A Constructive Approach to Reduced-Form Auctions with Applications to Multi-Item Mechanism Design
aa r X i v : . [ c s . G T ] N ov A Constructive Approach to Reduced-Form Auctionswith Applications to Multi-Item Mechanism Design
Yang Cai ∗ School of Computer Science, McGill [email protected]
Constantinos Daskalakis † EECS, MIT [email protected]
S. Matthew Weinberg ‡ Computer Science, Princeton [email protected]
May 21, 2018
Abstract
We provide a constructive proof of Border’s theorem [Bor91, HR15a] and its gener-alization to reduced-form auctions with asymmetric bidders [Bor07, MV10, CKM13].Given a reduced form, we identify a subset of Border constraints that are necessaryand sufficient to determine its feasibility. Importantly, the number of these constraintsis linear in the total number of bidder types. In addition, we provide a characterizationresult showing that every feasible reduced form can be induced by an ex-post allocationrule that is a distribution over ironings of the same total ordering of the union of allbidders’ types.We show how to leverage our results for single-item reduced forms to design auctionswith heterogeneous items and asymmetric bidders with valuations that are additive over ∗ Supported by NSERC Discovery RGPIN-2015-06127, FRQNT 2017-NC-198956 and NSF Awards CCF-0953960 (CAREER), CCF-1101491, and CCF-1617730. Work done in part while the author was a ResearchFellow at the Simons Institute for the Theory of Computing. † Supported by a Sloan Foundation Fellowship, a Microsoft Research Faculty Fellowship, and NSF AwardsCCF-0953960 (CAREER), CCF-1101491, and CCF-1617730. Work done in part while the author was aResearch Fellow at the Simons Institute for the Theory of Computing. ‡ Research completed in part while the author was supported by a NSF Graduate Research Fellowship,and in part while the author was a Microsoft Research Fellow at the Simons Institute for the Theory ofComputing. tems. Appealing to our constructive Border’s theorem, we obtain polynomial-time al-gorithms for computing the revenue-optimal auction. Appealing to our characterizationof feasible reduced forms, we characterize feasible multi-item allocation rules. Keywords:
Reduced forms, multi-dimensional mechanism design, revenue maximization
Consider a mechanism design setting with n bidders whose types lie in some finite set T and one copy of a single indivisible item. In this setting, explicitly describing an ex-postallocation rule requires | T | n probability distributions. In particular, for every type profileone needs to specify the probability that the item is allocated to each bidder. In applicationswhere a succinct characterization of optimal mechanisms is lacking (c.f. revenue optimalauctions with independent quasi-linear and risk-neutral bidders [Mye81]), it is desirable totake as a first step an optimization approach. However, optimizing over ex-post allocationrules is too expensive computationally and provides little structural insight into the optimalmechanism.The above considerations motivate the study of interim allocation rules , also called reduced-form auctions , or simply reduced forms . Formally, suppose that bidder i ’s type is dis-tributed according to some distribution D i over T , and that bidders’ types are independent.The reduced form of an allocation rule is a collection of functions R := { π i : T → [0 , } i ∈ [ n ] .For all bidders i and types τ ∈ T , π i ( τ ) is the probability that the item is allocated to bidder i conditioning on her report being τ . The conditional probability is defined with respect tothe reports of the other bidders, assumed to be drawn from the product distribution × j = i D j ,and any randomization used by the allocation rule itself.A key question surrounding reduced forms is: under what conditions is a reduced formfeasible? More specifically, given a reduced form does there exist an ex-post allocation ruleinducing it? This question was studied by Matthews [Mat84] and Maskin and Riley [MR84],and Border [Bor91] provided a collection of linear inequalities that are necessary and sufficientfor the feasibility of a symmetric reduced form, namely when D i = D j and π i ( · ) = π j ( · ) forall bidders i and j . In more recent work, Border [Bor07], Manelli and Vincent [MV10], andChe et al. [CKM13] extend Border’s conditions to the general (asymmetric) case. It follows Such applications include, for example, settings where bidders are risk-averse [MR84], budget-constrained [LR96], or have multi-dimensional preferences [RC98]. R is feasible if and only if ∀ x , . . . , x n : X i X τ i : π i ( τ i ) ≥ x i π i ( τ i ) Pr t i ∼D i [ t i = τ i ] ≤ − Y i (cid:18) − Pr t i ∼D i [ π i ( t i ) ≥ x i ] (cid:19) . (1)Intuitively, the left hand side of (1) represents the probability that the item is allocated tosome bidder i whose realized type τ i satisfies π i ( τ i ) ≥ x i , as computed by the reduced form R . The right hand side represents the probability that there exists some bidder i whoserealized type τ i satisfies π i ( τ i ) ≥ x i . Clearly, if R is feasible, it outght to satisfy (1) for anychoice of thresholds x , . . . , x n . What is less clear is that if the inequality is satisfied for allthresholds, then R is feasible (but indeed this is the case).From an optimization standpoint, one drawback of the afore-described conditions is thatthere are about | T | n inequalities that one needs to verify. Our first main result is that infact it suffices to only check a subset of n | T | constraints to verify the feasibility of a givenreduced form. Theorem 1.
A reduced form R is feasible if and only if ∀ x : X i X τ i ∈ S ( i ) x π i ( τ i ) Pr t i ∼D i [ t i = τ i ] ≤ − Y i (cid:18) − Pr t i ∼D i (cid:2) t i ∈ S ( i ) x (cid:3)(cid:19) , (2) where S ( i ) x = { τ i ∈ T | π i ( τ i ) · Pr t i ∼D i [ π i ( τ i ) ≥ π i ( t i )] > x } . In particular, one can test thefeasibility of a reduced form or obtain a hyperplane separating R from the set of feasiblereduced forms in time O ( | T | n · log( | T | n )) . Note that the collection of inequalities (2) are a subset of inequalities (1). In particular, wehave only kept n | T | out of about | T | n inequalities. One way to interpret our theorem is thatit coordinates which combinations of thresholds x , . . . , x n it suffices to check simultaneouslyin (1). A priori, the simplest approach that could conceivably work would be to set all the x i ’s equal, but we show that this does not suffice; see Example Three in Section 4.1.1. Insteadof comparing different bidders’ interim probabilities of allocation at face value, our theoremdescribes a way to shade these probabilities depending on each bidder’s type distribution.The resulting “shaded interim probabilities” can be compared at face value. We provideseveral examples showing how Theorem 1 can be used to turn the task of verifying thefeasibility of interim allocation rules analytically tractable in Section 4.1.1.In addition to determining the feasibility of reduced forms, it is also important to un-derstand the structure of ex-post allocation rules that induce them. To this end, Manelliand Vincent [MV10] provide an interesting characterization result, using the notion of a3 ierarchical allocation rule [Bor91]. A hierarchical allocation rule maintains a weak totalordering (cid:23) over the elements of [ n ] × T ∪ { (0 , ⊥ = τ ) } . On input ( τ , . . . , τ n ), the allocationrule computes the subset of indices W = { i | ( i, τ i ) (cid:23) ( j, τ j ) , ∀ j } , then selects a uniformlyrandom index i in W . If i >
0, the item is allocated to bidder i . If i = 0, the item isnot allocated. Manelli and Vincent show that, if a reduced form R is feasible, then thereexists a distribution over hierarchical allocation rules inducing it. Moreover, each hierarchi-cal allocation rule in the support of the distribution uses a weak total ordering (cid:23) satisfying π i ( τ ′ i ) ≥ π i ( τ i ) = ⇒ ( i, τ ′ i ) (cid:23) ( i, τ i ), i.e. “stronger types” of bidder i are ranked higher than“weaker types” of bidder i in every hierarchical allocation rule in the support. We strengthenthis characterization as follows. Theorem 2.
If a reduced form R is feasible then there exists a strict total ordering ≻ overthe elements of [ n ] × T ∪{ (0 , ⊥ ) } such that R can be induced by a distribution over hierarchicalallocation rules, each using a weak ordering that irons ≻ . In addition to the per-bidder notion of “strength” of types guaranteed by Manelli andVincent’s characterization, Theorem 2 guarantees the existence of a global notion of strengthof types in the sense that for every profile ( τ , . . . , τ n ) with τ i ≻ τ i ≻ . . . ≻ τ i n , the ex-postallocation probabilities p , . . . , p n of the item to the bidders satisfy p i ≥ p i ≥ . . . ≥ p i n . Designing revenue-optimal auctions in multi-item settings has been a challenging applicationdomain in mechanism design. A characterization theorem `a la Myerson [Mye81] is unknown,and it is well-understood that optimal multi-item mechanisms exhibit much richer structurecompared to optimal single-item auctions, involving bundling and randomization even in thecase of a single additive bidder; see discussion in Section 1.2. In light of this, it is valuableto develop optimization tools to compute optimal multi-item mechanisms.Towards this end we adopt a linear programming approach. It is easy to write a linearprogram optimizing expected revenue over ex-post allocation and price rules of feasible,Bayesian incentive compatible mechanisms. However, this approach has two drawbacks.First, describing ex-post allocation and price rules requires ( m + 1) n | T | n numbers (for eachof | T | n type profiles, and each of n bidders, one must list an allocation probability for eachof m items along with a price paid). The exponential dependence on n makes this approach We say that a weak ordering (cid:23) over some set S “irons” a strict ordering ≻ ′ over the same set S iff, forall i, j ∈ S , i ≻ ′ j = ⇒ i (cid:23) j . A bidder is additive if her valuation for a set of items is equal to the sum of her values for each item inthat set. n | T | linear constraints. What is not clear is how to concisely expressthe feasibility of a multi-item interim allocation rule.Our key observation is the following: a multi-item interim allocation rule is feasible if andonly if the single-item interim rules that it projects onto each item are all feasible. Therefore,it suffices to invoke our single-item results above to resolve this problem. To be absolutelyclear, even when the bidders are additive, it is folklore knowledge and well-understood thatin the revenue-optimal auction the interim probability that a bidder receives someitem must in principle depend on her values for the other items. It is exactly thisproperty that makes multi-dimensional mechanism design notoriously difficult and not justa product of tractable single-item problems, and we are not claiming otherwise. However,the very specific subproblem of determining whether a multi-item interim allocationrule is feasible can be solved separately across items . Making use of Theorem 1 as asubroutine inside a linear program solver, we obtain the following computational result:
Theorem 3.
There is a polynomial-time algorithm that finds a revenue-optimal, BIC mech-anism in multi-item settings with additive bidders. The algorithm takes as input the typedistributions D , . . . , D n of the bidders, and outputs a concise description of an optimalmechanism in time polynomial in the number of bidders n , the number of items m and thesize of the type-space, | T | . The bidders are assumed independent, but each D i may be anarbitrarily correlated distribution over item values. Besides Theorem 3, our key observation stated above, combined with Theorem 2, directlyimplies a characterization of feasible multi-item interim allocation rules, as follows.5 haracterization of Feasible Multi-Item Interim Allocation Rules
Every feasible multi-item interim allocation rule can be implemented as follows: • Every item is allocated independently of the other items. • The allocation rule of every item ℓ maintains: – a strict ordering ≻ ℓ over the elements of [ n ] × T ∪ { (0 , ⊥ = τ ) } ; and – a distribution over ironings of ≻ ℓ . • Each item is then allocated as follows. First, a random ironing (cid:23) ′ ℓ is sampled. On aninput of reported types ( τ , . . . , τ n ), the allocation rule computes the subset of indices W ℓ = { i | ( i, τ i ) (cid:23) ′ ℓ ( j, τ j ) , ∀ j } , then selects a uniformly random index i ℓ in W ℓ . If i ℓ >
0, item ℓ is allocated to bidder i ℓ . If i ℓ = 0, item ℓ is not allocated.Recall that the set T in the characterization above is the set of types a bidder may have.In particular, each element τ ∈ T determines the values of a bidder of type τ for each bundleof items. Hence, the content of the first bullet is that while the ex-post allocation rulefor item ℓ indeed must depend on bidders’ values for items = ℓ , it need notdepend on how items = ℓ are themselves allocated . Reduced Forms.
A necessary and sufficient condition for the feasibility of a bidder-symmetric reduced form was provided by Border [Bor91], building on prior work by Maskinand Riley [MR84] and Matthews [Mat84]. A simpler proof of Border’s theorem and alter-native criteria for feasibility were also provided by Hart and Reny [HR15a]. For all theseworks, the necessary and sufficient conditions take the form of | T | linear inequalities. Bor-der’s conditions for the symmetric setting were generalized to the asymmetric setting byBorder [Bor07], Manelli and Vincent [MV10], and Che et al. [CKM13]. For these works, thenecessary and sufficient conditions take the form of | T | n linear inequalities. In comparison,Theorem 1 shows that | T | n linear inequalities suffice.Let us review in more detail some of the most related works on reduced forms. Manelliand Vincent characterize the extreme points of the space of feasible, monotone reducedforms as monotone hierarchical allocation rules in both the bidder-symmeteric and asym-metric case when bidders have continuous type spaces. This implies that every monotonereduced form has an ex-post allocation rule inducing it that is also ex-post monotone. In To be more precise, Border’s [Bor07] conditions took the form of 2 | T | n linear inequalities, and Che etal. [CKM13] identified a necessary and sufficient subset of | T | n linear inequalities. global total ordering of all bidders’ types.Che et al. provide a clean network-flow interpretation of Border’s theorem for asymmet-ric bidders, and show how to also accommodate bidders’ capacity constraints in multi-unitgeneralizations (for example, that the set G of bidders must never receive more than C ( G )units or less than L ( G ) units on any profiles). Their necessary and sufficient conditionsremain in the form of | T | n linear inequalities (same as for a single item with asymmetric bid-ders) despite the significant increase in generality where their results apply. Independentlyfrom our work, Alaei et al. [AFH +
12] also provide a computationally efficient algorithm todetermine the feasibility of reduced forms via a “token-passing game.” Their work showsin fact that there exists a collection of roughly ( n | T | ) inequalities that define the space offeasible reduced forms. Their characterization is what is called an “extended formulation” -they introduce an additional ( n | T | ) variables and their inequalities are not of the form (1).The results of Alaei et al. further extend to settings where the allocation is subject to ma-troid constraints. We address neither capacity constraints nor matroid constraints in thiswork. One high-level distinction between the main contributions of these works and ourTheorems 1 and 2 is that their results extend Border’s theorem to more general settings,whereas our work provides deeper insight into the core single-item setting (and by extension,as observed earlier, the multi-item setting where each item can be feasibly allocated to anybidder, regardless of other items she is allocated).Finally, several recent papers have provided polynomial-time algorithms for determiningwhether a reduced form is approximately feasible in multi-item settings with more complexallocation constraints [CDW12b, CDW13a, CDW13b]. On the other hand, Gopalan et al.show essentially that approximation is the best one can hope for: their work identifies aformal barrier to the existence of exact and “computationally useful” Border-like theoremsbeyond single-item settings [GNR15]. Multi-Item Auctions.
Prior work on multi-dimensional mechanism design is extensive(see e.g. survey [MV07]), driven by the scarcity of settings where the optimal mecha-nism has a clean allocation rule (such as Myerson’s revenue-optimal auction for single-7imensional settings [Mye81], or the welfare-optimal VCG auction in quite general set-tings [Vic61, Cla71, Gro73]). Indeed, numerous formal barriers have been identified to the ex-istence of clean, revenue-optimal multi-item mechanisms, such as the necessity of randomiza-tion [RC98, Tha04, Pav11], large menu complexity [BCKW15, HN13, DDT13, DDT17], largedescription complexity [DDT14], and revenue non-monotonicity [HR15b, RW15]. Daskalakiset al. [DDT17] have recently provided a characterization of single-bidder revenue-optimalmechanisms using optimal transport theory. Older work of Rochet and Chon´e [RC98] hadprovided a characterization of optimal single-bidder mechanisms in the related setting wherethere is no bound on the number of units per item but the seller has a strictly convex pro-duction cost for generating more units. Finally, the problem has recently entered the Theoryof Computation, where the emphasis has mostly been in deriving computationally efficientalgorithms for computing optimal mechanisms. A number of results have emerged obtain-ing constant-factor approximations in polynomial time [CD11, Ala11, BGGM10, CHMS10,CMS15, KW12, CH13, BILW14, Yao15, CM16, CZ17].In comparison to these works, ours is the first to provide a poly-time algorithm and corre-sponding characterization of revenue-optimal multi-item auctions without any distributionalassumptions (such as a hazard rate condition or item-value independence). Indeed, followingthe announcement of portions of this work [CDW12a], several works (including some by theauthors) provided computationally efficient algorithms to find approximately-optimal mech-anisms in increasingly general multi-item settings [CDW12b, CDW13a, CDW13b, BGM13,DDW15, CDW16]. In comparison to these works, the present paper remains unique incontaining a computationally efficient algorithm to find the exact optimal mechanism in amulti-item setting without any approximation error.
Section 2 below makes clear the notation we use and formal questions we study with respectto reduced forms. Section 3 studies bidder-symmetric reduced forms as a warm-up forSection 4, which studies asymmetric reduced forms. Section 5 provides our results on multi-item auctions. Appendix A contains some omitted proofs.
Throughout the paper, we denote the number of bidders by n . We also use T to denotethe possible types of a bidder. In order to obtain computationally meaningful results, weassume that T is finite and use c as shorthand for | T | , but make no other assumptions on8 . In particular, it is not necessary to assume that T is a subset of R .We use τ to denote the type of a bidder, without emphasizing whether it is a vector ora scalar (or otherwise). The elements of T n are called type profiles , and specify a type forevery bidder. We assume type profiles are sampled from a distribution D = Q ni =1 D i over T n , where D i the marginal of this distribution on bidder i ’s type, and use D − i to denotethe marginal distribution over the types of all bidders, except bidder i . We use t i for therandom variable representing the type of bidder i . So when we write Pr[ t i = τ ], we meanthe probability that bidder i ’s type is τ . If bidders are i.i.d., because Pr[ t i = τ ] is the samefor all i , we will just write Pr[ τ ].The reduced form R of an allocation rule specifies a vector function π ( · ), specifying values π i ( τ ), for all bidders i and types τ ∈ T . π i ( τ ) is the probability that bidder i receives the itemwhen reporting type τ , where the probability is over the randomness of all other bidders’types and the internal randomness of the allocation rule, assuming that the other biddersreport their true types. We may think of R as a vector in [0 , nc , by simply listing π i ( τ ) forall i, τ , and will sometimes write ~π to emphasize this view.In Section 3, we consider settings where the bidders are i.i.d., i.e. D i = D j for all i, j ,and the reduced forms are bidder-symmetric, which satisfy π i ( τ ) = π j ( τ ), for all i, j, τ . Insuch cases, we will drop the subscript i , writing just π ( τ ), for the probability that a bidderof type τ ∈ T receives the item, over the randomness of the allocation rule and the types ofthe other bidders, assuming that the other bidders report their true types. Given a reducedform R , we will be interested in whether it is “feasible.” By this, we mean “does there existan ex-post allocation rule that never over-allocates the item whose reduced form is R ?” Ifthe answer to this question is “yes” then we will also say that the ex-post allocation rulewhose reduced form is R , “induces R ” or “implements R .” Note that there is some subtletyin defining feasibility of a reduced form if Pr[ t i = τ ] = 0 for some i ∈ [ n ] , τ ∈ T . There area couple reasonable choices that are qualitatively the same - we choose to define a reducedform to be feasible only if Pr[ t i = τ ] = 0 ⇒ π i ( τ ) = 0.We also note that the running times of the algorithms obtained in Sections 3 and 4 arequoted without accounting for the bit complexity of numbers involved. The bit complexityof a rational number x is the number of bits b required so that x can be expressed as theratio of two binary numbers with b bits of precision. If b upper bounds the bit complexityof Pr[ t i = τ ] for all i, τ , and all coordinates of the input reduced form ~π , then it suffices tomultiply all quoted running times by a factor that is polynomial in b . Order Notation.
Throughout the text we use the O ( · ) notation. Let f ( x ), g ( x ) be twopositive functions defined on some infinite subset of R + . Then we write f ( x ) = O ( g ( x )) iff9here exist some positive reals α and x such that f ( x ) ≤ αg ( x ), for all x > x . We also write f ( x ) = poly( x ) iff there exist positive reals α and x such that f ( x ) ≤ x α , for all x > x .Finally, we provide some brief geometric preliminaries. Definition 1. ( Corner ) Let P be a closed, convex subset of Euclidean space defined as theintersection of finitely many halfspaces. Namely, P = ∩ i ∈I { ~x | ~a i · ~x ≤ b i } , for some finiteindex set I . We say that ~x ∗ is a corner of P if ~x ∗ ∈ P , and the set of equations { ~a i · ~x = b i } i ∈ S has as a unique solution the point ~x ∗ , where S = { i ∈ I| ~a i · ~x ∗ = b i } . Definition 2. ( Separation Oracle ) Let P be a closed, convex subset of Euclidean space.Then a Separation Oracle for P is an algorithm that takes as input a point ~x and outputs“ Yes ” if ~x ∈ P , or a hyperplane ( ~w, c ) such that ~y · ~w ≤ c for all ~y ∈ P , but ~x · ~w > c . Notethat because P is closed and convex, such a hyperplane always exists whenever ~x / ∈ P .A separation oracle is poly-time if on inputs of bit complexity b , it terminates in timepoly ( b, x ) , where x is the maximum bit complexity of any coordinate in any halfspace defining P . We will also make use of the following theorem, reworded from [Kha79, GLS81, KP82].
Theorem 4. ([Kha79, GLS81, KP82]) Let P be a d -dimensional closed, convex subset of R d defined as the intersection of finitely many halfspaces, and SO be a poly-time separationoracle for P . Then it is possible to do the following: • Find an element in argmax ~x ∈ P { ~c · ~x } for any ~c ∈ Q d (i.e. solve linear programs) intime polynomial in d , and b , where b upper bounds the bit complexity of all coordinatesof the vector ~c , and all coordinates of the halfspaces defining P . • Decompose any ~x ∈ P into a convex combination of at most d + 1 corners of P in timepolynomial in d , and b , where b upper bounds the bit complexity of all coordinates ofthe vector ~x , and all coordinates of the halfspaces defining P . This section serves as a warm-up for our main results by viewing bidder-symmetric reducedforms through a computational lens. Some of the key ideas for our main results in Section 4can be more cleanly illustrated for symmetric bidders below.Let us begin by reviewing Border’s theorem [Bor91], which specializes (1) to the caseof i.i.d. bidders and bidder-symmetric reduced forms. A bidder-symmetric reduced form is10easible if and only if: ∀ x : n · X τ : π ( τ ) ≥ x π ( τ ) Pr[ τ ] ≤ − (cid:18) − Pr t ∼D [ π ( t ) ≥ x ] (cid:19) n . (3)The semantic meaning of the above equations are the same as those in Equation (1):the left-hand side denotes the probability that some bidder whose type τ satisfies π ( τ ) ≥ x receives the item, as promised by the reduced form, and the right-hand side denotes theprobability that some bidder i has π ( t i ) ≥ x as computed by the probability distribution.Note that there are drastically fewer inequalities to check of form (3): only | T | instead of | T | n . This is essentially because if there exist thresholds x , . . . , x n for which an equationof form (1) is violated, there is also a single x such that Equation (3) is violated at x . Asthere are only | T | inequalities to check, Equation (3) directly implies Corollary 1 below:one can determine the feasibility of a reduced form in time O ( c (log c + log n )) via a routinecomputation. A proof is included in Appendix A. Corollary 1 (of [Bor91]) . The feasibility of a given bidder-symmetric reduced form can bedetermined in time O ( c (log c + log n )) . If it is infeasible, a violated inequality of form (3) can be determined in the same time. Now that it is easy to determine the feasibility of a reduced form, we wish to understandhow to find an ex-post allocation rule inducing a given feasible reduced form in poly-time.To this end, let us formally define hierarchical allocation rules, again specialized to thebidder-symmetric case.
Definition 3. ([Bor91]) A hierarchical allocation rule consists of a weak total ordering (cid:23) over T ∪ {⊥} . On reports ( τ , . . . , τ n ) , the allocation rule computes the subset of indices W = { i ≥ | τ i (cid:23) ⊥ and τ i (cid:23) τ j , ∀ j } , then selects a uniformly random bidder i in W , if W is non-empty. If W is empty, the item is unallocated.We say that a hierarchical allocation rule induced by (cid:23) is well-ordered with respect to areduced form R if π ( τ ) ≥ π ( τ ′ ) ⇒ τ (cid:23) τ ′ . For every feasible reduced form R , Theorem 5 below characterizes the corners of a convexregion containing it - which is intimately connected to ex-post allocation rules inducing R . Theorem 5. (implied by [MV10]) Every feasible reduced form R lies inside a c -dimensionalpolytope P whose corners are all reduced forms of hierarchical allocation rules that are well-ordered w.r.t. R . Furthermore, there is a distribution over at most c + 1 hierarchical alloca-tion rules, all well-ordered w.r.t. R , that induces R . roof. For ease of notation, first relabel all types in T so that π ( τ ) ≥ π ( τ ) . . . ≥ π ( τ c ). Let S = { i | π ( τ i ) = π ( τ i +1 ) } (for notational convenience, denote by π ( τ c +1 ) = 0). Consider theconvex polytope P ⊆ [0 , c specified by the following constraints.˜ π ( τ i ) = ˜ π ( τ i +1 ) ∀ i ∈ S ; (4)˜ π ( τ i ) ≥ ˜ π ( τ i +1 ) ∀ i ∈ [ c ] − S ; (5) X j ≤ i n · Pr[ t j ]˜ π ( τ j ) ≤ − − X j ≤ i Pr[ t j ] ! n ∀ i ∈ [ c ]; (6)where for notational convenience we denote ˜ π ( τ c +1 ) = 0 (so in particular ˜ π ( τ ) , · · · , ˜ π ( τ c ) arethe free variables and ˜ π ( τ c +1 ) is not, and the afore-desribed polytope is a subset of [0 , c ).By (3), R is feasible if and only if π ∈ P . Consider the corners of this polytope. As thereare c variables, every corner must satisfy at least c of the above inequalities with equality.Refer to the constraint ˜ π ( τ i ) ≥ ˜ π ( τ i +1 ) or ˜ π ( τ i ) = ˜ π ( τ i +1 ) (whichever is included in thedefinition of P ) as the i th monotonicity constraint, and the constraint P j ≤ i n · Pr[ t j ]˜ π ( τ j ) ≤ − (cid:16) − P j ≤ i Pr[ t j ] (cid:17) n as the i th Border constraint.We first show that no feasible reduced form satisfies both the i th monotonicity constraint and the i th Border constraint with equality. Consider then a feasible reduced form ˜ π , andan ex-post allocation rule ˜ M inducing ˜ π . Recall that if the i th Border constraint is tight,then the probability that a type in { τ , . . . , τ i } receives the item is exactly the probabilitythat such a type is reported to ˜ M . Therefore, whenever one or more types from this set arereported to ˜ M , a bidder with a type from this set necessarily wins the item. In particular, thismeans that we must have ˜ π ( τ i ) ≥ (1 − P j ≤ i Pr[ t j ]) n − , as τ i must certainly win wheneverall other reported types have index strictly larger than i . In fact, we must have ˜ π ( τ i ) > (1 − P j ≤ i Pr[ t j ]) n − , as τ i must also win with non-zero probability in the disjoint eventthat there is at least one other reported type equal to τ i , and the remaining types haveindicies strictly larger than i (because R is bidder-symmetric). Similarly, we necessarilyhave ˜ π ( τ i +1 ) ≤ (1 − P j ≤ i Pr[ t j ]) n − , as τ i +1 can only win in the event that all other typeshave index strictly larger than i . Therefore, if the i th Border constraint is tight, the i th monotonicity constraint is not.Now let us consider a corner ˜ π of P . Since P lies in R c , there must be at least c constraintsthat ˜ π satisfies with equality. As there are 2 c constraints in total, and no feasible reducedform can satisfy both the i th Border constraint and the i th monotonicity constraint, wesee that every corner must satisfy either the i th Border constraint or the i th monotonicityconstraint with equality. Such a reduced form corresponds to a hierarchical allocation rulewith τ i (cid:23) τ j for all i < j , and τ j (cid:23) τ i for some i < j if and only if the k th monotonicity12onstraints are tight for all k ∈ { i, . . . , j − } . Additionally, if ˜ π ( τ i ) = 0, then ⊥ (cid:23) τ i and τ i . If ˜ π ( τ i ) >
0, then τ i (cid:23) ⊥ and ⊥ 6(cid:23) τ i . It is easy to see that this hierarchical allocationrule has a feasible reduced form that satisfies the desired equalities (hence it equals ˜ π ) andis well-ordered w.r.t. R .By Carath´eodory’s theorem, we can write any feasible reduced form π as π = P c +1 j =1 w j ˜ π j ,where P c +1 j =1 w j = 1, and for all j , w j ≥ π j is a corner of P . The last step of the proofis an immediate consequence of the following observation. Observation . If a reduced form R can be written as π = P j w j ˜ π j , where each w j ≥ P j w j = 1, and each ˜ π j is induced by the ex-post allocation rule M j , then R is induced bythe ex-post allocation rule P j w j M j (sample j with probability w j , then use M j ). Corollary 2.
Given a bidder-symmetric reduced form R we can determine if it is feasible,or find a hyperplane separating it from the set of feasible bidder-symmetric reduced formsin time O ( c (log c + log n )) . If R is feasible, we provide a succinct description of an ex-postallocation rule inducing it, in time polynomial in c and log n . In particular, the allocationrule is a distribution of at most c + 1 hierarchical allocation rules, all well-ordered w.r.t. R .Proof. The first sentence immediately follows from Corollary 1, and the observation that anyviolated Border inequality is exactly a hyperplane separating R from the space of feasiblebidder-symmetric reduced forms.We now need to describe how to computationally efficiently find an ex-post allocation ruleimplementing a reduced form π that is feasible. By Theorem 5, any feasible R lies inside a c -dimensional polytope P whose corners are all reduced forms of hierarchical allocation rulesthat are well-ordered w.r.t. R . We now observe that we have defined a separation oracle for P in the first paragraph that runs in time O ( c (log c + log n )). So Theorem 4 implies thatwe may decompose R into a convex combination of corners of P in time polynomial in c and c (log c + log n ) (resulting in a runtime polynomial in both c and log n ). Observation 1completes the proof.Notice in particular that in the proof of Theorem 5, we described an easy procedureto define a hierarchical allocation rule that implements any corner ˜ π of P in terms of theinequalities of the polytope P that are tight at ˜ π . Let’s begin this section by recapping the major components leading to Corollary 2 for sym-metric reduced forms: 13. First, we need a computationally-efficient algorithm that takes as input a prospectivereduced form and finds a violated Border constraint, if it exists (and otherwise claimsthat all Border constraints are satisfied). For symmetric reduced forms, Border’s The-orem [Bor91] gives us this for free as there are only | T | constraints to check. Forasymmetric reduced forms, it will take exponential time to check all | T | n constraints ofform (1), so we show in Section 4.1 that in fact it suffices to check only n | T | constraintsby properly shading the interim probabilities.2. Next, to implement feasible reduced forms, we need to understand the corners of aconvex region containing all feasible reduced forms. Work of Manelli and Vincentaccomplishes this for the symmetric case (Theorem 5) and the asymmetric case (The-orem 6) for continuous type spaces. Again, we include a proof for the asymmetric casein Section 4.2 for finite type spaces that follows the same intuitive approach as ourproof of Theorem 5.3. Finally, we want to gain more insights into the structure of the space of feasible reducedforms than only understanding the corners of the feasible region. In the symmetriccase, it is unclear what one might hope for beyond Theorem 5. In the asymmetric case,however, Theorem 6 doesn’t tell the whole story. More specifically, Theorem 6 provesthat every feasible reduced form can be induced by a distribution over hierarchicalallocation rules all of which respect the same partial ordering within a single bidder’stypes, but may not respect any global ordering across all bidders’ types .Indeed, a stronger charcaterization is possible in the form of Theorem 2: for everyfeasible reduced form R there exists a global total ordering of (bidder, type) pairs ≻ ,such that R can be induced by a distribution of hierarchical allocation rules that allrespect ≻ . We prove Theorem 2 in Section 4.3 using a combinatorial approach, andalso show that the algebraic approach of Sections 4.1 and 4.2 cannot possibly yieldsuch a theorem.Let us now proceed by first recalling Equation (1), which states that a reduced form R is feasible if and only if for all thresholds x , . . . , x n , the probability that R awards the itemto a bidder i whose reported type τ i satisfies π i ( τ i ) ≥ x i is at most the probability thatsuch a type is reported. When (1) is violated at some choice of thresholds ~x , we call thesethresholds constricting . Unfortunately, there are roughly c n relevant ~x to test, so testingeach of them separately is computationally/analytically intractable.What we would really like is a more structured subset of Border constraints that are suf-ficient to check. To this end, we show that properly shading interim allocation probabilities14ccording to a bidder’s type distribution allows interim probabilities to be compared acrossbidders at face value. In this section, we define our notion of a shaded reduced form . We first provide the defini-tion and main proposition, followed by some illustrative examples where analysis is greatlysimplified by our approach.
Definition 4.
For any s i ( · ) such that s i ( τ ) ∈ [Pr[ π i ( t i ) < π i ( τ )] , Pr[ π i ( t i ) ≤ π i ( τ )]] for all i, τ , and any reduced form π , we define the corresponding shaded reduced form ˆ π as follows:for all i and types τ ∈ T , ˆ π i ( τ ) := s i ( τ ) · π i ( τ ) .Observation . For all bidders i and any types τ, τ ′ ∈ T , ˆ π i ( τ ) ≥ ˆ π i ( τ ′ ) = ⇒ π i ( τ ) ≥ π i ( τ ′ ). Proof.
If ˆ π i ( τ ) ≥ ˆ π i ( τ ′ ) but π i ( τ ) < π i ( τ ′ ), clearly s i ( τ ) > s i ( τ ′ ). On the other hand, s i ( τ ) ≤ Pr [ π i ( t i ) ≤ π i ( τ )] ≤ Pr [ π i ( t i ) < π i ( τ ′ )] ≤ s i ( τ ′ ). Contradiction. Proposition 1.
Let π be an infeasible reduced form, and s i ( · ) be any shading satisfying s i ( τ ) ∈ [Pr[ π i ( t i ) < π i ( τ )] , Pr[ π i ( t i ) ≤ π i ( τ )]] for all i, τ . Then there exists a single threshold x such that: P i P τ i | ˆ π i ( τ i ) ≥ x π i ( τ i ) · Pr[ t i = τ i ] > − Q i (1 − Pr[ˆ π i ( t i ) ≥ x ]) .In other words, for any valid shading of the reduced form, one can determine feasibilityof a reduced form by checking Border’s constraints where the threshold for the shaded reducedform is constant across all bidders.Proof. If a reduced form R is infeasible, consider any maximal choice of constricting thresh-olds x , . . . , x n , i.e. a choice of x , . . . , x n such that ( x i + δ, x − i ) is not constricting for any i, δ >
0. Now let ( i, τ ) ∈ argmin j,µ : π j ( µ ) ≥ x j { ˆ π j ( µ ) } . Then by the maximality of x , . . . , x n ,it must be the case that decreasing from x i + δ to x i causes us to go from satisfying (1)to violating it, and therefore x i = π i ( τ ) and this change must increase the LHS by morethan it increases the RHS. The change in the LHS is easy to compute: Observation 2 impliesthat we are simply including additional (bidder, type) pairs in our calculations, namely ( i, τ )(and all ( i, τ ′ ) with π i ( τ ′ ) = π i ( τ )). The change in the RHS is also easy to compute: wehave increased the probability that some bidder k exists with π k ( t k ) ≥ x k by exactly theprobability that all bidders j = i have π j ( t j ) < x j and π i ( t i ) = π i ( τ ). This therefore implies:Pr[ π i ( t i ) = π i ( τ )] π i ( τ ) > Pr[ π i ( t i ) = π i ( τ )] Y j = i Pr[ π j ( t j ) < x j ] ⇐⇒ π i ( τ ) Q j = i Pr[ π j ( t j ) < x j ] > . τ ′ , k , π k ( τ ′ ) < x k and ˆ π k ( τ ′ ) ≥ ˆ π i ( τ ). Observe first that we musthave k = i , as Observation 2 would otherwise imply π i ( τ ′ ) ≥ x i . So we must have: π k ( τ ′ ) · Pr[ π k ( t k ) ≤ π k ( τ ′ )] ≥ ˆ π k ( τ ′ ) ≥ ˆ π i ( τ ) ≥ π i ( τ ) · Pr[ π i ( t i ) < π i ( τ )] (valid shading)= ⇒ π k ( τ ′ ) · Pr[ π k ( t k ) < x k ] ≥ π i ( τ ) · Pr[ π i ( t i ) < π i ( τ )] (because π k ( τ ′ ) < x k ) ⇐⇒ π k ( τ ′ ) · Pr[ π k ( t k ) < x k ] ≥ π i ( τ ) · Pr[ π i ( t i ) < x i ] (because π i ( τ ) = x i ) ⇐⇒ π k ( τ ′ ) Q j = k Pr[ π j ( t j ) < x j ] ≥ π i ( τ ) Q j = i Pr[ π j ( t j ) < x j ] . So by our choice of ( i, τ ) and the work above, we obtain: π k ( τ ′ ) Q j = k Pr[ π j ( t j ) < x j ] > ⇐⇒ Pr[ t k = τ ′ ] π k ( τ ′ ) > Pr[ t k = τ ′ ] Y j = k Pr[ π j ( t j ) < x j ] . This inequality (combined with Observation 2) tells us that we could lower x k to π k ( τ ′ )and still have constricting thresholds. In fact, it tells us something even stronger. If y , . . . , y n are constricting thresholds with y j ≤ x j for all j , then we could decrease y k to π k ( τ ′ ) and stillhave constricting thresholds. This is because lowering y k to π k ( τ ′ ), causes the LHS of (1)to increase by P τ k : π k ( τ k ) ∈ [ π k ( τ ′ ) ,y k ) Pr[ t k = τ k ] π k ( τ k ) ≥ Pr[ π k ( t k ) ∈ [ π k ( τ ′ ) , y k )] π k ( τ ′ ). Andthe RHS increases by exactly Pr [ π k ( t k ) ∈ [ π k ( τ ′ ) , y k )] times the probability that π j ( t j ) < x j (respectively, y j ) for all j = k . As we decrease from x j to y j , this probability will clearlynever increase.So starting from any constricting thresholds x , . . . , x n and a bidder type pair ( i, τ ) asabove, we can lower each bidder k ’s threshold x k to the lowest π k ( τ ′ ) such that π k ( τ ′ ) ≥ ˆ π i ( τ ).The resulting thresholds remain constricting and have the desired form.The proof of Theorem 1 now readily follows from Proposition 1 and similar routinecomputation to Corollary 1. A complete proof of Theorem 1 appears in Appendix A. We nowproceed with a few examples to clearly illustrate the benefits of an improved characterization. Consider a reduced form where T = [0 ,
1] and each D i = U ( T ) (the uniform distribution on [0 , i let π i ( τ ) = τ α i , for someconstants (but not necessarily identical) α i . Using prior work, one could check all constraintsof the form (1), which is a multi-variate optimization problem: maximize P ni =1 R y i τ α i dτ − (1 − Q ni =1 y i ), over all ~y ∈ [0 , n (to map onto the variables used in Equation (1), set16 i = y α i i ). If the maximum happens to yield a value ≤
0, then all constraints are satisfied.If the maximum yields a value >
0, then we have explicitly found a violated constraint.In turn, identifying the maximum requires considering second-order conditions at all localoptima in addition to all points on the boundary, and is tedious. With Theorem 1 in hand, we observe that we need not optimize over all settings of thresh-olds ~y ∈ [0 , n . Rather, it suffices to only consider thresholds that are jointly parametrizedvia a single threshold x ∈ [0 ,
1] on the shaded reduced form. For all x ∈ [0 , ~y satisfy y α i +1 i = x , for all i : This is because for any τ ∈ [0 , π i ( t i ) ≤ π i ( τ )] = τ , and π i ( τ ) = τ α i , so Pr[ π i ( t i ) ≤ π i ( τ )] · π i ( τ ) = τ α i +1 . So we see that our problem nowreduces to a single-variate optimization: maximize P ni =1 R x / ( αi +1) τ α i dτ − (1 − Q ni =1 x / ( α i +1) ),over all x ∈ [0 , P ni =1 1 − xα i +1 − (cid:0) − x ( P ni =1 / ( α i +1)) (cid:1) . We can then take a derivative with respect to x , whichis (cid:16)P ni =1 1 α i +1 (cid:17) · (cid:0) x ( P ni =1 / ( α i +1) − − (cid:1) . At this point, we can observe that if P ni =1 / ( α i +1) >
1, then the derivative is negative on the entire interval [0 , x = 0. At x = 0, the objective function evaluates to P i / ( α i + 1) − >
0, andtherefore such reduced forms are infeasible. If instead, P i / ( α i + 1) ≤
1, then the derivativeis non-negative on the entire interval [0 , x = 1. At x = 1, the objective function evaluates to 0, and therefore such reduced forms are feasible.In conclusion, Theorem 1 combined with the single-variable optimization above providesa complete proof that reduced forms of the above form are feasible if and only if they promiseat most one item in expectation ex ante, which occurs if and only if P ni =1 / ( α i + 1) ≤ Example Two: Uniform Distributions of Support Two.
Now, consider a reducedform where T = { H, L } , and each D i = U ( { H, L } ). Then if π i ( H ) ≥ π j ( H ) / i, j ,a valid shading sets ˆ π i ( H ) = min j { π j ( H ) } , and ˆ π i ( L ) = 0 for all i . Proposition 1 thenguarantees that we only need to check two constraints: P ni =1 π i ( H ) / ≤ − / n , and P ni =1 π i ( H ) / π i ( L ) / ≤
1. On the other hand, using prior work would require checking 2 n equations of the form (1) (with some extra thought, this can be reduced to n by appealingto the fact that the type distributions are iid. But getting all the way down to 2 requiresreasoning `a la Proposition 1).Similarly, if each D i assigns probability p i to L , and π i ( H ) ≥ p j π j ( H ) for all i, j , avalid shading sets ˆ π i ( H ) = min j { π j ( H ) } , and ˆ π i ( L ) = 0 for all i . Proposition 1 again Note that Che et al. extend Border’s Theorem to continuous type spaces (the theorem statement isidentical to that for discrete type spaces) [CKM13]. Given this, it is easy to see that our proof of Proposition 1extends to continuous type spaces as well as-is (subject to some change in notation). P ni =1 (1 − p i ) · π i ( H ) ≤ − Q ni =1 p i ,and P ni =1 (1 − p i ) · π i ( H ) + p i · π i ( L ) ≤
1. On the other hand, using prior work would againrequire checking 2 n equations of the form (1) (and this time it is not obvious how to checkany fewer). Example Three: A Correct Ordering is Necessary.
The above two examples clearlyillustrate why a tighter characterization is helpful, but it is not a priori clear that the evensimpler approach of just considering thresholds where x i = x j for all i, j fails. To see thatindeed this approach fails, consider the following example with two bidders, and two typesper bidder, H and L . For bidder 1, we set Pr[ t = H ] = 1 /
8, Pr[ t = L ] = 7 / π ( H ) = 5 / π ( L ) = 0. For bidder 2, we set Pr[ t = H ] = 1 /
2, Pr[ t = L ] = 1 / π ( H ) = 1, π ( L ) = 3 / τ = H , which happens with probability 1 /
2. So if we have π ( H ) = 1, wecannot also have π ( H ) > /
2. So (1) is violated when x = 5 / x = 1.However, one can check that Border’s conditions are satisfied whenever x = x . Essen-tially the problem is that to find constricting thresholds, we need x to be small enough toinclude (1 , H ), yet x big enough to exclude (2 , L ), which is not possible when x = x . Examples: Summary.
Examples One and Two demonstrate the usefulness of a tightercharacterization. On the other hand, Example Three shows that such a tighter character-ization must be constructed carefully. Indeed, the shaded reduced form is exactly what isnecessary to determine whether a reduced form is feasible or not.
Theorem 1 tightens the necessary and sufficient conditions of Border’s theorem in a waythat allows for computationally efficient determination of the feasibility of reduced forms.We proceed by examining which ex-post allocation rules are neccessary to induce all feasiblereduced forms, similar to Theorem 5, first formally stating the definition of a hierarchicalallocation rule in asymmetric settings.
Definition 5. A hierarchical allocation rule consists of a weak total ordering (cid:23) on T × [ n ] ∪ { (0 , ⊥ ) } . On reported types ( τ , . . . , τ n ) , the allocation rule computes the subset of There are four inequalities of this form: Pr[ t = H ] · π ( H ) = 1 / ≤ − π ( L ), Pr[ t = H ] · π ( H )+Pr[ t = L ] · π ( L ) = 7 / ≤
1, Pr[ t = H ] · π ( H ) + Pr[ t = L ] · π ( L ) + Pr[ t = H ] · π ( H ) = 61 / ≤
1, andPr[ t = H ] · π ( H ) + Pr[ t = L ] · π ( L ) + Pr[ t = H ] · π ( H ) + Pr[ t = L ] · π ( L ) = 61 / ≤
1, all of whichare satisfied. ndices { i | ( i, τ i ) (cid:23) ( j, τ j ) , ∀ j } , then selects a uniformly random index i ∈ W . If i > , theitem is allocated to bidder i . If i = 0 , the item is not allocated.We say that a hierarchical allocation rule (cid:23) for non-identical bidders is partially-ordered with respect to R if for all i and τ, τ ′ ∈ T , π i ( τ ) ≥ π i ( τ ′ ) ⇒ ( i, τ ) (cid:23) ( i, τ ′ ) . We saythat a hierarchical allocation rule is strict if for all bidders i = j and types τ, τ ′ ∈ T : ( i, τ ) (cid:23) ( j, τ ′ ) ∧ ( i, τ ) (cid:23) (0 , ⊥ ) ⇒ ( j, τ ′ ) ( i, τ ) , and ( i, τ ) (cid:23) (0 , ⊥ ) ⇒ (0 , ⊥ ) ( i, τ ) (i.e. |W| = 1 on all inputs). Similar to the symmetric case, a simple counting argument shows that every feasiblereduced form can be implemented as a distribution over strict, partially-ordered hierarchicalallocation rules. The proof for the asymmetric case follows the same outline, but requiresone additional technical lemma whose proof is deferred to the Appendix A.
Theorem 6. (implied by [MV10]) Every feasible reduced form R lies in a cn -dimensionalpolytope whose corners are all strict, partially-ordered w.r.t. R hierarchical allocation rules.Furthermore, there is a distribution over at most cn +1 hierarchical allocation rules, all strictand partially-ordered w.r.t. R , that induces R .Proof. For ease of notation, relabel all types in T (differently for each bidder) so that π i ( τ i, ) ≥ . . . ≥ π i ( τ i,c ) for all i , and so that Pr[ t i = τ i,j ] = 0 ⇒ Pr[ t i = τ i,k ] = 0 forall k > j (i.e. since π i ( τ ) = 0 for all τ such that Pr[ t i = τ ] = 0, we are free to put them atthe end of the list). Let c i ≤ c denote the number of types τ ∈ T such that Pr[ t i = τ ] > π i ( τ i,c +1 ) = 0. Let S i denote the set of j suchthat π i ( τ i,j ) = π i ( τ i,j +1 ). Consider the closed, convex polytope P ⊆ [0 , cn specified by thefollowing constraints. ˜ π i ( τ i,j ) = ˜ π i ( τ i,j +1 ) ∀ i ∈ [ n ] , j ∈ S i (7)˜ π i ( τ i,j ) ≥ ˜ π i ( τ i,j +1 ) ∀ i ∈ [ n ] , j ∈ [ c ] − S i (8) X i X j
Lemma 1.
For (cid:23) defined as above for some ˜ π ∈ P , the ( i, j ) th monotonicity constraint canbe tight and linearly independent of the tight Border constraints only if ( i, τ i,j +1 ) (cid:23) ( i, τ i,j ) . The remainder of the proof is just a counting argument. Define an equivalence relation( i, τ ) ∼ ( j, τ ′ ) ⇔ ( i, τ ) (cid:23) ( j, τ ′ ) (cid:23) ( i, τ ). Then because the tight Border constraints arenested, the number of tight Border constraints is exactly the number of equivalence classesunder ∼ among (type, bidder) pairs (cid:23) (0 , ⊥ ). It is also now clear, from Lemma 1, that amonotonicity constraint can be tight and linearly independent of the tight Border constraintsonly if it is between two types ( i, τ i,j ) and ( i, τ i,j +1 ) of the same bidder i and ( i, τ i,j ) ∼ ( i, τ i,j +1 ). Therefore, there are no tight monotonicity constraints across equivalence classes,and the number of tight monotonicity constraints in each equivalence class is at most thenumber of types in that class minus one. Moreover, the number of tight monotonicityconstraints in each equivalence class can only be equal to the number of types in that classminus one if all types in that class are from the same bidder. We simply observe that if( i, τ ) ∼ ( j, τ ′ ) (cid:23) (0 , ⊥ ) for any i = j , that the above counting shows we can’t possibly have cn tight linearly independent constraints. So we may conclude that ( i, τ ) ( j, τ ′ ) for any( i, τ ) , ( j, τ ′ ) (cid:23) (0 , ⊥ ).Finally, we now want to conclude that the hierarchical allocation rule defined by (cid:23) inducesthe proposed corner ˜ π . We make one slight modification to (cid:23) to fit exactly Definition 5,and merge adjacent equivalence classes that contain types from the same bidder. Moreformally, if there exist two types ( i, τ i,j ), ( i, τ i,j +1 ) (cid:23) (0 , ⊥ ) that do not lie in the sameequivalence class and no bidder k = i has a type τ with ( i, τ i,j ) (cid:23) ( k, τ ) (cid:23) ( i, τ i,j +1 ), wemerge the two equivalence classes by modifying (cid:23) so that ( i, τ i,j +1 ) (cid:23) ( i, τ i,j ) (but keeping (cid:23) otherwise the same). Now, it is clear that (cid:23) is a weak total-ordering that is strict (because( i, τ ) ( j, τ ′ ) for any ( i, τ ) , ( j, τ ′ ) (cid:23) (0 , ⊥ )) and partially-ordered w.r.t. to ˜ π , and the This adjustment is technically necessary to claim that when ˜ π i ( τ i,j ) = ˜ π i ( τ i,j +1 ), we have τ i,j +1 (cid:23) τ i,j .This adjustment doesn’t affect the implementation of the hierarchical allocation rule according to (cid:23) at all,since there is only ever one type per bidder present at the auction. (cid:23) uniquely implements the corner ˜ π (becausethe tight Border constraints uniquely determine a winner on every possible type profile,exactly the strongest type according to (cid:23) ).The final sentence of the theorem statement is again a consequence of Carath´eodory’sTheorem and Observation 1.Now that we know that every corner of the polytope can be implemented as a strict,partially-ordered w.r.t. R hierarchical allocation rule, we want to ensure that the hierarchy (cid:23) can be found computationally efficiently. Lemma 2.
Let ˜ π be any corner of P and (cid:23) be the strict and partially-ordered w.r.t. R hierarchical allocation rule that implements ˜ π . Then the shaded reduced form defined as ˆ˜ π i ( τ ) = ˜ π i ( τ ) · Pr [˜ π i ( t i ) ≤ ˜ π i ( τ )] respects (cid:23) . Specifically, for any two types ( i, τ ) and ( j, τ ′ ) , ( i, τ ) (cid:23) ( j, τ ′ ) ⇐⇒ ˆ π i ( τ ) ≥ ˆ π j ( τ ′ ) . Therefore, given ˜ π we can construct the ordering (cid:23) intime O ( cn log( cn )) .Proof. First, observe that for any type ( i, τ ), ˜ π i ( τ ) = Q k = i Pr[( i, τ ) (cid:23) ( k, t k )] and Pr [˜ π i ( t i ) ≤ ˜ π i ( τ )] =Pr[( i, τ ) (cid:23) ( i, t i )]. Therefore, ˆ˜ π i ( τ ) = Q nk =1 Pr[( i, τ ) (cid:23) ( k, t k )]. We may therefore immedi-ately conclude that ( i, τ ) (cid:23) ( j, τ ′ ) ⇔ ˆ˜ π i ( τ ) ≥ ˆ˜ π j ( τ ′ ). So in order to find the ordering (cid:23) , weonly need to compute the shaded reduced form and sort its components, which can clearlybe done in time O ( cn log( cn )).And now, we can draw the main conclusion of this section: given as input any reducedform, we can computationally efficiently determine whether or not it is feasible. If it isfeasible, we can computationally efficiently output an implementation. Corollary 3.
Given an asymmetric reduced form R , one can determine if it is feasible orfind a hyperplane separating it from the set of feasible reduced forms, in time O ( cn log( cn )) .If R is feasible, a succinct description of an allocation rule implementing R can be found intime polynomial in c and n . The output allocation rule is a distribution over at most cn + 1 hierarchical allocation rules, all strict and partially-ordered w.r.t. R .Proof. We first observe that Theorem 1 provides an algorithm that determines if R is fea-sible, or provides a hyperplane separating it from the space of feasible reduced forms (theviolated Border constraint) that runs in time O ( cn log( cn )). We now have to describe howto efficiently find an ex-post allocation rule implementing a reduced form R that is feasible.Theorem 6 implies that π lies inside a cn -dimensional polytope, P , whose corners are thereduced forms of the strict, partially-ordered w.r.t. R hierarchical allocation rules. Theo-rem 1 provides a separation oracle for P , so Theorem 4 implies that we can decompose π P in time polynomial in cn and cn log( cn ) (resultingin a runtime polynomial in both c and n ). Lemma 2 shows how to implement a hierarchicalallocation rule in time O ( cn log( cn )) given a corner. Observation 1 completes the proof. Let’s first briefly recall the goal of this section. Theorem 6 provides a nice characterizationresult: every feasible reduced form can be induced by a distribution over hierarchical alloca-tion rules, all of which respect the same partial ordering within a single bidder’s types, butmay not respect any global ordering across all bidders’ types. The purpose of this section isto prove Theorem 2 and show that in fact the distribution over hierarchical allocation rulesmay be taken to respect the same global ordering over all bidders’ types.At this point, we note that it would be great if the shaded reduced form provided aneasy way to extend Theorem 5 to the asymmetric setting. Specifically, we can say thata hierarchical allocation rule (cid:23) is shaded-ordered if ˆ π i ( τ ) ≥ ˆ π j ( τ ′ ) ⇒ ( i, τ ) (cid:23) ( j, τ ′ ), andhope it is the case that every feasible reduced form can be induced by a distribution overshaded-ordered hierarchical allocation rules. Unfortunately, although the shaded reducedform provides a nice structural theorem about feasible reduced forms and a near-linear timealgorithm for determining feasibility, the following example shows that distributions overshaded-ordered hierarchical allocation rules are not sufficient to implement every feasiblereduced form when the bidders are non-i.i.d. For completeness, we rule out all possibleshadings. Proposition 2.
There exist feasible reduced forms that cannot be induced by distributionsover shaded-ordered hierarchical allocation rules.Proof.
Consider the following example with two bidders and two types, with ǫ < /
4. Bidderone has Pr[ t = H ] = 1 − ǫ , Pr[ t = L ] = ǫ , π ( H ) = 1 − ǫ and π ( L ) = 1 − ǫ . Biddertwo has Pr[ t = H ] = ǫ, Pr[ t = L ] = 1 − ǫ , π ( H ) = ǫ and π ( L ) = 0.Then for any shading, we have ˆ π ( H ) ≥ (1 − ǫ ) > ǫ ≥ ˆ π ( H ) ≥ ǫ · (1 − ǫ ) > ǫ · (1 − ǫ ) ≥ ˆ π ( L ). So any shaded-ordered reduced form necessarily has (2 , H ) (cid:23) (1 , L ), and thereforecannot possibly have π ( L ) > − ǫ/ ≥ − ǫ .Observe also that this reduced form is clearly feasible: consider the allocation rule thatawards the item to bidder one whenever t = L , awards the item to bidder one with prob-ability 1 − ǫ when t = t = H (and bidder two otherwise), and awards the item to bidderone with probability 1 − ǫ , to bidder two with probability ǫ when t = L, t = H (andthrows the item away otherwise). Then bidder one receives the item with probability 1 − ǫ t = H , with probability 1 − ǫ when t = L , and bidder two receives the item withprobability ǫ when her type is H , and 0 otherwise.In light of Proposition 2, it seems that geometric techniques will not get us all the way toa proof of Theorem 2 (which provides a global ordering instead of just a partial ordering asin Theorem 6), so our proof below appeals more to analytical tools. Throughout the proof,we will use the term ≻ -ordered hierarchical allocation rule to denote a hierarchical allocationrule corresponding to some weak total ordering (cid:23) that irons the strict total ordering ≻ . Proof of Theorem 2:
The high-level approach is to find a strict total ordering, ≻ , andan allocation rule that is a distribution over ≻ -ordered hierarchical allocation rules, M , thatis “closest” to inducing R by some measure (over all ≻ , M ). We will then argue that if M does not already induce R , then the fact that we cannot improve M witnesses a violatedBorder constraint.We first formally introduce a dummy bidder 0 with Pr[ t = ⊥ ] = 1 and π ( ⊥ ) = 1 − P i> P τ i Pr[ t i = τ i ] π i ( τ i ). With this addition, we now have P i ≥ P τ i Pr[ t i = τ i ] π i ( τ i ) = 1.So if we find a feasible allocation rule M whose reduced form π M satisfies π Mi ( τ ) ≥ π i ( τ ) forall i ≥ , τ , then we must have π M = π , and M induces R .Let ≻ be a strict total ordering over all possible types that respects all the per-bidderpartial orderings induced by π . Namely, for all i , if ( i, τ ) ≻ ( i, τ ′ ), then π i ( τ ) ≥ π i ( τ ′ ).Define the unhappiness F ≻ ( M ) of a distribution over ≻ -ordered hierarchical allocation rules, M (with reduced form π M ), as follows: F ≻ ( M ) = max i ≥ ,τ ∈ T ( π i ( τ ) − π Mi ( τ )) .F ≻ can be viewed as a continuous function over a compact set: There are finitely many ≻ -ordered hierarchical allocation rules, so their convex hull is a compact set (and ex-actly the space of distributions over ≻ -ordered hierarchical allocation rules). Each func-tion π i ( τ ) − π Mi ( τ ) is linear in this space (and therefore continuous), and the maximumof continuous functions is continuous. Hence, F ≻ achieves its infimum. Let then M ≻ ∈ argmin M F ≻ ( M ) (where the minimization is over all distributions over ≻ -ordered hierarchi-cal allocation rules) and define the set S ≻ to be the set of maximally unhappy types under M ≻ ; formally, S ≻ = argmax i,τ { π i ( τ ) − π M ≻ i ( τ ) } . If for some ≻ there are several minimizers M ≻ , choose one that minimizes | S ≻ | . Now, let M O (stands for Minimal Orderings) be theset of the orderings ≻ that minimize F ≻ ( M ≻ ), further refined to only contain ≻∈ M O thatalso minimizing | S ≻ | . Formally, first set M O = argmin ≻ { F ≻ ( M ≻ ) } and then refine M O as M O new = argmin ≻∈ MO {| S ≻ |} . We drop the subscript “new” for the rest of the proof.From now on, we call a (bidder, type) pair ( i, τ ) happy if π Mi ( τ ) ≥ π i ( τ ), otherwise we24all ( i, τ ) unhappy . Intuitively, here is what we have already done: For every ordering ≻ , wehave found a distribution over ≻ -ordered hierarchical allocation rules M ≻ that minimizesthe maximal unhappiness and subject to this, the number of maximally unhappy types. Wethen choose from these ( ≻ , M ≻ ) pairs those that minimize the maximal unhappiness, andsubject to this, the number of maximally unhappy types. We have made these definitionsbecause we want to eventually show that there is an ordering ≻ , such that F ≻ ( M ≻ ) = 0,and it is natural to start with the ordering that is “closest” to satisfying this property.What we will show next is that, if ∃ ≻∈ M O that does not make every (bidder, type) pairhappy, then there also exists some ≻ ′ ∈ M O , such that F ≻ ′ ( M ≻ ′ ) = F ≻ ( M ≻ ), | S ≻ ′ | = | S ≻ | ,and ( i, τ ) ≻ ′ ( j, τ ′ ) for all ( i, τ ) ∈ S ≻ ′ , ( j, τ ′ ) / ∈ S ≻ ′ . In other words, only the top | S ≻ ′ | typesin ≻ ′ are maximally unhappy. From here, we will show that because ≻ ′ ∈ M O , that S ≻ ′ isa constricting set for R , contradicting its feasibility. We begin by showing the existence of ≻ ′ , beginning with an arbitrary ≻∈ M O .Before we begin, we introduce some terminology. We say that two (bidder, type) pairs( i, τ ) , ( j, τ ′ ) are adjacent if ( i, τ ) ≻ ( k, τ ′′ ) ⇔ ( j, τ ′ ) ≻ ( k, τ ′′ ) for all ( k, τ ′′ ) / ∈ { ( i, τ ) , ( j, τ ′ ) } .For any (cid:23) , we also define an equivalence relation ∼ (cid:23) with ( i, τ ) ∼ (cid:23) ( j, τ ′ ) ⇔ ( i, τ ) (cid:23) ( j, τ ′ ) ∧ ( j, τ ′ ) (cid:23) ( i, τ ). Finally, we say that there is a cut between two adjacent types ( i, τ )and ( j, τ ′ ) in (cid:23) if ( i, τ ) ≻ ( j, τ ′ ) and ( j, τ ′ ) ( i, τ ). When we talk about adding a cutbelow ( i, τ ), we mean modifying (cid:23) so that ( j, τ ′ ) ( i, τ ) for all ( i, τ ) ≻ ( j, τ ′ ) (but otherwisekeeping (cid:23) the same). When we talk about removing a cut between two equivalence classes A and B , we mean modifying (cid:23) so that ( i, τ ) ∼ (cid:23) ( j, τ ′ ) for all ( i, τ ) , ( j, τ ′ ) ∈ A ∪ B (butotherwise keeping (cid:23) the same).Now, if S ≻ is not the highest | S ≻ | (bidder, type) pairs, let ( i, τ ) be the maximal elementunder ≻ in S ≻ such that there exists some ( k, τ ′′ ) / ∈ S ≻ with ( k, τ ′′ ) ≻ ( i, τ ). Then theadjacent (bidder, type) pair ( j, τ ′ ) with ( j, τ ′ ) ≻ ( i, τ ) is necessarily / ∈ S ≻ . We proceed toshow that we can change ≻ to swap ( i, τ ) ≻ ( j, τ ′ ) (keeping M , S ≻ and F ≻ ( M ) as-is). Wecan repeat these swaps iteratively and they will terminate in the ≻ ′ we want (with S ≻ ′ equalto the first | S ≻ ′ | (bidder, type) pairs).We now proceed with a case analysis, for fixed ( j, τ ′ ) / ∈ S ≻ , ( i, τ ) ∈ S ≻ , ( j, τ ′ ) ≻ ( i, τ )and ( j, τ ′ ) , ( i, τ ) adjacent. • Case 1: i = j .Since ≻ is a linear extension of the bidder’s own ordering, we must have π i ( τ ′ ) ≥ π i ( τ ),but we know that π i ( τ ′ ) − π M ≻ i ( τ ′ ) < π i ( τ ) − π M ≻ i ( τ ) , thus π M ≻ i ( τ ′ ) > π M ≻ i ( τ ) ≥
0. In any hierarchical mechanism (cid:23) , if there is no cut25etween ( i, τ ′ ) and ( i, τ ), then they would receive the item with the same probability.Therefore, there must exist some (cid:23) in the support of M ≻ with a cut below ( i, τ ′ ), andin which ( i, τ ′ ) gets the item with non-zero probability. We modify M ≻ by modifyingthe hierarchical allocation rules (cid:23) in its support as follows.Let (cid:23) be a hierarchical allocation rule in the support of M ≻ . If there is no cutbelow ( i, τ ′ ), we do nothing. If all (bidder, type) pairs equivalent to ( i, τ ′ ) and thoseequivalent to ( i, τ ) are of bidder i , we remove the cut below ( i, τ ′ ). This does not affectthe allocation probabilities at all, because it was impossible for two types equivalentto either ( i, τ ′ ) or ( i, τ ) to show up together anyway. So after this “modification,” wehaven’t changed M ≻ at all, meaning that there must still exist some (cid:23) in the support of M ≻ with a cut below ( i, τ ′ ), and in which ( i, τ ′ ) gets the item with non-zero probability,and clearly it is not one of the allocation rules we just modified by removing the cutbelow ( i, τ ′ ). For such an (cid:23) , there is at least one (bidder, type) pair with bidder = i equivalent to ( i, τ ′ ) or ( i, τ ). We distinguish two sub-cases: – Every bidder k = i has at least one type τ k such that ( i, τ ) (cid:23) ( k, τ k ) (in otherwords, every (bidder, type) pair equivalent to ( i, τ ) wins the item with non-zeroprobability). Consider now moving the cut from below ( i, τ ′ ) to right above ( i, τ ′ ).Clearly, ( i, τ ′ ) will be less happy if we do this. Every (bidder, type) pair withbidder = i that was formerly equivalent to ( i, τ ′ ) will be strictly happier, as nowthey do not have to share the item with ( i, τ ′ ), whereas previously they did withpositive probability. Every (bidder, type) pair with bidder = i that is equivalentto ( i, τ ) will be strictly happier, as they now get to share the item with ( i, τ ′ ),whereas previously they always lost to ( i, τ ′ ). It is also clear to see that all ( i, τ ′′ ),for τ ′′ = τ ′ are unaffected by this change. So in particular ( i, τ ) is unaffected.Consider instead moving the cut from below ( i, τ ′ ) to right below ( i, τ ). Then( i, τ ) is clearly strictly happier, every (bidder, type) pair with bidder = i that wasformerly equivalent to ( i, τ ) is less happy than before (as they now don’t get toshare with ( i, τ )), every (bidder, type) pair with bidder = i that is equivalent to( i, τ ′ ) is also less happy than before (because now they have to share with ( i, τ ′ )),and all ( i, τ ′′ ), for τ ′′ = τ are not affected by the change.To summarize, we have argued that when we move the cut from below ( i, τ ′ ) tojust below ( i, τ ), ( i, τ ) becomes strictly happier, and every (bidder, type) pairthat becomes less happy by this change becomes strictly happier if we insteadmove the cut to just above ( i, τ ′ ) instead. Also, ( i, τ ) is unaffected by moving thecut to just above ( i, τ ). So with a tiny probability ǫ , move the cut from below26 i, τ ′ ) to just above ( i, τ ′ ), whenever (cid:23) is sampled from M ≻ . This makes all ofthe (bidder, type) pairs with bidder = i that were equivalent to either ( i, τ ′ ) or( i, τ ) strictly happier. With a tinier probability δ , move the cut from below ( i, τ ′ )to below ( i, τ ), whenever (cid:23) is sampled from M ≻ . Choose ǫ to be small enoughthat we don’t make ( i, τ ′ ) maximally unhappy, and choose δ to be small enoughso that we don’t make any (type, bidder) pairs besides ( i, τ ′ ) less happy than theywere in (cid:23) . Then we have strictly increased the happiness of ( i, τ ) without making( i, τ ′ ) maximally unhappy, or decreasing the happiness of any other (bidder, type)pairs. Therefore, we have reduced | S ≻ | , contradicting the choice of M ≻ . – If there is a bidder k such that ( i, τ ) ( k, τ k ) for all τ k , (call such bidders high ),then no (bidder, type) pair equivalent to ( i, τ ) can possibly win the item. We alsoknow that every high bidder k has at least one type τ k such that ( k, τ k ) ∼ (cid:23) ( i, τ ′ )by our choice of (cid:23) (otherwise ( i, τ ′ ) would get the item with probability 0). Nowwe can basically use the same argument as above. The only difference is thatwhen we move the cut to just above ( i, τ ′ ) or just below ( i, τ ), (bidder, type)pairs formerly equivalent to ( i, τ ) (other than ( i, τ ) itself) will remain unaffected.But since every high bidder k has a type τ k with ( k, τ k ) ∼ (cid:23) ( i, τ ′ ), ( i, τ ) will bestrictly happier if we move the cut to just below ( i, τ ). Therefore, it is still thecase that every (bidder, type) pair who is made unhappier by moving the cut tojust below ( i, τ ) is made strictly happier by moving the cut to just above ( i, τ ′ ).So we can carry over the same reasoning as above (choosing ǫ, δ sufficiently small),and again contradict the choice of M ≻ .Therefore, it can not be the case that i = j . • Case 2: i = j and there is never a cut below ( j, τ ′ ).This case is easy. If we switch ( j, τ ′ ) and ( i, τ ) in ≻ , then the set S ≻ is exactly thesame, and the distribution M ≻ is exactly the same. However, we have now relabeledthe types in S ≻ to get closer to the top | S ≻ | elements being in S ≻ . Note that all (cid:23) withno cut below ( j, τ ′ ) are all ≻ -ordered for the new ≻ as well, so this is a valid swap. • Case 3: i = j and there is sometimes a cut below ( j, τ ′ ).Pick a (cid:23) in the support of M ≻ that has a cut between ( j, τ ′ ) and ( i, τ ) and in which( j, τ ′ ) gets the item with positive probability. Note that if such a (cid:23) doesn’t exist, wecan remove the cut below ( j, τ ′ ) in all (cid:23) in the support of M ≻ without changing theallocation probabilities and return to Case 2. From here, there are again two subcases:27 ( k, τ ′′ ) / ∈ S ≻ for all ( k, τ ′′ ) ∼ (cid:23) ( j, τ ′ ). This means that all (bidder, type) pairsequivalent to ( j, τ ′ ) are not maximally unhappy. Therefore, if we pick a tiny ǫ and remove the cut below ( j, τ ′ ) with probability ǫ , only the types equivalent to( j, τ ′ ) will become unhappier. So there is a sufficiently small ǫ > i = j and ( j, τ ′ ) receives the item with non-zero probability under (cid:23) ,this operation makes ( i, τ ) strictly happier, as she now sometimes shares the itemwith ( j, τ ′ ) (whereas previously she always lost). So this operation will create nonew maximally unhappy (bidder, type) pairs, while making ( i, τ ) strictly happier,decreasing the size of | S ≻ | and contradicting the choice of M ≻ . – There exists a ( k, τ ′′ ) ∈ S ≻ with ( k, τ ′′ ) ∼ (cid:23) ( j, τ ′ ) (and ( k, τ ′′ ) = ( j, τ ′ )). Let( k, τ ′′ ) be the minimal such (bidder, type) pair under ≻ . Note that by our choiceof ( i, τ ), that all ( ℓ, τ ′′′ ) ≻ ( k, τ ′′ ) are also in S ≻ (maximally unhappy). Nowconsider introducing a cut below ( k, τ ′′ ) with some tiny probability ǫ . Then theonly (bidder, type) pairs who may become unhappier with this change are thosethat are still equivalent to ( j, τ ′ ), and all such types are not maximally unhappy.The only (bidder, type) pairs who may become happier with this change are thosethat are ( k, τ ′′ ) or those that are ≻ ( k, τ ′′ ), all of which are in S ≻ . So if any ofthese types become happier at all with this change, there is a sufficiently smallprobability ǫ with which we can make this change without introducing any newmaximally unhappy types and therefore decreasing | S ≻ | , a contradiction. So wemust not make any (bidder, type) pairs happier with this change, and therefore wemust also not make any (bidder, type) pairs unhappier (note that it is impossibleto make any (bidder, type) pair unhappier without making some other (bidder,type) pair happier, since we are treating (0 , ⊥ ) as a regular type). So we mayin fact introduce a cut below ( k, τ ′′ ) with probability 1 whenever M ≻ samples (cid:23) without affecting π M ≻ at all, but removing the original (cid:23) from the supportof M ≻ , and replacing it with a (cid:23) in which all (bidder, type) pairs equivalent to( j, τ ′ ) are not maximally unhappy. After doing this for all such (cid:23) , we must returnto the previous sub-case, after which we again obtain a contradiction.Hence, it can not be the case that i = j with a cut sometimes below ( j, τ ′ ).At the end of all three cases, we see that if we ever have ( j, τ ′ ) / ∈ S ≻ and ( i, τ ) ∈ S ≻ ,with ( j, τ ′ ) ≻ ( i, τ ) and ( j, τ ′ ) , ( i, τ ′ ) adjacent, then we must have i = j , and no (cid:23) in thesupport of M ≻ ever places a cut directly below ( j, τ ′ ). Hence, we can simply swap the order28f these types in ≻ without affecting S ≻ or F ≻ ( M ) (as we described in Case 2 above), andwe do that repeatedly until S ≻ is equal to the top | S ≻ | (bidder, type) pairs according to ≻ .Now that we have shown the existence of such a ≻ , we show that it implies a constrictingset. Label the elements in S ≻ as ( i , τ ) ≻ . . . ≻ ( i k , τ k ) ( k = | S ≻ | ). Now consider a (cid:23) in thesupport of M ≻ that has no cut below ( i k , τ k ), and consider putting a cut there with sometiny probability ǫ whenever (cid:23) is sampled. The only effect this might have is that when theitem is awarded to a (bidder, type) pair outside S ≻ , it is now awarded to a (bidder, type)pair inside S ≻ instead with some probability. Therefore, if anyone gets happier, it is someonein S ≻ . However, if we make anyone in S ≻ happier and choose ǫ small enough so that wedon’t make anyone outside of S ≻ maximally unhappy, we decrease | S ≻ | , contradicting thechoice of M ≻ . Therefore, putting a cut below ( i k , τ k ) cannot possibly make anyone happier,and therefore cannot make anyone unhappier. So we may w.l.o.g. assume that there is a cutbelow ( i k , τ k ) in all (cid:23) in the support of M ≻ . But now we get that the item always goes tosomeone in S ≻ whenever a (bidder, type) pair in S ≻ is reported, yet all (bidder, type) pairsin this set are unhappy. Therefore, S ≻ is a constricting set, certifying that the given R isinfeasible.Putting everything together, we have shown that if there is no ≻ with F ≻ ( M ≻ ) = 0then the reduced form is infeasible. So there must be some ≻ with F ≻ ( M ≻ ) = 0, and suchan M ≻ induces the reduced form by sampling only ≻ -ordered hierarchical allocation rules,completing the proof. (cid:3) In this section, we show how our results above on reduced forms can be useful for multi-item mechanism design as well. Essentially, our key observation is that an m -item interimallocation rule is feasible if and only if the m projected single-item reduced forms are feasible,so the question is simply whether or not an m -item reduced form contains enough usefulinformation for mechanism design. When buyers are additive, this information indeed sufficesto guarantee that a mechanism inducing to this interim allocation rule is Bayesian IncentiveCompatible , which allows us to formulate and solve an optimization problem. We makethese statements more precise shortly, but first provide some notation specific to multi-itemmechanism design not covered in Section 2. 29 .1 Notation
For Section 5, there are n bidders and m items. All bidders’ valuation functions are additive.We write ~v i to denote the type of bidder i , with the convention that v ij represents her valuefor item j and that her value for a bundle S of items is simply P j ∈ S v ij . We still let T denotethe space of possible types, which is now a subset of R n .To fully specify a (direct-revelation) multi-item mechanism for additive bidders, we needto describe, potentially succinctly, for all type profiles ~v ∈ T n , and for every bidder i , theoutcome M i ( ~v ) = ( ~φ i ( ~v ) , p i ( ~v )) given by M to bidder i when the reported bidder types are ~v . Here, φ ij ( ~v ) is the ex-post probability that item j is given to bidder i when the reportedtypes are ~v , and p i ( ~v ) is the ex-post price that i pays. The value of bidder i for outcome M i ( ~w ) is just her expected value ~v i · ~φ i ( ~w ), while her utility is quasi-linear, meaning thatbidder i ’s utility for the same outcome is U ( ~v i , M i ( ~w )) := ~v i · ~φ i ( ~w ) − p i ( ~w ). The relationbetween the ex-post probabilities φ and interim probabilities π is just the following: for all i , j , ~v i ∈ T : π ij ( ~v i ) = E ~v − i ∼D − i [ φ ij ( ~v i ; ~v − i )]. We now formally define Bayesian IncentiveCompatibility (BIC): Definition 6. (Bayesian Incentive Compatible Mechanism) A mechanism M is called BICiff the following inequality holds for all i ∈ [ n ] , ~v i , ~w i ∈ T : E ~v − i ∼D − i [ U ( ~v i , M i ( ~v ))] ≥ E ~v − i ∼D − i [ U ( ~v i , M i ( ~w i ; ~v − i ))] . We begin this section with our key observation, essentially stating that some single-itemresults (specifically, those in Section 4) can be extended for free to some multi-item settings.Let us begin by being clear what we mean by an interim allocation rule projecting a reducedform onto item j . Definition 7. (Projected reduced form) Let there be m heterogeneous items, T be some arbi-trary type space, and R = { π ij ( · ) } i ∈ [ n ] ,j ∈ [ m ] be some interim allocation rule of a mechanism.Then the projected reduced form of R onto item j , R j , is just R j = { π ij ( · ) } i ∈ [ n ] . Note that R j still takes as input types in the original type space T .Observation . An m -item interim allocation rule R is feasible if and only if for all j , theprojected reduced form R j onto item j is feasible. Furthermore, if R is feasible, R is inducedby the ex-post allocation rule that allocates each item j according to R j independently ofthe others. 30 roof. First, assume that R is feasible, and let M be an ex-post allocation rule that induces R . We wish to come up with an allocation rule for item j that induces R j . Define M j to be the single-item allocation rule that runs M and awards the single item to whicheverbidder was awarded item j under M . Clearly, M j implements R j , so if R is feasible, so iseach projection R j .Next, assume that each R j is feasible, and let M j be an ex-post allocation rule thatinduces R j . Then let M be the allocation rule that on every input type profile, runs M j on that type profile for all j and awards item j to whoever receives the single item under M j . Clearly, the projection of the reduced form of M onto item j will be exactly R j , so thereduced form of M is exactly R . Therefore, M induces R , and R is feasible.So R is feasible if and only if each projection R j is feasible. Furthermore, the aboveargument shows that when R is feasible, R can be induced by an ex-post allocation rulethat allocates each item separately.Observation 3 combined with Theorem 2 immediately yields our characterization of fea-sible multi-item interim allocation rules (from Section 1). Replacing Theorem 2 with Theo-rem 5 provides a tighter characterization in the symmetric case. Note that at this point wehave absolutely not addressed the issue of when this characterization is useful for multi-itemmechanism design - all we have done is observed (somewhat trivially) that the structure ofreduced forms is preserved under concatenation.So now, let’s address this issue and discuss the multi-item settings in which Observation 3is useful for mechanism design. Essentially, we observe that the interim allocation rule as wehave defined it provides sufficient information to determine whether or not a mechanism isBIC if and only if bidders’ valuations are additive. We first present an example illustratingthat this fails, for instance, when bidders are instead unit-demand. Example 1.
Consider a setting with a single unit-demand bidder, two items, and one possibletype, (1 , (value one for each item). Consider the following two ex-post allocation rules: • Pick j uniformly random from { , } and award item j . • Award the set of items { , } with probability / , and the set ∅ with probability / .Then these two ex-post allocation rules have the same interim allocation rule: π (1 ,
1) = π (1 ,
1) = 1 / . But the bidder’s expected value under the first ex-post allocation rule is ,whereas under the second it is / . Therefore, the interim allocation rule simply does not Note that M j takes the whole type ~v i of each bidder i as input, and not just v ij . A bidder is unit-demand if whenever they have value v j for item j , their value for set S is equal tomax j ∈ S { v j } . ontain enough information for the bidder to compute her expected value for reporting a giventype to an ex-post allocation rule inducing it - because it depends on which ex-post allocationrule is chosen.Note that if instead the bidder were additive, she would have expected value under bothex-post allocation rules, and this would not be an issue.Observation . When bidders are additive, the per-item interim allocation rule R containsenough information for a bidder to compute her expected value when reporting type ~w i to any ex-post allocation rule inducing R . It is exactly P j v ij · π ij ( ~w i ). Proof.
Let M be any ex-post allocation rule with ex-post allocation probabilities φ ij ( · ). Thenwe can write the expected value of a buyer with type ~v i for reporting ~w i to M as: E ~v − i ∼D − i [ X j v ij · φ ij ( ~w i ; ~v − i )] = X j E ~v − i ∼D − i [ v ij · φ ij ( ~w i ; ~v − i )]= X j v ij · E ~v − i ∼D − i [ φ ij ( ~w i ; ~v − i )] = X j v ij · π ij ( ~w i ) . Essentially what makes additive buyers unique in comparison to other multi-dimensionalvaluation functions is that the marginal value for item j is completely independent of the setit is being added to. Let us again emphasize that an ex-post allocation rule M implementingan interim allocation rule R absolutely takes into consideration the value of bidders for items ℓ = j when determining how to allocate item j , as this information is stored in their types.However, Observations 3 shows that it need not consider how items ℓ = j are themselves allocated when determining how to allocate item j . It is well-known that even when there isjust a single additive bidder and two items, and even if the bidder’s values for these itemsare distributed i.i.d., that the allocation rule of the revenue-optimal mechanism necessarilyconsiders values for items ℓ = j when deciding the allocation of item j . Below is a folkloreexample (that appears concretely, for instance, in [DDT14]). This shows that even if weare willing to restrict to a characterization only of interim allocation rules that are revenue-optimal for simple multi-item instances, we should not hope for a stronger characterizationthat (say) allocates each item j independent of bidders’ values for items ℓ = j . Example 2.
There is a single additive bidder and two items. Each v j is drawn independentlyand uniformly from the set { , } . Then the revenue optimal mechanism awards both itemsto the bidder whenever v + v ≥ , and charges price .Note that when v = 1 , whether or not the bidder receives item depends on v (namely,she receives item iff v = 2 ). BIC Multi-item auction.
Given as input n distributions D , . . . , D n over valuationvectors for m items, output a BIC mechanism M whose expected revenue is optimalrelative to any other, possibly randomized, BIC mechanism, when played by n additivebidders whose valuation vectors are sampled independently from D , . . . , D n . Note thateach D i need not be a product distribution, a single bidder’s values for different items maybe arbitrarily correlated.Our approach to solving proving Theorem 3 is to use the separation oracle for checking thefeasibility of a reduced form developed in Corollary 3 inside a linear program that optimizesover all feasible interim allocation rules and interim price rules. We begin with the LPformulation in Figure 1 below, followed by a proof that the LP is correct. Variables: • π ij ( ~v i ), for all bidders i ∈ [ n ], items j ∈ [ m ], and ~v i ∈ T , the interim probability thatbidder i gets item j when reporting type ~v i ( mnc variables). • q i ( ~v i ), for all bidders i ∈ [ n ], ~v i ∈ T , the interim expected price that bidder i payswhen reporting type ~v i ( nc variables). Constraints: • ≤ π ij ( ~v i ) ≤
1, for all i ∈ [ n ], j ∈ [ m ], ~v i ∈ T , guaranteeing that each π ij ( ~v i ) is aprobability ( mnc constraints). • P j ∈ [ m ] v ij π ij ( ~v i ) − q i ( ~v i ) ≥
0, for all i ∈ [ n ], ~v i ∈ T , guaranteeing that the mechanismis interim Individually Rational (interim IR) ( nc constraints). • P j ∈ [ m ] v ij π ij ( ~v i ) − q i ( ~v i ) ≥ P j ∈ [ m ] v ij π ij ( ~v ′ i ) − q i ( ~v ′ i ), for all i ∈ [ n ] , ~v i , ~v ′ i ∈ T , guaran-teeing that the mechanism is Bayesian Incentive Compatible (BIC) ( nc constraints). • SO ( ~π, D ) =“ Yes ”, guaranteeing that there is an ex-post allocation rule inducing π ; Maximizing: • P i ∈ [ n ] ,~v i ∈ T q i ( ~v i ) Pr[ ~v i ← D i ], the expected revenue.Figure 1: A folklore LP (that appears concretely, e.g., in [DW12]), where we use a separationoracle to determine feasibility of the interim allocation rule. In parentheses at the end ofeach line is the number of such variables/constraints . Proposition 3.
Provided that SO acts as a valid separation oracle for the space of feasibleinterim allocation rules, the LP of Figure 1 outputs the revenue optimal interim allocation ule and interim price rule for BIC multi-item auction in time polynomial in n, m, c and theruntime of SO .Proof. First, it is clear that any output of the LP of Figure 1 (henceforth, just “the LP”) isinterim IR, BIC, and feasible, as long as SO is correct. This is because the constraints on aninterim allocation rule/price rule pair to be interim IR are linear and explicitly included inthe LP. The same holds for BIC. Therefore, as long as SO is correct, any interim allocationrule/price rule pair accepted by the LP must be interim IR, BIC, and feasible. It is alsoclear that the objective function correctly computes the expected revenue of any interim pricerule considered. So the LP outputs exactly the feasible, interim IR, BIC interim allocationrule/price rule pair that maximizes expected revenue with respect to all feasible, interim IR,BIC interim allocation rule/price rule pairs.Second, it is clear that every feasible, IR, BIC mechanism has an interim allocationrule/price rule pair, and that this pair is also interim IR, feasible, and BIC. So the revenue-optimal interim allocation/price rule pair output by the LP is indeed optimal with respectto all feasible, IR, BIC mechanisms.Finally, it is clear that the number of variables and constraints (excluding SO ) is polyno-mial in nmc . Therefore, the LP can be solved in time polynomial in n, m, c and the runtimeof SO via a direct application of Theorem 4. Proof of Theorem 3:
Theorem 3 now follows immediately from Proposition 3, Obser-vation 3, and Corollary 3. Corollary 3 combined with Observation 3 guarantees that we candesign the desired separation oracle, which terminates in time poly( n, m, c ), and Proposi-tion 3 guarantees that we can use this to solve the LP in time poly( n, m, c ). After findingthe optimal interim allocation/price rule pair ( ~π ∗ , ~q ∗ ), Corollary 3 shows how to find in timepoly( n, m, c ) a succinct description of an ex-post allocation rule implementing ~π ∗ (as a dis-tribution over at most cn + 1 strict hierarchical allocation rules). So one can implement theoptimal mechanism by using this ex-post allocation rule, and charging bidder i price q ∗ i ( ~v i )when her bid is ~v i . (cid:3) We also note that the LP of Figure 1 only requires that the mechanism be interimindividually rational. A well-known simple trick (shown e.g. in [DW12]) converts any BIC,interim IR mechanism into one that is BIC and ex-post IR with no loss in revenue, so this isw.l.o.g. For completeness, we describe the trick below. Note that the trick does not requireany assumptions on the valuation functions of bidders except that they are quasi-linear andrisk-neutral. 34 roposition 4.
Let M be any BIC and interim IR mechanism for quasi-linear and risk-neutral bidders. There exists another mechanism M ′ that uses exactly the same ex-postallocation rule and interim price rule as M that is also ex-post IR.Proof. For any bidder i and valuation function/type v i ( · ) : 2 [ m ] → R ( v i takes as input aset of items and outputs a value), let V Mi ( v i ) denote the expected value of bidder i whentruthfully reporting type v i to M (over any randomness in M , and the randomness in otherbidders’ types). Let also q Mi ( v i ) denote the interim price paid by i when reporting type v i to M (over the same randomness). As M is interim IR, we must have q Mi ( v i ) ≤ V Mi ( v i ).Now define the following ex-post price rule for M ′ : whenever bidder i receives items S after reporting type v i , charge her q Mi ( v i ) V Mi ( v i ) · v i ( S ). Observe that we have not changed theex-post allocation rule, so clearly M and M ′ have the same ex-post allocation rule. As q Mi ( v i ) V Mi ( v i ) ≤ M ′ is clearly ex-post IR. All that remains to verify is that M and M ′ havethe same interim price rule. But this is also clear: if we define φ S ( ~v ) to be the probabilitythat M allocates set S to bidder i on input ~v , then the interim price paid by bidder i whenreporting type v i to M ′ is exactly: E ~v − i ∼D − i [ X S ⊆ [ m ] φ S ( ~v ) · q Mi ( v i ) V Mi ( v i ) · v i ( S )] = q Mi ( v i ) V Mi ( v i ) · E ~v − i ∼D − i [ X S ⊆ [ m ] φ S ( ~v ) · v i ( S )]= q Mi ( v i ) V Mi ( v i ) · V Mi ( v i ) = q Mi ( v i ) . Motivated by settings where an optimization approach is necessary to develop optimal auc-tions, we study single-item reduced-form auctions for asymmetric bidders. We provide alinear-sized subset of necessary and sufficient Border conditions (down from exponential-sized of prior work) that can be checked in nearly-linear time, and also show that everyfeasible reduced form can be implemented as a distribution of hierarchical allocation rulesthat all respect the same total ordering over all bidders’ types. We further show that our re-sults imply both polynomial-time algorithms and structural characterizations for multi-itemauctions with additive bidders.Our work demonstrates how a better understanding of reduced-form auctions in thecore single-item setting can be utilized for much more general settings, and also that acomputational lens can lead not only to tractable optimization algorithms, but also improved35tructural understanding.We conclude with a very brief discussion on how one could extend our approach formulti-item auction design beyond additive buyers. Indeed, the core issue is that the interimallocation probabilities no longer contain enough information to verify that a mechanism isBIC (Example 1), so in order to get mileage out of our approach one first needs to come upwith a new interim description of auctions. It is not hard to come up with definitions thatwork for, say, unit-demand buyers: first observe that it is w.l.o.g. to only consider auctionswhich award each bidder at most one item. Then simply define π ij ( ~v i ) to be the interimprobablity that bidder i receives item j when submitting valuation vector ~v i , and exactlythe same LP as written in Figure 1 finds the revenue-optimal BIC mechanism, providedwe can design a computationally efficient separation oracle for this new space of feasibleinterim allocation rules (which never allocate the same item to multiple bidders or the samebidder multiple items). However, obtaining such a separation oracle is no longer simplythe product of n single-item problems, and in fact Gopalan et. al. prove that, under well-believed complexity-theoretic assumptions, no computationally efficient separation oracle forthis space exists [GNR15].Still, our approach has proven useful for the design of nearly -optimal multi-item auc-tions in settings with unit-demand buyers (and in fact significantly more general settings aswell) in follow-up works by the authors and others [CDW12b, CDW13a, CDW13b, BGM13,DDW15, CDW16], essentially by making use of computationally-efficient but approximate characterization of the feasible interim allocation probabilities in these settings. The presentpaper remains unique in this line of works for finding the optimal mechanism without ap-proximation error. References [AFH +
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Omitted Proofs
The first omitted proof is of Corollary 1, which simply claimed that we can find a violatedBorder constraint in the symmetric case in time O ( c (log c + log n )). Proof of Corollary 1.
First, sort the types τ in T in decreasing order according to π ( τ )in time O ( c log c ), and label the distinct values of π ( τ ) as x , . . . , x k ( k ≤ c ), and de-fine X i = { τ | π ( τ ) = x i } . Next, compute P τ : π ( τ ) ≥ x i π ( τ ) Pr[ τ ] for all i in the follow-ing way: First, compute P τ ∈ X i π ( τ ) Pr[ τ ] for all i . This can clearly be done in time O ( | X i | ) for all i , and therefore the total computation takes time O ( c ). Next, observe that P τ : π ( τ ) ≥ x i +1 π ( τ ) Pr[ τ ] = P τ : π ( τ ) ≥ x i π ( τ ) Pr[ τ ] + P τ ∈ X i +1 π ( τ ) Pr[ τ ]. So we can compute P τ : π ( τ ) ≥ x i +1 π ( τ ) Pr[ τ ] from P τ : π ( τ ) ≥ x i π ( τ ) Pr[ τ ] using just O (1) additional computation.Therefore, we can compute P τ : π ( τ ) ≥ x i π ( τ ) Pr[ τ ] for all i in total time O ( c ). This imme-diately gives us the LHS for all inequalities (3) in total time O ( c log c ). Similarly, we cancompute P τ ∈ X i Pr[ τ ] for all i in total time O ( c ), and use these in the same way to computePr t ∼ D [ π ( t ) ≥ x i ] = P τ : π ( τ ) ≥ x i Pr[ τ ] for all i in total time O ( c log c ). With this, we canthen compute the RHS of each inequality (3) using repeated squaring in time O (log n ) perinequality.So all together, we have a pre-processing stage which takes time O ( c log c ), and an addi-tional O ( c log n ) to compute each RHS, resulting in a total runtime of O ( c (log c + log n )).The next omitted proof is of (the algorithmic portion of) Theorem 1, which claims thatwe can find a violated Border constraint in the asymmetric case in time O ( cn · log( cn )). Proof of Theorem 1:
Proposition 1 immediately yields the first part of Theorem 1 (thata reduced form is feasible if and only if it satisfies Equation 2 for all x ). The only remainingdetail to check is that a violated Border constraint (if it exists) can be found in the desiredruntime, which follows from a similar routine computation as in Corollary 1. For the sakeof completeness, we include with a full proof of this.First, sort all (bidder, type) pairs ( i, τ ) ∈ [ n ] × T in decreasing order according to ˆ π i ( τ )in time O ( nc log( nc )) ( c = | T | ), label the distinct values of ˆ π i ( τ ) as x , . . . , x k ( k ≤ nc ),and store X j = { ( i, τ ) | ˆ π i ( τ ) = x j } (i.e. X j is the set of (bidder, type) pairs for whichˆ π i ( τ ) = x j ). Next, to compute all left-hand sides of Equation (2), begin by computing P ( i,τ ) ∈ X j Pr[ t i = τ ] · π i ( τ ) for all j . This can clearly be done in time O ( | X j | ) for all j , andtherefore the total computation takes time O ( nc ). Next, observe that P ( i,τ ):ˆ π i ( τ ) ≥ x j +1 Pr[ t i = τ ] · π i ( τ ) = P ( i,τ ):ˆ π i ( τ ) ≥ x j Pr[ t i = τ ] · π i ( τ ) + P ( i,τ ) ∈ X j +1 Pr[ t i = τ ] · π i ( τ ). So we can compute P ( i,τ ):ˆ π i ( τ ) ≥ x j +1 Pr[ t i = τ ] · π i ( τ ) from P ( i,τ ):ˆ π i ( τ ) ≥ x j Pr[ t i = τ ] · π i ( τ ) with O (1) additionalcomputation (by making use of our pre-processed values). So starting with x , we can41ompute all left-hand sides of Equation (2) with O (1) additional computation per constraint,for a total computation time of O ( nc ).Computing all right-hand sides is a touch trickier, because we don’t actually want tomultiply n numbers together for each of nc inequalities. So we will make use of the factthat most of the terms going into the product on the RHS don’t change between successiveinequalities. To compute all right-hand sides, additionally for all i , sort all types τ ∈ T in decreasing order of ˆ π i ( τ ) in time O ( c log c ) per bidder, or O ( nc log c ) in total, label thedistinct values of ˆ π i ( τ ) as x i , . . . , x ik i ( k i ≤ c ), and store X ij = { τ | ˆ π i ( τ ) = x ij } . Compute P τ ∈ X ij Pr[ t i = τ ] for all i, j in total time O ( nc ). Recall S ( i ) x = { τ | ˆ π i ( τ ) ≥ x } , and observethat Pr t i ∼D i [ t i ∈ S ( i ) x ij ] = P τ :ˆ π i ( τ ) ≥ x ij Pr[ t i = τ ] can be computed for fixed i and all j intime O ( c ), and therefore for all i and all j in total time O ( nc ). With this, finally compute − Pr ti ∼D i [ t i ∈ S ( i ) xi ( j +1) ]1 − Pr ti ∼D i [ t i ∈ S ( i ) xij ] for all i and j ∈ { , . . . , k i − } (denoting S ( i ) x i = ∅ ), again in time O ( c ) perbidder, or O ( nc ) in total (it is just at most one division per ( i, τ ) ∈ [ n ] × T ). Now, startingfrom j = 1, we will inductively compute Q i (1 − Pr t i ∼D i [ t i ∈ S ( i ) x j ]) for all j , using the resultfor j − Q i (1 − Pr t i ∼D i [ t i ∈ S ( i ) x j ]) = Q i (1 − Pr t i ∼D i [ t i ∈ S ( i ) x j − ]) · Q i − Pr ti ∼D i [ t i ∈ S ( i ) xj ]1 − Pr ti ∼D i [ t i ∈ S ( i ) xj − ] . For many i , it might be that − Pr ti ∼D i [ t i ∈ S ( i ) xj ]1 − Pr ti ∼D i [ t i ∈ S ( i ) xj − ] = 1(namely, all i such that ( i, τ ) / ∈ X j for all τ ∈ T ). So to compute Q i − Pr ti ∼D i [ t i ∈ S ( i ) xj ]1 − Pr ti ∼D i [ t i ∈ S ( i ) xj − ] simplymultiply together for all i such that ∃ τ ∈ T, ( i, τ ) ∈ X j the corresponding − Pr ti ∼D i [ t i ∈ S ( i ) xj ]1 − Pr ti ∼D i [ t i ∈ S ( i ) xj − ] ,which we have precomputed. This can clearly be done in time O ( | X j | ). So the total timeto compute Q i (1 − Pr t i ∼D i [ t i ∈ S ( i ) x j ]) for all j is O ( P j | X j | ) = O ( nc ). From here, we justtake 1 − Q i (1 − Pr t i ∼D i [ t i ∈ S ( i ) x j ]) to get the corresponding right-hand sides for all x j . So allinequalities of the form (2) can be checked in total time O ( nc log( nc )). (cid:3) The final omitted proof is a technical lemma used inside the proof of Theorem 6. Thelemma states that monotonicity constraints can be tight (and linearly independent) only ifthey are between two types in the same equivalence class.
Proof of Lemma 1:
There are six cases to consider. Below we are studying whichmonotonicity constraints might potentially be tight in any possible ex-post allocation rulethat induces ˜ π . • What if ( i, τ i,j +1 ) (cid:23) ( i, τ i,j )? Then ( i, j ) th monotonicity constraint might be tight. • What if ( i, τ i,j ) (cid:23) (0 , ⊥ ) (cid:23) ( i, τ i,j +1 )? Then by definition of (cid:23) , ˜ π i ( τ i,j ) > π i ( τ i,j +1 ),so the ( i, j ) th monotonicity constraint can’t be tight. • What if (0 , ⊥ ) ( i, τ i,j +1 ) ( i, τ i,j ) and there exists a bidder k = i and type τ ′ with ( i, τ i,j +1 ) ( k, τ ′ ) ( i, τ i,j )? First, it is clear that bidder i with type τ i,j +1 only win when ( τ i,j +1 ) (cid:23) ( ℓ, t ℓ ) for all ℓ = i (because on any other profile, thereexists a (bidder, type) pair present in a tight Border constraint without ( i, τ i,j +1 )), andthat bidder i with type τ i,j must win in all these cases (exactly because on all suchprofiles, ( i, τ i,j ) is present in a tight Border constraint without ( ℓ, t ℓ ) for all ℓ ). So forevery opposing profile where τ i,j +1 has a chance of winning, τ i,j certainly wins. Nowwe show that an opposing profile exists where bidder i loses with type τ i,j +1 but winswith type τ i,j , meaning that the ( i, j ) th monotonicity constraint can’t be tight. Because( i, τ i,j +1 ) (cid:23) (0 , ⊥ ), we necessarily have ( i, τ i,j +1 ) (cid:23) ( ℓ, τ ℓ,c ℓ ) for all bidders ℓ . So considerthe opposing profile where bidder k has type τ ′ , and all other bidders ℓ / ∈ { i, k } havetype τ ℓ,c ℓ , which arises with non-zero probability by our choice of c ℓ . Clearly, bidder i with type τ i,j must win the item against these opponents, while bidder i with type τ i,j +1 must lose. Therefore, the ( i, j ) th monotonicity constraint can’t be tight. • What if (0 , ⊥ ) ( i, τ i,j +1 ) ( i, τ i,j ) and there exists a bidder k = i and type τ ′ with ( i, τ i,j ) (cid:23) ( k, τ ′ ) (cid:23) ( i, τ i,j )? We know that bidder i with type τ i,j +1 must losewhenever there exists a bidder ℓ with ( ℓ, τ ℓ ) (cid:23) ( i, τ i,j ). However, if bidder i also loseswith type τ i,j against all such profiles, then the Border constraint for the set of all(type, bidder) pairs ( ℓ, τ ) = ( i, τ i,j ) with ( ℓ, τ ) (cid:23) ( i, τ i,j ) would necessarily be tight aswell, meaning that the type ( k, τ ′ ) is present in a tight Border constraint that ( i, τ i,j )is not, contradicting ( i, τ i,j ) (cid:23) ( k, τ ′ ). So bidder i with type τ i,j must win with non-zero probability against some such opposing profile, meaning again that the ( i, j ) th monotonicity constraint can’t be tight. • What if (0 , ⊥ ) ( i, τ i,j +1 ) ( i, τ i,j ) and there exists a bidder k = i and type τ ′ with ( i, τ i,j +1 ) (cid:23) ( k, τ ′ ) (cid:23) ( i, τ i,j +1 )? The reasoning is symmetric to the above case.We know that bidder i with type τ i,j must win whenever all other bidders ℓ have( i, τ i,j +1 ) (cid:23) ( ℓ, t ℓ ). However, if bidder i also wins with type τ i,j +1 against all suchprofiles, then the Border constraint for the set ( i, τ i,j +1 ) and all (type, bidder) pairs( ℓ, τ ) with ( i, τ i,j +1 ) ( ℓ, τ ) (i.e. the set { ( ℓ, τ ) | ( i, τ i,j +1 ) ( ℓ, τ ) } ∪ { ( i, τ i,j +1 ) } )would necessarily be tight as well, meaning that the type ( i, τ i,j +1 ) is in a tight Borderconstraint that ( k, τ ′ ) is not, contradicting ( k, τ ′ ) (cid:23) ( i, τ i,j +1 ). So bidder i with type τ i,j +1 must lose with non-zero probability against some such opposing profile, meaningagain that the ( i, j ) th monotonicity constraint can’t be tight. Observe that all ( ℓ, τ ) with ( ℓ, τ ) (cid:23) ( i, τ i,j ) are necessarily have τ = τ ℓ,c , as such types are present in atmost one tight Border constraint (namely (11)), in which ( i, τ i,j +1 ) is itself present. So the specified Borderconstraint is indeed one that remains in Equation (10). Again, observe that the specified Border constraint is indeed one that remains in Equation (10), becauseif j + 1 = c , then ˜ π k ( τ ′ ) will be 0 contradicting the assumption that ( k, τ ′ ) (cid:23) ( i, τ i,j +1 ) (cid:23) (0 , ⊥ ). What if (0 , ⊥ ) ( i, τ i,j +1 ) ( i, τ i,j ) and for all bidders k = i and types τ ′ , ( i, τ i,j ) ( k, τ ′ ) or ( k, τ ′ ) ( i, τ i,j +1 )? In this case, the ( i, j ) th monotonicity constraint is guar-anteed to be tight, but linearly dependent on the set of tight Border constraints.We first argue that the Border constraint is tight for the set A = { ( k, τ ′ ) | ( k, τ ′ ) (cid:23) ( i, τ i,j ) } − ( i, τ i,j ): on any profile ( t , . . . , t n ) where t i = τ i,j , but some ( k, t k ) (cid:23) ( i, τ i,j )(possibly k = i ), clearly some bidder k with ( k, t k ) ∈ A must win (as the Border con-straint for B = { ( k, τ ′ ) | ( k, τ ′ ) (cid:23) ( i, τ i,j ) } is tight, by definition of (cid:23) ). Moreover, when t i = τ i,j and the profile ( t , . . . , t n ) has some type ( k, t k ) ∈ A then k = i , and we musthave ( i, τ i,j ) ( k, t k ) (by hypothesis defining this case). Therefore, the winner muststill be some ( k, t k ) ∈ A , and we conclude that the Border constraint for A must betight.Similar reasoning implies that the Border constraint for set C = { ( k, τ ′ ) | ( k, τ ′ ) (cid:23) ( i, τ i,j ) } ∪ { ( i, τ i,j +1 ) } is tight. Finally, one can take a linear combination of the tightBorder constraints for the three sets { ( k, τ ′ ) | ( k, τ ′ ) (cid:23) ( i, τ i,j ) }− ( i, τ i,j ), { ( k, τ ′ ) | ( k, τ ′ ) (cid:23) ( i, τ i,j ) } and { ( k, τ ′ ) | ( k, τ ′ ) (cid:23) ( i, τ i,j ) } ∪ { ( i, τ i,j +1 ) } to recover the ( i, j ) th monotonicityconstraint. To be thorough, we do the calculation below: we first list the three tightBorder’s constraints. X k X j | ( k,τ k,j ) ∈ A Pr[ t k = τ k,j ]˜ π k ( τ k,j ) = 1 − Y k − X j | ( k,τ k,j ) ∈ A Pr[ t k = τ k,j ] (13) X k X j | ( k,τ k,j ) ∈ B Pr[ t k = τ k,j ]˜ π k ( τ k,j ) = 1 − Y k − X j | ( k,τ k,j ) ∈ B Pr[ t k = τ k,j ] (14) X k X j | ( k,τ k,j ) ∈ C Pr[ t k = τ k,j ]˜ π k ( τ k,j ) = 1 − Y k − X j | ( k,τ k,j ) ∈ C Pr[ t k = τ k,j ] (15)Clearly, Equation (13) and (14) imply thatPr[ t i = τ i,j ] · ˜ π i ( τ i,j ) = Y k = i − X j | ( k,τ k,j ) ∈ A Pr[ t k = τ k,j ] · Pr[ t i = τ i,j ] . (16) Again, all three Border constraints remain in Equation (10). t i = τ i,j +1 ] · ˜ π i ( τ i,j +1 ) = Y k = i − X j | ( k,τ k,j ) ∈ B Pr[ t k = τ k,j ] · Pr[ t i = τ i,j +1 ]= Y k = i − X j | ( k,τ k,j ) ∈ A Pr[ t k = τ k,j ] · Pr[ t i = τ i,j +1 ] . (17)Hence, ˜ π i,j ( τ i,j ) = ˜ π i,j +1 ( τ i,j +1 ) = Y k = i − X j | ( k,τ k,j ) ∈ A Pr[ t k = τ k,j ] . From this case analysis, we now conclude that the ( i, j ) th monotonicity constraint can betight and linearly independent of the tight Border constraints only if ( i, τ i,j +1 ) (cid:23) ( i, τ i,j ). (cid:3)(cid:3)