR(v) that is IDHR and μf(0) < 1:

i) If HR'(v^*) > 0 and 1/HR(v^*) ≥ μ, then ECS(p) is maximized by pure lottery assignment of Q^*.

ii) If HR'(v^*) < 0 and 1/HR(v^*) ≥ μ, then ECS(p) is maximized by market assignment of Q^*.

iii) If lim v → ∞ 1 / H R ( v ) ≤ μ , then ECS(p) is maximized by pure lottery assignment of Q^*.

Proof: In all cases, if HR(v) is IDHR and μf(0) < 1, the MRL(v) is DIMRL.

i) As MRL'(v) < 0 if HR'(v) > 0 and 1/HR(v) ≥ μ, it follows that if HR'(v^*) > 0 and 1/HR(v^*) ≥ μ, then μ > MRL(v^*). Therefore, ECS(0) > ECS(p) for p Î (0, v^*] and ECS(p) is maximized by pure lottery assignment of Q^*.

ii) If HR'(v^*) < 0 and 1/HR(v^*) ≥ μ, then MRL(v^*) > μ. It follows that ECS(v^*) > ECS(p) for p Î [0, v^*) and ECS(p) is maximized by market assignment of Q^*.

iii) From [16] , lim v → ∞ M R L ( v ) = lim v → ∞ 1 / H R ( v ) if the latter limit is finite. Therefore,

lim v → ∞ M R L ( v ) ≤ μ if lim v → ∞ 1 / H R ( v ) ≤ μ . It follows that ECS(0) > ECS(p) for p Î (0, v^*] and ECS(p) is maximized by pure lottery assignment of Q^*.

Figure 1 depicts cases i and ii of Proposition 2.⁵ Case i applies to the lower interval v Î [0, v_L]. As HR'(v) > 0 and 1/HR(v) ≥ μ, it follows that μ > MRL(v). Therefore, if Q^* is sufficiently abundant such that v^* ≤ v_L, then ECS(0) > ECS(v^*), and ECS(p) is maximized by allocating Q^* by pure lottery. For case ii, as HR'(v) < 0 and 1/HR(v) ≥ μ in the upper interval v Î [v_U, ∞), it follows that MRL(v) > μ. Therefore, if Q^* is sufficiently scarce such that v^* ≥ v_U, then ECS(v^*) > ECS(0) and market assignment is optimal. Similar to Proposition 1, examples of distributions for which Proposition 2 applies are reported in Table 1. Included are the lognormal, inverse Gaussian, log-logistic, and Burr XII distributions and variants of the Weibull distribution.

Lastly, the case in which HR(v) implies MRL(v) that is IDMRL is covered in Proposition 3.

Proposition 3: Given HR(v) that is DIHR and μf(0) > 1:

i) If HR'(v^*) < 0 and 1/HR(v^*) ≤ μ, then ECS(p) is maximized by market assignment of Q^*.

ii) If HR'(v^*) ≥ 0, then ECS(p) is maximized by hybrid lottery assignment of Q^*.

Proof: In both cases, if HR(v) is DIHR and μf(0) > 1, then MRL(v) is IDMRL.

i) If HR'(v^*)< 0 and 1/HR(v^*) ≤ μ, then v^* < v₁. As Q^* > NF⁺(v) for v>v^* and MRL'(v) > 0 for v < v₁, it follows that ECS(v^*) > ECS(p) for p Î [0, v^*) and ECS(p) is maximized by market assignment of Q^*.

ii) If HR'(v^*) ≥ 0, then v^* > v₁ and MRL(v) > MRL(v^*) for v Î [v₁, v^*). Therefore, ECS(p) > ECS(v^*) for p Î [v₁, v^*) and ECS(p) is maximized by hybrid lottery assignment of Q^*.

Results from Proposition 3 are depicted in Figure 2. Case i applies to the interval v Î [0, v_L]. As HR'(v) < 0 and 1/HR(v) ≤ μ it follows that MRL(v) > μ and MRL'(v) > 0. Therefore, if the scarcity of Q^* is such that v^* ≤ v_L, then ECS(0) < ECS(v^*), and ECS(p) is maximized by selling Q^* outright. For case ii, as MRL'(v) < 0 for v Î (v₁, ∞), if HR'(v^*) ≥ 0 (or v^* ≥ v₀) then MRL(v₀) ≥ MRL(v^*). It follows that MRL(v) > MRL(v^*) for some values in the interval (0, v^*). Therefore, if Q^* is sufficiently abundant such that v^* ≥ v₀, then ECS(p) attainable from allocating Q^* by hybrid lottery exceeds that from market assignment. Table 1 report several distributions for which Proposition 3 applies. Included are the beta and exponential power distributions and variants of the Weibull distribution.

The survival and hazard rate functions of the log-logistic distribution are given by:

F + ( v ) = 1 1 + ( α v ) β ,     H R ( v ) = α β β v β − 1 1 + ( α v ) β (3)

where α > 0 and β > 0. Derivation of MRL(v) yields:

M R L ( v ) = 1 + ( α v ) β α β [ B ( 1 β , 1 − 1 β ) − B F ( v ) ( 1 β , 1 − 1 β ) ] (4)

where F(v) = 1 − F⁺(v), B(1/β, 1 − 1/β) is the beta function, B_F(v)(1/β, 1 − 1/β) is the incomplete beta function, and μ = M R L ( 0 ) = B ( 1 / β ,1 − 1 / β ) / α β (see [11] ). Despite the complexity of MRL(v), if β > 1, then HR(v) is IDHR and μf(0) − 1 < 0 and therefore MRL(v) is DIMRL.

To determine the optimal mechanism directly from MRL(v) or indirectly from HR(v) and μ (Proposition 2), and referencing Figure 1, if MRL(v^*) > MRL(v_μ) = μ, then F⁺(v^*) < F⁺(v_μ) and ECS(p) is maximized by selling Q^* outright. If instead MRL(v^*) < MRL(v_μ) = μ, then F⁺(v^*) > F⁺(v_μ) and ECS(p) is maximized by pure lottery. F⁺(v_μ) is calculated from (3) and (4) across a range of parameter values and compared against F⁺(v^*) = Q^*/N (or NF⁺(v^*) = Q^*) to determine from MRL(v) whether market or pure lottery assignment is dominant.

For evaluation of Proposition 2, if 1/HR(v^*) > 1/HR(v_L) = μ and HR'(v^*) > 0, then F⁺(v^*) > F⁺(v_L) and MRL(v^*) < MRL(v) for v Î [0, v^*). It follows that ECS(p) is maximized by allocating Q^* by pure lottery. If instead 1/HR(v^*) > 1/HR(v_U) = μ and HR'(v^*) < 0, then F⁺(v_U) > F⁺(v^*) and MRL(v^*) > MRL(v) for v Î [0, v^*). Therefore, ECS(p) is maximized by selling Q^* outright. F⁺(v_L) and F⁺(v_U) are calculated and compared against F⁺(v^*) = Q^*/N to determine the relative scarcity or abundance of the good for the mechanism to be identified from HR(v) and μ.

The first half of Table 2 reports F⁺(v_μ), F⁺(v_L), F⁺(v_U), and the Gini coefficient for the log-logistic distribution across a range of (α, β) values. The Gini coefficient indicates that individual private values become increasingly homogenous as β increases from 1. Considering F⁺(v_μ), the results indicate lim β → 0 F + ( v μ ) = 1 and lim β → 0 F + ( v μ ) = 0 . It follows that given the relative scarcity of the good, market (pure lottery) assignment will tend to dominate as β → 1 (β → ∞). For example, if β ≤ 2 then F⁺(v_μ) ≥ 0.50. Therefore, MRL(v^*) > μ and ECS(v^*) > ECS(0) if F⁺(v^*) < 0.50. It follows that market assignment dominates pure lottery assignment if Q^* is less than half of N. In contrast, if β ≥ 3 then F⁺(v_μ) < 0.10, so pure lottery assignment will dominate if N is less than ten times greater than Q^*.

^aFrom F⁺(v) = 1 − F(v), where F(v) is the cumulative distribution function, F⁺(v_μ) references MRL(v_μ) = μ; F⁺(v_L) references 1/HR(v_L) = μ; similarly, F⁺(v_U) references 1/HR(v_U) = μ; F⁺(v₀) references HRʹ(v₀) = 0; and F⁺(v₁) references MRLʹ(v₁) = 0. The values and equalities also appear in Figure 1 and Figure 2.

In contrast to the log-logistic distribution, the upper bound of the beta distribution is finite and the survival and hazard rate functions do not appear in closed form. The functions may be written:

F + ( v ) = B ( α , β ) − B v ( α , β ) B ( α , β ) ,     H R ( v ) = v α − 1 ( 1 − v ) β − 1 B ( α , β ) − B v ( α , β ) (5)

where α > 0 and β > 0, and B(α, β) and B_v(α, β) are the beta function and incomplete beta function, respectively. Derivation of MRL(v) yields:

M R L ( v ) = α α + β 1 − B v ( α + 1 , β ) / B ( α + 1 , β ) 1 − B v ( α , β ) / B ( α , β ) − v (6)

where μ = M R L ( 0 ) = α / ( α + β ) . [15] showed that HR(v) is DIHR and MRL(v) is IDMRL for α < 1.

To determine the mechanism that maximizes ECS(p) directly from MRL(v), and referencing Figure 2, note that MRL(v) attains an interior maximum atv₁, so if v₁ ≥ v^* then F⁺(v^*) ≥ F⁺(v₁) and ECS(p) is maximized by market assignment of Q^*. If insteadv₁ < v^* then F⁺(v^*) +(v ₁) and ECS(p) is maximized by allocating Q ^* by hybrid lottery at price p = v ₁. F ⁺(v ₁) is calculated from (5) and (6) across a range of parameter values and compared against F ⁺(v ^*) = Q ^*/N to determine from MRL(v) whether market or hybrid lottery assignment is optimal.

For evaluating Proposition 3, if 1/HR(v^*) < 1/HR(v_L) = μ and HR'(v^*) < 0, then F⁺(v^*) > F⁺(v_L) and MRL(v^*) > MRL(v) for v Î [0, v^*). It follows that ECS(p) is maximized by selling Q^* outright. If instead 1/HR(v^*) < 1/HR(v₀) and HR'(v) > 0, then F⁺(v^*) +(v ₀) and MRL(v ^*) < MRL(v) for v Î [0, v ^*). ECS(p) is therefore maximized by allocating Q ^* by hybrid lottery. F ⁺(v _L) and F ⁺(v ₀) are calculated from (5) and (6) and compared against F ⁺(v ^*) to determine the relative scarcity or abundance of the good for the optimal mechanism to be identified from HR(v) and μ.

⁶The Gini coefficient of the beta distribution is given by (α/2)B(α + β, α + β)/(B(α, α)B(β, β)).

The second half of Table 2 reports F⁺(v₁), F⁺(v_L), F⁺(v₀), and the Gini coefficient across a range of (α, β) values⁶. In contrast to the log-logistic distribution, the homogeneity of individual private values is relatively constant across the cases considered as indicated by the Gini coefficients. Referencing F⁺(v₁), the results indicate the interior maximum of MRL(v) is located well within the upper and lower bounds of the distribution, with F⁺(v₁) ranging from 0.66 to about 0.40. As ECS(p) is maximized by hybrid lottery assignment at price p = v₁ if F⁺(v₁) > F⁺(v^*) and by market assignment at p = v^* if F⁺(v₁) ≤ F⁺(v^*), it follows that across the cases reported in Table 2 the hybrid lottery is optimal if Q^* is less than fourth tenths of N and market assignment is optimal if Q^* is at least two thirds as large as N.

The results in Table 2 also identify the relative scarcity or abundance of Q^*needed for the optimal allocation mechanism to be determined from HR(v) and μ, as formalized in Proposition 3. As F⁺(v_L) ranges from 0.73 to 0.63 and F⁺(v₀) ranges from about 0.40 to 0.22, it follows that allocation of Q^* by hybrid lottery is optimal over all cases in Table 2 if F⁺(v^*) < 0.22, or N is at least five times greater than Q^*, whereas market assignment is optimal if F⁺(v^*)> 0.73, or Q^* is at least three-fourths as large as N. Overall, in settings in which the agency has a limited number of units to be allocated over a large population of individuals, if private values are beta distributed with α < 1, then the expected consumer surplus from screening low-valued individuals by posting a non-zero price and assigning units by lottery will exceed that resulting from selling the units outright.