SSelling Two Identical Ob jects ∗ Sushil Bikhchandani † and Debasis Mishra ‡ November 9, 2020
Abstract
It is well-known that optimal (i.e., revenue-maximizing) selling mechanisms in mul-tidimensional type spaces may involve randomization. We study mechanisms for sellingtwo identical, indivisible objects to a single buyer. We analyze two settings: (i) de-creasing marginal values (DMV) and (ii) increasing marginal values (IMV). Thus, thetwo marginal values of the buyer are not independent. We obtain sufficient conditionson the distribution of buyer values for the existence of an optimal mechanism that isdeterministic.In the DMV model, we show that under a well-known condition, it is optimal tosell the first unit deterministically. Under the same sufficient condition, a bundlingmechanism (which is deterministic) is optimal in the IMV model. Under a strongersufficient condition, a deterministic mechanism is optimal in the DMV model.Our results apply to heterogenous objects when there is a specified sequence inwhich the two objects must be sold. ∗ We are grateful to Moshe Babaioff, Tilman B¨orgers, Rahul Deb, Bhaskar Dutta, Jingtao Li, AroonNarayanan, Kolagani Paramahamsa, and seminar participants at Ashoka University, Delhi Economic The-ory Workshop, Essex University, University of Michigan, Penn State University, Stony Brook University,University of Toronto, and UCLA for helpful comments. † Anderson School at UCLA, Los Angeles ( [email protected] ). ‡ Indian Statistical Institute, Delhi ( [email protected] ). a r X i v : . [ ec on . T H ] N ov Introduction
We consider optimal, i.e., expected revenue maximizing, mechanisms for selling two identicalunits of an object to a buyer. The buyer’s type (values for the units) is two dimensionaland privately known to the buyer. We focus on two cases: decreasing marginal values andincreasing marginal values. Thus, the buyer’s values for the units are not independent.The assumption of homogenous objects reduces the dimensionality of the price space andtherefore the dimensionality of random allocation rules (compared to heterogenous objects).While this represents a simplification of the problem of finding an optimal mechanism, thecorrelation of values in our paper increases complexity.A general solution to the optimal mechanism design problem for the sale of multipleindivisible products is unknown. Unlike the single product case, the optimal mechanism forselling two or more products may involve randomization (see Thanassoulis (2004), Manelliand Vincent (2006), Pycia (2006), and Hart and Reny (2015)). Our objective is to findsufficient conditions under which a deterministic mechanism is optimal among all mechanismsfor selling two identical units, including random mechanisms.We assume that the seller can commit to a mechanism. Implicit in this is the assumptionthat the mechanism can be objectively verified by both parties. As Laffont and Martimort(2002) emphasize, it is easier to verify a deterministic mechanism than a random mechanism.For instance, commitment by a seller to a random mechanism may not be credible in a one-shot interaction with a buyer. Perhaps this is a reason for the limited use of randomizedselling methods. Under our sufficient conditions, the seller does not sacrifice optimality forcredible commitment to a (deterministic) mechanism.With homogenous objects, there is a natural order of transactions: the second unit canbe sold only after the first unit is sold. In some settings with two heterogenous objects, oneof the two objects can be sold only after the other object is sold. For instance, the warrantyon a product is only sold to a buyer who purchases the product. Another example is when aseller offers two versions of a product, basic or premium. The premium version of a productcan be viewed as the basic version plus an upgrade. That is, the upgrade is sold only if the Random selling methods, called opaque selling, are used by travel websites such as Hotwire and Priceline. See Armstrong (2016) for a discussion of this issue. v , v ), plays a key role in the analysis. Thefunction Φ acts as a guidepost for making revenue improvements to any incentive compatibileand individual rational mechanism. If Φ satisfies certain single-crossing conditions, thenincentive compatibility and individual rationality are maintained in the improved mechanism.The function Φ depends only on the distribution of types and not on any specific mechanism.With decreasing marginal values, we show that if Φ satisfies single-crossing in the hor-izontal direction (which corresponds to changes in v only), then there exists an optimalselling mechanism in which the first unit is sold deterministically. We refer to a mechanismin which the first unit is sold with probability 0 or 1 as a line mechanism . Line mechanismsare completely described by the payment for the first unit and the probability of allocatingthe second unit to types on the vertical line (1 , v ), where 0 ≤ v ≤ a . If, in addition to hori-zontal single-crossing, Φ satisfies single-crossing in the vertical direction (which correspondsto changes in v only) then there is an optimal mechanism which is semi-deterministic , i.e.,a line mechanism with at most one probabilistic value for allocating the second unit. Fi-nally, if Φ satisfies diagonal single-crossing (along the diagonal boundary of the support ofthe distribution), in addition to horizontal and vertical single-crossing, then there exists anoptimal mechanism that is deterministic.Our results for increasing marginal values are under weaker conditions, in that single-crossing of Φ in the horizontal direction is sufficient for the existence of an optimal mechanismthat is deterministic. In this optimal mechanism, the two units are bundled together andsold at a take-it-or-leave-it price.While decreasing marginal values is a common assumption, there are scenarios wheremarginal values are increasing. For instance, if the buyer is unfamiliar with the product andincurs a learning cost before using it, the marginal value for the second unit may be higherthan the marginal value for the first unit. Alternatively, if there is a fixed cost of production,then the model resembles increasing marginal values. As described later, increasing marginal The buyer’s marginal valuations (or type) for the two units are v ∈ [0 ,
1] and v ∈ [0 , a ]. Related Literature:
Early work on mechanism design with multidimensional typesincludes Rochet (1987), McAfee and McMillan (1988), Wilson (1993), Armstrong (1996),and Rochet and Chon´e (1998). As these papers focused primarily on divisible products,existence of deterministic mechanisms was not an issue.Thanassoulis (2004), Manelli and Vincent (2006), Manelli and Vincent (2007), Pycia(2006), Pavlov (2011), and Hart and Reny (2015) investigate the sale of indivisible, het-erogenous objects with independent values that are additive. As already noted, it may beoptimal to randomize in this setting. Moreover, as Hart and Reny (2015) show, revenue maynot be monotone in the distribution of the buyer’s type. Correlation between marginal valuesadds another layer of complexity and may increase the desirability of randomization. In amodel with two heterogenous goods and correlated values, Hart and Nisan (2019) show thatmechanisms of bounded menu size, such as deterministic mechanisms, may yield a negligiblefraction of the optimal revenue.The sale of homogenous objects is analyzed by Malakhov and Vohra (2009) and byDevanur et al. (2020). These models differ from ours in that buyers have the same privatelyknown value for all units, but the number of units desired is privately known. Pavlov (2020)investigates a model in which buyers buy one of two heterogenous objects, each of which hastwo units with the same value to the buyer.Apart from Malakhov and Vohra (2009) and Devanur et al. (2020), two other papersobtain sufficient conditions for the existence of optimal mechanisms that are deterministic.Manelli and Vincent (2006) obtain sufficient conditions in a model with two heterogenousgoods with independent, additive values. Haghpanah and Hartline (2020) obtain sufficient3onditions for bundling to be optimal in a general model.When there are two or more buyers, Chen et al. (2019) provide sufficient conditions forthe existence of an optimal Bayesian incentive compatible mechanism that is deterministic.These conditions do not apply to our setting, where there is one buyer, or to dominantstrategy incentive compatible mechanisms. Daskalakis et al. (2017) and Kleiner and Manelli(2019) characterize optimality for a multi-product monopolist using duality theory.There is a literature on approximately optimal mechanism design, starting with thework of Chawla, Hartline, and Kleinberg (2007) and Hartline and Roughgarden (2009).Recent contributions include Dhangwatnotai, Roughgarden, and Yan (2015), Hart and Nisan(2017), Hart and Nisan (2019), Hart and Reny (2019), Bhattacharya et al. (2020), andBabaioff, Immorlica, Lucier, and Weinberg (2020). These papers identify simple mechanisms,which are often deterministic mechanisms, that guarantee a constant fraction of the optimalmechanism revenue. These guarantees are usually independent of the prior. Another relatedpaper is Carroll (2017), which shows that posted-prices are robustly optimal for heterogenousobjects with additive values.The rest of the paper is organized as follows. We investigate the decreasing marginalvalues model in Section 2. In Section 2.1, we provide a sufficient condition under whichit is optimal to sell the first unit deterministically. Line mechanisms are characterized inSection 2.2 and sufficient conditions for the existence of an optimal mechanism that is de-terministic are provided in Section 2.3. Necessary conditions for a specific deterministicmechanism to be optimal are presented in Section 2.4. A special case of decreasing marginalvalues, the ordered decreasing values model, is introduced in Section 2.5. In Section 2.6, wedescribe the application of the results in the decreasing marginal values model to heteroge-nous objects. Our results for increasing marginal values are in Section 3. All proofs are inan Appendix. 4
Decreasing Marginal Values
We begin with an example with identical objects and decreasing marginal values in which itis optimal to randomize. Example 1
There are two units and the buyer has three possible types, A, B and C. Themarginal valuations of the three types and the probability distribution over types are pro-vided below: Type ( v , v ) Probability A (3 ,
1) 0 .
5B (4 ,
3) 0 .
25C (10 ,
1) 0 . Bundling.
The three candidate optimal prices for the two units bundled together are4, 7, and 11. The highest expected revenue (of 4) is achieved at bundle price 4. Alltypes buy the two units at this price.
Unbundling.
The prices of the two units, p and p , are strictly positive. The mostthat can be earned by selling only to type C is . The most that can be earned byselling only to types B and C is . Therefore, consider prices at which type A buyseither one or two units. If type A buys exactly one unit then p ≤ p > p = 3 , p = 3. If type A buys two units then p ≤ p ≤ . This generates an expected revenue of2 × . × . > . We are not aware of examples of optimal random mechanisms that involve homogenous objects withdecreasing marginal values in the literature. The example can be modified to a type space with continuous density as modeled in this paper. For simplicity, we assume that when a buyer is indifferent between two options, she chooses the optionthe seller prefers. That is, the mechanism is seller favorable as defined in Hart and Reny (2015). rice Package , and unit 2 with probability 06 (1,1), i.e., sell both units with probability 1Thus, randomization yields higher expected revenue than any deterministic mechanism. (cid:3) The Model
We describe a model with decreasing marginal values over two identical units of anindivisible object. The buyer’s (marginal) value for the i th unit is v i , i = 1 ,
2. The jointdensity function of v = ( v , v ) is f ( v ), which has support D ≡ { ( v , v ) ∈ [0 , × [0 , a ] : v ≤ a v } Marginal values are decreasing if a ≤ a ≤ f ( · ) is strictly positive on its support and is absolutelycontinuous. As the support of the marginal distribution of v depends on the realized valueof v , the values v and v are not independent.An allocation rule is a function q = ( q , q ), where q i : D → [0 , i = 1 , i th unit is allocated to the buyer. If buyer type ( v , v )obtains a second unit, then this buyer must also obtain the first unit. Therefore, the followingfeasibility constraint is implied: q ( v ) ≥ q ( v ) , ∀ v A transfer is a function t : D → (cid:60) , a payment by the buyer to the seller. A mechanism is ( q, t ).The payoff of a buyer who truthfully reports v is u ( v ) ≡ v · q ( v ) − t ( v )6 mechanism ( q, t ) is individually rational if u ( v ) ≥ v ; it is incentive compatible if u ( v ) ≥ u ( v (cid:48) ) + ( v − v (cid:48) ) · q ( v (cid:48) ) , ∀ v, v (cid:48) It is well-known (see Rochet (1987) or B¨orgers (2015)) that a necessary and sufficient con-dition for incentive compatibility is that u ( v ) is a convex function and q i ( v ) = ∂u ( v ) ∂v i , a.e. , i = 1 , t ( v ) = (cid:53) u ( v ) · v − u ( v ) , a.e.The seller’s expected revenue is Rev ( q, t ) ≡ E[ t ( v )] = (cid:90) D (cid:104) (cid:53) u ( v ) · v − u ( v ) (cid:105) f ( v ) dv (1)The integral of the first term in the integrand is the expected welfare from the mechanism.Subtracting the expected payoff of the buyer yields the seller’s expected revenue.A mechanism ( q ∗ , t ∗ ) is optimal if it is incentive compatible (IC) and individually ratio-nal (IR), and for any other IC and IR mechanism ( q, t ) we have Rev ( q ∗ , t ∗ ) ≥ Rev ( q, t )It is easy to show that in any optimal mechanism ( q ∗ , t ∗ ), if q ∗ ( v ) = (0 , t ∗ ( v ) = 0.Thus, in an optimal mechanism, the payoff of a buyer type who received zero units is zero.A mechanism ( q, t ) is deterministic if its allocation rule is deterministic, i.e., q i ( v ) ∈ { , } for all v and i . If a mechanism is not deterministic, it is random . A random mechanism (orallocation rule) is a lottery over deterministic mechanisms (or allocation rules).Let Q be the set of IC and IR mechanisms. If mechanisms ( q a , t a ) and ( q b , t b ) are IC andIR then so is λ ( q a , t a ) + (1 − λ )( q b , t b ), λ ∈ [0 , Q is a convex set. The set Q iscompact. As the expected revenue is a continuous, linear functional of q , it is maximized See Manelli and Vincent (2007) for a proof of compactness of the set of mechanisms for the sale ofheterogenous objects. A similar proof applies for the case of homogenous objects considered in this paper.
7t an extreme point of Q . When two or more indivisible objects are for sale, the extremepoints of Q may be random mechanisms. This contrasts with the sale of one object to onebuyer, where all extreme points of the set of IC, IR mechanisms are deterministic. Hence, adeterministic optimal mechanism always exists when a single object is sold to a buyer but arandom mechanism might be optimal if two or more objects are sold. In Proposition 1 below, we show that under a sufficient condition on the density, there existsan optimal mechanism in which the first unit is sold deterministically. The following lemmais required for the proposition. The proof, which is in the Appendix, is similar to that ofa result in McAfee and McMillan (1988); however, our assumption of decreasing marginalvalues yields a simpler expression for expected revenue.
Lemma 1
The seller’s expected revenue from an IC and IR mechanism ( q, t ) is Rev ( q, t ) = a (cid:90) u (1 , v ) f (1 , v ) dv − a (cid:90) (cid:90) v a u ( v , v ) (cid:104) f ( v , v ) + ( v , v ) · (cid:53) f (( v , v )) (cid:105) dv dv The following condition on density, introduced by McAfee and McMillan (1988), is ofteninvoked in the multidimensional mechanism design literature.
The density f satisfies Condition SC-H if f ( v ) + v · (cid:53) f ( v ) ≥ for all v ∈ D . In the single-object case, Condition SC-H becomes 2 f ( v ) + v df ( v ) dv ≥
0, which (i) is equiv-alent to the assumption that the expected revenue, v [1 − F ( v )], is concave and (ii) impliesthat Myerson’s virtual value function satisfies single crossing. As shown next, under SC-Hthe first unit is allocated deterministically in an optimal mechanism. Condition SC-H is one of three conditions we impose on a function Φ that is defined in Section 2.3; SC-His a single-crossing assumption in the horizontal direction on Φ. Together, the three conditions imply theexistence of an optimal mechanism that is deterministic. If the inequality in SC-H is strict, then in any optimal mechanism q ( v ) ∈ { , } for almost all v . roposition 1 If the density function f satisfies Condition SC-H, then there exists an op-timal mechanism ( q, t ) in which q ( v ) ∈ { , } for all v . That is, there exists an optimalmechanism such that for each v ( q ( v ) , q ( v )) ∈ { (0 , , (1 , q ) } where q ∈ [0 , may depend on v . Remark 1:
Proposition 1 holds if the seller’s costs for the two units, c and c , are positivewith 1 > c /a ≥ c ≥ f ( v ) + ( v − c ) · (cid:53) f ( v ) ≥ ∀ v ∈ D . Remark 2:
The proof of Proposition 1 does not appeal to the constraint q ( v ) ≥ q ( v )implied by the sale of identical objects. Therefore, it applies to heterogenous objects as well.We elaborate on this in Section 2.6.Proposition 1 is proved as follows. Lemma 1 implies that if, for any IC and IR mechanism,the payoff function u ( v , v ) is decreased when v <
1, without decreasing u (1 , v ) andCondition SC-H is satisfied, then expected revenue increases. One can make such decoupledchanges to the buyer payoff function of a mechanism in which the first unit is allocatedrandomly to some buyer types, thereby creating a new IC and IR mechanism which hasgreater expected revenue. The argument is similar to the proof of Proposition 2 in Pavlov(2011), who showed that in the unit-demand case and in the additive, heterogenous objectscase, there is an optimal mechanism in which any positive allocation belongs to the upperboundary of the feasible allocation set.Thus, under SC-H we may restrict our search for optimal mechanisms to those thatallocate the first unit deterministically. This reduces the dimensionality of the problem aspotentially optimal mechanisms are specified by a price for the first unit and an allocationrule for types (1 , v ) only. We refer to such potentially optimal allocation mechanisms asline mechanisms. 9 .2 Line Mechanisms Let Y ≡ { (1 , v ) : v ∈ [0 , a ] } be the one-dimensional subset of the type space along the v -axis. Definition 1
A mechanism ( q, t ) is a line mechanism ifi. its restriction to Y is IC and IRii. for every (1 , v ) ∈ Y , q (1 , v ) = 1 iii. for every v ≡ ( v , v ) ∈ D \ Y , (cid:16) q ( v ) , q ( v ) , t ( v ) (cid:17) = (0 , , , if v + v q (1 , v ) < t (1 , v )(1 , q (1 , v ) , t (1 , v )) , otherwise . The first unit is allocated deterministically in a line mechanism, with types (1 , v ) obtainingthe first unit with probability one. A type ( v , v ) is allocated ( q (1 , v ) = 1 , q (1 , v ) , t (1 , v ))if it is IR; otherwise type ( v , v ) gets (0 , , Thus, u ( v , v ) = max (cid:104) , v + v q (1 , v ) − t (1 , v ) (cid:105) = max (cid:104) , u (1 , v ) − (1 − v ) (cid:105) (2) Lemma 2
Every line mechanism is IC and IR on D . For any IC and IR mechanism, the proof of Proposition 1 constructs a line mechanismwhich generates at least as much expected revenue. Hence, we have the following corollaryto Proposition 1.
Corollary 1
If the density function f satisfies Condition SC-H, then there is an optimalmechanism that is a line mechanism. A line mechanism is seller favorable. In particular, if v + v q (1 , v ) = t (1 , v ), then (1 , q (1 , v ) , t (1 , v ))is the outcome for type ( v , v ). t (1 , ,
0) (i.e., the price for the first unit), and q (1 , v ),the allocation rule for the second unit for types with v = 1. However, the problem does notbecome one-dimensional. Two line mechanisms with the same allocation rule on Y will havedifferent allocation rules on D if the price for the first unit is different in the two mechanisms.Moreover, the set of line mechanisms is not convex. The Structure of Line Mechanisms
For any line mechanism ( q, t ), define Z ( q, t ) := { ( v , v ) : u (1 , v ) − (1 − v ) < } Eq. (2) implies that the set of buyer types who do not receive any unit in the line mech-anism is Z ( q, t ). The closure of Z ( q, t ) consists of ( v , v ) such that u ( v , v ) = 0. A linemechanism is shown in Figure 1. a ( α ; a α ) t (1 ;
0) ( v ; v ) (1 ; v ) t ( v ; v ) = t (1 ; v ) q ( v ; v ) = 1 q ( v ; v ) = q (1 ; v ) @Z ( q; t ) : v + u (1 ; v ) = 1 Z ( q; t ) Figure 1: A line mechanismFor any line mechanism ( q, t ), define α ≡ { x ∈ [0 ,
1] : u (1 , ax ) − (1 − x ) = 0 } (3)That α exists and is unique follows from 1 + u (1 , a ) ≥ u (1 , ≤
1, and the fact that x + u (1 , ax ) is strictly increasing and continuous in x . The dependence of α on ( q, t ) is11uppressed in the notation. From (3) we have α = 1 − u (1 , aα ) ≤ − u (1 ,
0) = t (1 ,
0) (4)The upper boundary of Z ( q, t ) is ∂Z ( q, t ) := { ( v , v ) : u (1 , v ) − (1 − v ) = 0 } Note that ∂Z ( q, t ) is a curve with slope − q (1 ,v ) (see Lemma 7 in Appendix A.2 for aproof) that connects the points ( t (1 , ,
0) and ( α, aα ). A deterministic mechanism is aline mechanism in which ∂Z ( q, t ) is piecewise linear with (at most) two line segments, onevertical and the other with slope − q, t ), let¯ q := sup v , if α = 0 Definition 2
A line mechanism ( q, t ) is a constrained line mechanism ifeither (i) q (1 , aα ) = ¯ q and for all v > aα , q (1 , v ) ∈ { ¯ q , } or (ii) q (1 , aα ) = 1 . If ¯ q = 1 then (i) and (ii) mean the same thing. Thus, in a constrained line mechanismfor v ≥ aα the probability of allocating the second unit takes at most two values, one ofwhich may be less than 1 (see Figure 2, where the case q (1 , aα ) = ¯ q < q (1 , v ) at v = aα , then q (1 , aα ) = 1. However, for v < aα any(increasing) value for q (1 , v ) is possible. Lemma 3
If the density function f satisfies Condition SC-H, then there exists an optimalmechanism which is a constrained line mechanism. Lemma 3 is proved by showing that there is an optimal mechanism which is an extremepoint of a convex, compact subset of line mechanisms. As we are maximizing a linear function12 a ( α ; a α ) t (1 ; Z ( q; t ) (1 ; q (1 ; v )) Figure 2: A constrained line mechanismon this subset, the maximum is attained at an extreme point of this subset. Every extremepoint of the subset is a constrained line mechanism.
Remark 3:
From Definition 2, we conclude that if in a constrained line mechanism ¯ q = 0,then the mechanism is deterministic as q (1 , v ) = 0 , ∀ v < aα and q (1 , v ) ∈ { , } , ∀ v ≥ aα . Therefore, in the sequel we restrict attention to constrained line mechanisms in which¯ q >
0. This, and the definition of ¯ q , implies α > t (1 , > In this section, we provide sufficient conditions under which there is an optimal mechanismthat is deterministic.It is useful to split the expected revenue of a constrained line mechanism ( q, t ) into twoparts,
Rev ( q, t ) = Rev α − ( q, t ) + Rev α + ( q, t )where Rev α − ( q, t ) is the expected revenue from types v ≤ aα and Rev α + ( q, t ) is theexpected revenue from types v > aα . Define for every ( v , v ),Φ( v , v ) := f (1 , v ) − (cid:90) v (cid:104) f ( x, v ) + ( x, v ) · ∇ f ( x, v ) (cid:105) dx Note that the function Φ depends only on f and not on any mechanism. The role of Φ is13iscussed after the next lemma. Lemma 4 If ( q, t ) is a constrained line mechanism, then Rev ( q, t ) = Rev α − ( q, t ) + Rev α + ( q, t ) , where Rev α − ( q, t ) := aα (cid:90) (cid:90) − u (1 ,v ) Φ( v , v ) dv dv (5) Rev α + ( q, t ) := a (cid:90) aα (cid:90) v a Φ( v , v ) dv dv − a (cid:90) aα (1 − v a − u (1 , v ))Φ( v a , v ) dv (6)As noted immediately after Proposition 1, decreasing u ( v , v ) when v < u (1 , v ) increases expected revenue (provided Condition SC-H is satisfied). These decoupledchanges in u are not possible in constrained line mechanisms. Whenever u (1 , v ) is increased, u ( v , v ) either increases or stays the same (see eq. (2)). Thus, in a constrained line mecha-nism the net change in expected revenue by increasing u (1 , v ), and the consequent increasein u ( v , v ), may be positive or negative. This trade-off is captured by the function Φ (whichwe reiterate is independent of the mechanism and is only a function of the density).To see the role of Φ, consider a constrained line mechanism ( q, t ) with buyer payoff u (1 , · )for types in Y . First, consider v ≤ aα . Differentiating Rev α − ( q, t ) with respect to u (1 , v ),we see from (5) that if Φ(1 − u (1 , v ) , v ) > u (1 , v ) increases expected revenue. This is the process of “straightening”described later. If, instead, Φ(1 − u (1 , v ) , v ) < u (1 , v ) increases expected revenue. This is the process of “covering” amechanism described later. The single-crossing property SC-V, introduced below, allowschanges in u (1 , v ) for a range of v ≤ aα in a manner that preserves incentive compatibility.14imilarly, differentiating Rev α + ( q, t ) with respect to u (1 , v ), we see from (6) that for v > aα if u (1 , v ) is increased [decreased] when Φ( v a , v ) > v a , v ) < u (1 , v ) for a range of v > aα in a manner that preserves incentive compatibility.Thus, Φ indicates the direction of revenue improvements, if any, for an arbitrary mecha-nism. Consider the following single-crossing properties of Φ in the horizontal, vertical, anddiagonal directions in the type space: Definition 3
The density function f satisfies Condition SC ifSC-H: for every v , Φ is increasing in v SC-V: for every v , Φ( v , · ) crosses zero at most once (from above). That is, for all ( v , v ) (cid:104) Φ( v , v ) > (cid:105) = ⇒ (cid:104) Φ( v , v (cid:48) ) > , ∀ v (cid:48) < v (cid:105) SC-D: for every v , (cid:82) av Φ( ya , y ) dy crosses zero at most once (from below). That is, for all v (cid:104) a (cid:90) v Φ( ya , y ) dy ≥ (cid:105) = ⇒ (cid:104) a (cid:90) v (cid:48) Φ( ya , y ) dy ≥ , ∀ v (cid:48) > v (cid:105) Note that SC-H is equivalent to 3 f ( v ) + v · ∇ f ( v ) ≥ v . Further, SC-V is implied ifΦ is decreasing in v and SC-D is implied if Φ( ya , y ) satisfies single crossing. In Section 2.5,we provide a class of distributions that satisfy Condition SC.These restrictions on the prior yield our main result. Theorem 1
If the density function f satisfies Condition SC, then there is an optimal mech-anism that is deterministic. The proof consists of two steps. • Step 1.
SC-H implies that there is an optimal mechanism that is a constrained linemechanism (Lemma 3). In a constrained line mechanism, q (1 , v ) takes at most two15alues for v ≥ aα but may take any number of values for v < aα . Under SC-H andSC-V, Proposition 2 in Section 2.3.1 shows that there is an optimal mechanism whichis semi-deterministic ; that is, a constrained line mechanism in which q (1 , v ) takes atmost three values for v ∈ [0 , a ], and only one of these three values is strictly between0 and 1. • Step 2.
If SC holds, a deterministic line mechanism is optimal in the class of semi-deterministic line mechanisms, completing the proof of Theorem 1.Next, we explain Step 1 in some detail. The proof of Step 2 is in Appendix A.3.
For a constrained line mechanism ( q, t ), define v := inf { v ∈ [0 ,
1] : q (1 , v ) > } (7)¯ v := sup { v ∈ [0 ,
1] : q (1 , v ) < } As IC implies that q (1 , v ) is increasing in v , we have v ≤ ¯ v with equality only if ( q, t ) isa deterministic mechanism.Consider the following definition. Definition 4
A constrained line mechanism ( q s , t s ) straightens another constrained linemechanism ( q, t ) at v s ∈ ( v , aα ] if u s (1 , v ) = u (1 , v s ) , ∀ v ≤ v s u s (1 , v ) = u (1 , v ) , ∀ v ≥ v s . Note that ( q s , t s ) is completely specified by u s (1 , · ). Moreover, t s (1 ,
0) = 1 − u s (1 ,
0) = 1 − u (1 , v s ) In a deterministic mechanism, for any v , q (1 , v ) is either 0 or 1. The dependence of v , ¯ v on q is suppressed in the notation. By Remark 3, we may assume that ¯ q >
0. Therefore, for small positive (cid:15) , q (1 , aα − (cid:15) ) > v < aα . s (1 , v ) = , if v < v s q (1 , v ) , if v ≥ v s as illustrated in Figure 3. By construction, α = α s and u s (1 , v ) ≥ u (1 , v ) for all v < v s .In a straightening, the payoff of types (1 , v ), v < v s increases compared to their payoffin ( q, t ). Therefore, the price of the first unit is lower, t s (1 , < t (1 , v < v s is never allocated a second unit. Consequently, Z ( q s , t s ) (cid:40) Z ( q, t ). a ( α ; a α ) t (1 ; Z ( q; t ) v ¯ v v s @Z ( q s ; t s ) Figure 3: Straightening a line mechanism
Lemma 5
Suppose that the density function f satisfies Conditions SC-H and SC-V. Con-sider a constrained line mechanism ( q, t ) . If Φ( t (1 , , v ) > , then there exists a straightening ( q s , t s ) of ( q, t ) , such that Rev ( q s , t s ) > Rev ( q, t ) . Consider the following definition.
Definition 5
A constrained line mechanism ( q, t ) is semi-deterministic if (cid:16) q ( v ) , q ( v ) (cid:17) ∈ (cid:110) (0 , , (1 , , (1 , q (1 , aα )) , (1 , (cid:111) ∀ v ∈ D q, t ) is a semi-deterministic mechanism if either it isa deterministic mechanism or if q ( v ) ∈ (0 ,
1) for some v , then q ( v ) = q (1 , aα ). The menusize of a semi-deterministic mechanism is at most four.Next, consider the following definition. Definition 6
A mechanism ( q c , t c ) is a cover of a constrained line mechanism ( q, t ) if ( q c , t c ) is semi-deterministic and α c = α (8) t c (1 ,
0) = t (1 , u c (1 , v ) ≤ u (1 , v ) , ∀ v < aαu c (1 , v ) = u (1 , v ) , ∀ v ≥ aα (9)In a cover, the payoff of types (1 , v ), v < aα decreases compared to their payoff in( q, t ). Several implications of the definition are worth noting. First, t c (1 ,
0) = t (1 ,
0) implies u c (1 ,
0) = u (1 , u c ( v , v ) = u ( v , v ) for all v ≥ aα .Therefore, (8) and (9) imply q c (1 , aα ) = q (1 , aα ). As ( q c , t c ) is semi-deterministic, we have q c (1 , v ) ∈ { , q (1 , aα ) , } for all v . Thus, if q (1 , aα ) = 1, then its cover is deterministic.Figure 4 shows the boundaries of type sets where 0, 1, 1 + q (1 , aα ) and 2 units are sold inthe cover ( q c , t c ) of a constrained line mechanism ( q, t ). Let ( q (cid:48) , t (cid:48) ) be any semi-deterministicmechanism such that for any v , q i ( v ) = 0 implies q (cid:48) i ( v ) = 0. Then Z ( q c , t c ) ⊆ Z ( q (cid:48) , t (cid:48) ). Thatis, among all semi-deterministic mechanisms that never allocate an object when ( q, t ) doesnot, the cover of ( q, t ) has the smallest set of types to which the object is never allocated.The cover, ( q c , t c ), of a constrained line mechanism, ( q, t ), is constructed from Z ( q, t )using two line segments: (i) a straight line with slope − q (1 ,aα ) at ( α, aα ) and (ii) a verticalline at ( t (1 , , Z ( q, t ) (see Lemma 7 in Appendix A.2) and the factthat the slope of Z ( q, t ) equals − q (1 ,v ) , which is less than or equal to − q (1 ,aα ) , ensures that18 a ( α, aα ) t (1 , v c ∂Z ( q c , t c ) ∂Z ( q, t ) ¯ v (1 , q (1 , aα ))(1 , , Figure 4: Cover of a constrained line mechanism Z ( q, t ) ⊆ Z ( q c , t c ) and therefore u (1 , v ) ≥ u c (1 , v ) , ∀ v < aα . This construction gives us( q c (1 , v ) , q c (1 , v )) := (cid:0) , (cid:1) , if v < v c (cid:0) , q (1 , aα )) , if v c ≤ v < ¯ v (cid:0) , (cid:1) , otherwise . where v c := q (1 ,aα ) (cid:104) (1 + aq (1 , aα )) α − t (1 , (cid:105) . Note that ¯ v c = ¯ v .By Lemma 5, Φ( t (1 , , v ) ≤ q, t ). Under this condition, we show that the revenue of any constrained linemechanism is no more than the revenue of its (semi-deterministic) cover. Lemma 6
Suppose that the density function f satisfies Conditions SC-H and SC-V. Con-sider a constrained line mechanism ( q, t ) . If Φ( t (1 , , v ) ≤ , (10) then Rev ( q c , t c ) ≥ Rev ( q, t ) where ( q c , t c ) is the cover of ( q, t ) . This leads to the main result of this section.19 roposition 2
Suppose that the density function f satisfies Conditions SC-H and SC-V.Then there exists an optimal mechanism that is semi-deterministic. Proof:
Condition SC-H and Lemma 3 imply that there is an optimal mechanism ( q, t ) whichis a constrained line mechanism. Therefore, Lemma 5 implies that (10) is satisfied for ( q, t ).Let ( q c , t c ) be the (semi-deterministic) cover of ( q, t ). By Lemma 6 Rev ( q c , t c ) ≥ Rev ( q, t ) (11)Hence, ( q c , t c ) is an optimal mechanism that is semi-deterministic. (cid:4) Proposition 2 is used in the proof of Theorem 1 in Appendix A.3.
We provide necessary conditions for a deterministic mechanism to be optimal in the classof all deterministic mechanisms. If Condition SC is satisfied, then these conditions arenecessary for optimality of a deterministic mechanism in the class of all mechanisms.A deterministic mechanism is described by prices p and p for the two units. If atoptimality ap ≤ p then the analysis is straightforward. The optimal price p ∗ i , i = 1 , v i .Therefore, assume that ap > p and that p , p are in the interior of the domain D , i.e., p ∈ (0 , , p ∈ (0 , ap ). The following necessary conditions are implied.
Proposition 3 If ( p ∗ , p ∗ ) are optimal prices in the interior of the domain D , then p ∗ (cid:90) Φ( p ∗ , v ) dv = 0 (12) The assumption that p ≤ p , p ) with p >
1, the prices (ˆ p , ˆ p ), with ˆ p = 1 and ˆ p = p + p −
1, yield the same expected revenueas ( p , p ). α ∗ (cid:90) p ∗ Φ((1 + a ) α ∗ − v , v ) dv + (cid:90) aaα ∗ Φ( v a , v ) dv = 0 (13) Further, Φ( p ∗ , p ∗ ) ≤ and Φ( p ∗ , ≥ . We describe a model in which marginal valuations are based on the order statistics of twodraws from the same distribution. Let X , X be two i.i.d. random variables with cdf G ( · )and density function g ( · ) that is strictly positive on its support [0 , v = max { X , X } , v = a min { X , X } Thus av ≥ v . We call this an ordered decreasing values model. Note that f ( v , v ) = 2 a g ( v ) g ( v a ) , ≥ v ≥ v a ≥ X i ’srepresent the values from the different ways of using the object; the buyer will put the objectto its best possible use. A similar interpretation applies to the ordered decreasing valuesmodel, where, if the buyer obtains one unit of the object, she will deploy it in its best usageand if she obtains two units, she will deploy them in the two best usages.Another interpretation is that the buyer in the ordered decreasing values model is anintermediary who resells the units to two final consumers. The seller does not have accessto the final consumers and can only sell the units to the intermediary. The final consumershave unit demand, their values are distributed i.i.d. and the realizations are known to theintermediary. If the intermediary purchases only one unit, she will resell it to the finalconsumer with a higher value. These two interpretations assume that a = 1. η ( x ) := xg ( x ) dg ( x ) dx be the elasticity of g . For every ( v , v ), define W ( v , v ) := v − − G ( v ) g ( v ) (cid:104) η ( v a ) (cid:105) W min ( v ) := v − − G min ( v ) g min ( v ) , where G min ( v ) = 1 − [1 − G ( v )] is the cumulative distribution function of the minimumof two independent random variables drawn from G . Definition 7
For every v , W ( v , · ) crosses zero at most once (from above) if forevery v (cid:104) W ( v , v ) > (cid:105) = ⇒ (cid:104) W ( v , v (cid:48) ) > , ∀ v (cid:48) < v (cid:105) W min crosses zero at most once (from below) if for every v (cid:104) W min ( v ) ≥ (cid:105) = ⇒ (cid:104) W min ( v (cid:48) ) ≥ , ∀ v (cid:48) > v (cid:105) The following proposition gives equivalent conditions for Condition SC in the ordereddecreasing model.
Proposition 4
In an ordered decreasing model,SC-H is satisfied if and only if η ( x ) ≥ − ;SC-V is satisfied if and only if W ( v , · ) crosses zero at most once for all v ;SC-D is satisfied if and only if W min crosses zero at most once. Examples of densities that satisfy the sufficient conditions of Proposition 4 include theuniform family g ( x ) = αx α − with α > g ( x ) = e x e − , and some beta distributions. Note that this is the condition that density g min satisfies the usual regularity condition. It is satisfied if g has increasing hazard rate. xample 2 Uniform Distribution
We describe the optimal mechanism for a uniform distribution on the domain D . Thedensity is f ( v , v ) = a , if 1 ≥ v ≥ v a ≥ , otherwiseIt may be verified that there is an optimal solution ( p , p ) where prices satisfy ap > p .For a uniform distribution, Φ( v , v ) = v − a . As Φ(2 / , v ) = 0 for any value of v , the onlysolution to eq. (12) is p ∗ = 2 /
3. Therefore, the unique prices in the interior of D that satisfynecessary conditions (12) and (13) for an internal optimal solution are p ∗ = 23 , p ∗ = 13 (cid:16) a − (cid:112) a (1 + a ) (cid:17) Unbundled prices
A direct calculation reveals that the unique prices on the boundary of D that are a candidatefor an optimal solution are p ∗ = (cid:114) a , p ∗ = 0 Bundle price
As we are maximizing a continuous function on a compact set, an optimal solution exists.Therefore, one of these two prices is optimal. A calculation reveals that if a > , thenthe optimal prices (i.e., optimal mechanism among all deterministic mechanisms) are theunbundled prices above. If, instead, a < , then it is optimal to sell the two units as abundle at the price (cid:113) a .In the limit as a →
0, the buyer has positive value for one object only with density f ( v ) = 2 v . The limit of the optimal bundling price as a → (cid:113) , which is the optimalprice for selling one object to a buyer with density f ( v ) = 2 v .The optimal prices are shown in Figure 5a for a ≥ and in Figure 5b for a < .That there is no random mechanism that yields greater expected revenue than thesedeterministic mechanisms follows from our results. First, note that the uniform model is an Armstrong (2016) shows that these are optimal prices for the case a = 1. ;
0) (1 ; ; a ) p ∗ = p ∗ = (2 a − q a (1 + a )) (a) a ≥ . (0 ;
0) (1 ; ; a )( q a ; (b) a < . Figure 5: Optimal mechanism for uniform distribution.ordered decreasing values model with v = max { X , X } and v = a min { X , X } , where X i are i.i.d. uniform on [0 , ,
1] has elasticity 0 and has increasinghazard rate. Thus, the single-crossing conditions in Definition 7 are satisfied. By Proposi-tion 4, Condition SC is satisfied and by Theorem 1 there is a deterministic mechanism thatis optimal. (cid:3)
As noted in Remark 2, Proposition 1 applies to heterogenous objects as the constraint q ( v ) ≥ q ( v ) is not used in the proof. The analysis after Proposition 1 is restricted to linemechanisms, which also satisfy q ( v ) ≥ q ( v ). Therefore, our results apply to heterogenousobjects. That is, consider a model with two heterogenous objects where the buyer values aredistributed with a density function f with support { ( v , v ) ∈ [0 , × [0 , a ] : v ≤ av } . Then,a constrained line mechanism is optimal if f satisfies Condition SC-H, a semi-deterministicline mechanism is optimal if f satisfies Conditions SC-H and SC-V, and a deterministicmechanism is optimal if f satisfies Condition SC.In general, there exist IC and IR mechanisms with q ( v ) > q ( v ) in a heterogenous objectsmodel described above. We describe two scenarios where that is not the case.– After a decision to purchase a product, the buyer might be offered a related productor service. For instance, after purchasing a new car the buyer might also purchase anextended warranty. Such add-on sales fit our model.24 A seller who offers two versions of a product, basic or premium. The buyer’s value forthe basic product is v and for the premium product is v + v . In these settings, one of the two heterogenous objects can be sold only after the other objectis sold. Consequently, q ( v ) ≥ q ( v ) in all feasible mechanisms. We begin with an example with identical objects and increasing marginal values in which itis optimal to randomize.
Example 3
The probability distribution of buyer marginal values is in the table below:
Type ( v , v ) Probability A (6 ,
8) 0 .
1B (3 ,
12) 0 . × . × . . > Price Package
12 (1 , . × . × . . > .
1. Hence, there is no optimal mechanism that is deterministic. (cid:3) The additional value a buyer places on a premium product might be more than the value for a basicproduct, v > v . This is admissible in our model as we do not assume a ≤ v is the following triangle D := { ( v , v ) ∈ [0 , a ] × [0 ,
1] : v ≤ av } . The definitions of a mechanism and its properties remain as in Section 2. In particular,the constraint q ( v ) ≥ q ( v ) , ∀ v is imposed. The density function f has support D and isassumed to be absolutely continuous.The counterpart of Proposition 1 is the following. The constraint q ( v ) ≥ q ( v ) , ∀ v isused in the proof. Proposition 5
If the density function f satisfies Condition SC-H, then there exists an op-timal mechanism in which q ( v ) = q ( v ) for all v . There is an optimal mechanism in which probability of selling each unit is the same. In other words, the seller bundles the two units and sells them as one object. Hence, Rileyand Zeckhauser (1983) and Myerson (1981) imply the following:
Theorem 2
If the density function f satisfies Condition SC-H, then it is optimal to bundlethe two units and sell them at a take-it-or-leave-it price. Thus, a deterministic mechanism is optimal in this model under weaker conditions thanin the model with decreasing marginal values.In a general class of models, Haghpanah and Hartline (2020) show that bundling isoptimal if the value of any subset of objects relative to the value of the grand bundle isstochastically increasing in the value of the grand bundle. Their condition does not imply,nor is it implied by, Condition SC-H.Let T be the cdf and τ the density of w ≡ v + v . If T is regular in the sense of Myerson(1981), then the optimal bundle price solves B = − T ( B ) τ ( B ) . If there are n units, then the conclusion of Proposition 5 generalizes to “there exists an optimal mecha-nism in which q n − ( v ) = q n ( v ) for all v .” rdered Increasing Values Model This is the counterpart of the order decreasing values model. Let X , X be two i.i.d.random variables with density g ( · ) that is strictly positive on its support [0 , v = a min { X , X } and v = max { X , X } . Thus v ≤ av . The following is a sufficient conditionfor the regularity of T . Proposition 6
Let v and v be from an ordered increasing values model. If g , the densityof X i , has increasing hazard rate and η ( x ) ≥ − , then T is regular. Hence, Theorem 2 implies that the optimal mechanism for an ordered increasing val-ues model is to the sell the two units as a bundle under the conditions on g specified inProposition 6. A Two-period Model with Increasing Marginal Values
As with decreasing marginal values, the results here also apply when there is an orderin which two objects can be sold. Consider a two-period model in which a durable productmay be sold either at the beginning of the first period or at the beginning of the secondperiod. If the product is sold in the first period, the buyer consumes it in both periods. If,instead, it is sold in the second period, then only second-period consumption is possible.It is convenient to number time by the number of periods left, including the currentperiod. Thus, the first period is period and second period is period
1. If the buyerpurchases the product in period 2, she consumes it in periods 2 and 1.The buyer’s values are v for consumption in period 1 (the latter period) and v for con-sumption in period 2 (the earlier period). The restriction v ≤ v follows from discounting.Thus, we have increasing marginal values. The values ( v , v ) are known to the buyer at thebeginning (no dynamics).An allocation rule Q determines two things: Q ( v ), the probability of selling the productin period 1 (the latter period), and Q ( v ), the probability of selling the product in period 2(the earlier period). A natural restriction is that Q ( v ) + Q ( v ) ≤
1, as the buyer who buysin period 2 will not also buy later in period 1. The expected value to buyer type ( v , v ) Recall that η is the elasticity of g . v + v ) Q ( v ) + v Q ( v ) = v [ Q ( v ) + Q ( v )] + v Q ( v )Define q ( v ) := Q ( v ) + Q ( v ) and q ( v ) := Q ( v ). So, q ( v ) is the probability with which thebuyer consumes the product in the first period only and q ( v ) is the probability with whichthe buyer consumes the product in both periods, with q ( v ) , q ( v ) ∈ [0 ,
1] and q ( v ) ≥ q ( v ). There are several directions we hope to explore in future work. An obvious one is generalizingthe results to more than two units. Proposition 1 generalizes to the sale of n > n units for sale with D = { ( v , v , . . . , v n ) | ≤ v ≤ , ≤ v i ≤ a i v i − , i ≥ } . If the inequality in Condition SC-H is changed to( n + 1) f ( v ) + v · (cid:53) f ( v ) ≥ , for all v ∈ D, then it is optimal to sell the first unit deterministically. A generalization of condition SCwould be required to obtain a deterministic optimal mechanism.Another direction to build on our results would be to obtain optimal dominant-strategyincentive compatible auctions. As noted on Remark 1, Proposition 1 generalizes to allowfor positive seller costs. With a modification of the definition of Φ to include seller costs,Theorem 1 also generalizes. With two buyers, the seller’s cost for providing a unit to buyer Ais the lost revenue from buyer B. This may be useful in constructing optimal auctions.The strategy of proofs developed in this paper may be useful in other models. Ourpreliminary investigations indicate that the approach used here can be adapted to somesettings with heterogenous objects. 28 Appendix
A.1 Proofs of Section 2.1
Proof of Lemma 1:
From (1), the seller’s expected revenue is
Rev ( q, t ) = (cid:90) D (cid:20) (cid:53) u ( v ) · v − u ( v ) (cid:21) f ( v ) dv = (cid:90) a (cid:90) v a (cid:20) (cid:53) u ( v , v ) · ( v , v ) − u ( v , v ) (cid:21) f ( v , v ) dv dv = (cid:90) (cid:90) av (cid:20) (cid:53) u ( v , v ) · ( v , v ) − u ( v , v ) (cid:21) f ( v , v ) dv dv Observe that (cid:90) v a ∂u ( v ) ∂v v f ( v ) dv = v u ( v ) f ( v ) (cid:12)(cid:12)(cid:12)(cid:12) v a − (cid:90) v a u ( v ) (cid:104) f ( v ) + v ∂f ( v ) ∂v (cid:105) dv = u (1 , v ) f (1 , v ) − v a u ( v a , v ) f ( v a , v ) − (cid:90) v a u ( v ) (cid:104) f ( v ) + v ∂f ( v ) ∂v (cid:105) dv = ⇒ a (cid:90) (cid:90) v a ∂u ( v ) ∂v v f ( v ) dv dv = a (cid:90) u (1 , v ) f (1 , v ) dv − a (cid:90) v a u ( v a , v ) f ( v a , v ) dv − a (cid:90) (cid:90) v a u ( v ) (cid:104) f ( v ) + v ∂f ( v ) ∂v (cid:105) dv dv Similarly, av (cid:90) ∂u ( v ) ∂v v f ( v ) dv = v u ( v ) f ( v ) (cid:12)(cid:12)(cid:12)(cid:12) av − av (cid:90) u ( v ) (cid:104) f ( v ) + v ∂f ( v ) ∂v (cid:105) dv = av u ( v , av ) f ( v , av ) − av (cid:90) u ( v ) (cid:104) f ( v ) + v ∂f ( v ) ∂v (cid:105) dv ⇒ (cid:90) av (cid:90) ∂u ( v ) ∂v v f ( v ) dv dv = (cid:90) av u ( v , av ) f ( v , av ) dv − (cid:90) av (cid:90) u ( v ) (cid:104) f ( v ) + v ∂f ( v ) ∂v (cid:105) dv dv By a change of variable v = av , we have (cid:90) av u ( v , av ) f ( v , av ) dv = a (cid:90) v a u ( v a , v ) f ( v a , v ) dv Thus, (cid:90) D [ (cid:53) u ( v ) · v ] f ( v ) dv = a (cid:90) u (1 , v ) f (1 , v ) dv − a (cid:90) (cid:90) v a u ( v ) (cid:104) f ( v ) + v · (cid:53) f ( v ) (cid:105) dv dv and Rev ( q, t ) = a (cid:90) u (1 , v ) f (1 , v ) dv − a (cid:90) (cid:90) v a u ( v ) (cid:104) f ( v ) + v · (cid:53) f ( v ) (cid:105) dv dv (cid:4) Proof of Proposition 1:
Condition SC-H and Lemma 1 imply that if u is modified to ˆ u (while maintaining IC and IR) such thatˆ u (1 , v ) ≥ u (1 , v ) , ∀ v and ˆ u ( v , v ) ≤ u ( v , v ) , ∀ ( v , v ) s.t. v < Rev (ˆ q, ˆ t ) ≥ Rev ( q, t ).Let ( q, t ) be any IC and IR mechanism. WLOG, assume that q (0 ,
0) = (0 , , t (0 ,
0) = 0.Let Y = { (1 , v ) : v ≤ a } . Defineˆ q (1 , v ) = 1 , ˆ q (1 , v ) = q (1 , v )ˆ t (1 , v ) = t (1 , v ) + (1 − q (1 , v ))30nd ˆ q (0 ,
0) = (0 , , ˆ t (0 ,
0) = 0. In the mechanism (ˆ q, ˆ t ), the probability of getting the firstunit is increased to 1 for types (1 , v ) and the payment increased so as to leave such typesindifferent between ( q, t ) and (ˆ q, ˆ t ). Extend (ˆ q, ˆ t ) from Y ∪ { (0 , } to v ∈ D \ [ Y ∪ { (0 , } ]as follows: (cid:16) ˆ q ( v ) , ˆ q ( v ) , ˆ t ( v ) (cid:17) = (0 , , , if v + v ˆ q (1 , v ) < ˆ t (1 , v )(1 , ˆ q (1 , v ) , ˆ t (1 , v )) , otherwise . (15)So, the range of (ˆ q, ˆ t ) is { (0 , , } and the outcomes for types (1 , v ) ∈ Y . Clearly, (ˆ q, ˆ t ) isIR on D \ Y .In the mechanism (ˆ q, ˆ t ), type (1 , v ) obtains payoff equal to that in ( q, t ) asˆ u (1 , v ) = (1 , v ) · ˆ q (1 , v ) − ˆ t (1 , v )= (1 , v ) · q (1 , v ) + (1 − q (1 , v )) − [ t (1 , v ) + (1 − q (1 , v ))]= u (1 , v )Thus, (ˆ q, ˆ t ) is IR on Y . That (ˆ q, ˆ t ) is IC on Y follows fromˆ u (1 , v ) − ˆ u (1 , v (cid:48) ) = u (1 , v ) − u (1 , v (cid:48) ) ≥ ( v − v (cid:48) ) q (1 , v (cid:48) ) = ( v − v (cid:48) )ˆ q (1 , v (cid:48) )where the inequality follows from IC of ( q, t ).We use the fact that (ˆ q, ˆ t ) is IC on Y to prove that (ˆ q, ˆ t ) is IC on D \ Y . Consider anytype ( v , v ) ∈ D \ Y . The payoff to this type from outcome (1 , ˆ q (1 , v (cid:48) ) , ˆ t (1 , v (cid:48) )) is v + v ˆ q (1 , v (cid:48) ) − ˆ t (1 , v (cid:48) ) = ( v −
1) + ( v − v (cid:48) )ˆ q (1 , v (cid:48) ) + 1 + v (cid:48) ˆ q (1 , v (cid:48) ) − ˆ t (1 , v (cid:48) )= ( v −
1) + ( v − v (cid:48) )ˆ q (1 , v (cid:48) ) + ˆ u (1 , v (cid:48) ) ≤ ( v −
1) + ˆ u (1 , v )= v + v ˆ q (1 , v ) − ˆ t (1 , v ) , where the inequality follows since (ˆ q, ˆ t ) is IC for any (1 , v ) ∈ Y . But v + v ˆ q (1 , v ) − ˆ t (1 , v )is the payoff of type ( v , v ) from the outcome (1 , ˆ q (1 , v ) , ˆ t (1 , v )). Hence, the payoff of type( v , v ) is maximized at the outcome (1 , ˆ q (1 , v ) , ˆ t (1 , v )). The payoff from this outcome is31 + v q (1 , v ) − t (1 , v ).To summarize, if v + v ˆ q (1 , v ) < ˆ t (1 , v ), then type ( v , v ) strictly prefers (0 , ,
0) toall other outcomes in the range of (ˆ q, ˆ t ); otherwise, this type’s payoff is maximized at theoutcome (1 , ˆ q (1 , v ) , ˆ t (1 , v )). From (15) we see that (ˆ q, ˆ t ) is IC on D \ Y .Finally, the payoff of type ( v , v ) ∈ D \ Y that is allocated (1 , ˆ q (1 , v ) , ˆ t (1 , v )) in themechanism (ˆ q, ˆ t ) is ˆ u ( v , v ) = ˆ u (1 , v ) − (1 − v )= u (1 , v ) − (1 − v ) ≤ u ( v , v ) + (1 − v ) q (1 , v ) − (1 − v )= u ( v , v ) − (1 − v )(1 − q (1 , v )) ≤ u ( v , v )where the first inequality follows from the IC of ( q, t ) and the second from v <
1. If, instead,( q ( v , v ) , q ( v , v ) , t ( v , v )) = (0 , ,
0) then ˆ u ( v , v ) = 0 ≤ u ( v , v ) by IR of ( q, t ).Hence, ˆ u (1 , v ) = u (1 , v ) for all (1 , v ) ∈ Y and ˆ u ( v ) ≤ u ( v ) for all v ∈ D \ Y . Asthe conditions in (14) are satisfied, we conclude that Rev (ˆ q, ˆ t ) ≥ Rev ( q, t ). Therefore, as( q, t ) was arbitrary, there is an optimal mechanism in which the allocation of the first unitis deterministic. (cid:4) A.2 Proofs of Section 2.2
Proof of Lemma 2:
Fix a line mechanism ( q, t ). By definition, q (1 , · ) = 1 and ( q, t )is IC and IR on Y . The rest of the proof is identical to the second part of the proof ofProposition 1. (cid:4) The following lemma is needed in the sequel.
Lemma 7
For any line mechanism ( q, t ) , the set Z ( q, t ) satisfies the following properties.i. Z ( q, t ) is convex.ii. Further, α ≤ t (1 , ≤ . If t (1 ,
0) = α , then q (1 , y ) = 0 for all y ∈ [0 , aα ) . ii. The slope of the boundary ∂Z ( q, t ) is − q (1 ,v ) . Proof: i. Take v, v (cid:48) ∈ Z ( q, t ) and let v (cid:48)(cid:48) = λv + (1 − λ ) v (cid:48) for some λ ∈ (0 , v (cid:48)(cid:48) + u (1 , v (cid:48)(cid:48) ) = λv + (1 − λ ) v (cid:48) + u (cid:0) , λv + (1 − λ ) v (cid:48) (cid:1) ≤ λv + (1 − λ ) v (cid:48) + λu (1 , v ) + (1 − λ ) u (1 , v (cid:48) )= λ ( v + u (1 , v )) + (1 − λ )( v (cid:48) + u (1 , v (cid:48) )) < u is convex and the second from the factthat v, v (cid:48) ∈ Z ( q, t ). Therefore, v (cid:48)(cid:48) ∈ Z ( q, t ). ii. That α ≤ t (1 ,
0) follows from (4) and t (1 , ≤ u (1 ,
0) = 1 − t (1 , ≥
0. If α = t (1 , u (1 ,
0) = u (1 , aα ) = u (1 ,
0) + (cid:82) aα q (1 , y ). As q isnon-negative, we must have q (1 , y ) = 0 for all y ∈ [0 , aα ). iii. Differentiating along the boundary, v + u (1 , v ) = 1, we get1 + ∂u (1 , v ) ∂v dv dv = 1 + q (1 , v ) dv dv = 0= ⇒ dv dv = − q (1 , v ) (cid:4) Proof of Lemma 3:
We know that the buyer’s payoff u from any IC, IR mechanism ( q, t )satisfies ∇ u = ( q , q ) a.e. WLOG we restrict attention to mechanisms with u (0 ,
0) = 0.Therefore, u ( v , v ) = v (cid:90) q ( s , ds + v (cid:90) q ( v , s ) ds Thus, (1) implies that the expected revenue functional is linear in the allocation rule q .Let ( q ∗ , t ∗ ) be a line mechanism that is optimal. We know from Corrolary 1 that such amechanism exists. Let Q α ∗ be the set of line allocation rules that use q ∗ for allocating the33rst unit and, for v < aα ∗ , use q ∗ for allocating the second unit. That is, Q α ∗ := { q (cid:48) : q (cid:48) ( v ) = q ∗ ( v ) for all v and q (cid:48) ( v , v ) = q ∗ ( v , v ) for all ( v , v ) such that v < aα ∗ } Hence, for every line mechanism ( q (cid:48) , t (cid:48) ) such that q (cid:48) ∈ Q α ∗ , we have t (cid:48) (1 ,