Reduced-form mechanism design and ex post fairness constraints

This paper incorporates fairness constraints into the classic single-unit reduced-form implementation problem (Border in Econometrica, 59(4):1175–1187, 1991, Econ Theory 31(1):167–181, 2007; Che et al. in Econometrica 81(6): 2487–2520, 2013; Manelli and Vincent in Econometrica, 78(6):1905–1938, 2010) with two agents. To do so, I use a new approach that utilizes the results from Kellerer (Math Ann 144(4):323–344, 1961) and Gutmann et al. (Ann Prob 19(4):1781–1797, 1991). Under realistic assumptions on the constraints, the conditions are transparent and can be verified in polynomial time.


Introduction
The traditional objectives of mechanism design include aggregate welfare maximization, profit maximization, and budget balance. However, it can be desirable to add some fairness constraints to this list, as well. In allocation mechanisms with money, fairness constraints restrict ex post allocations, and they may be carried out even if they I would especially like to thank Paata Ivanisvili for his advice on the proofs provided in this work. My thanks also go out to Igor Kopylov for providing useful discussions and comments and Stergios Skaperdas for his suggestions on this paper. Finally, I am grateful to the two anonymous referees and the editor, whose insightful suggestions greatly improved this work. Any errors that remain are my own.
undermine aggregate welfare maximization, profit maximization, or budget balance. 1 For example, many government procurement and allocation programs are required by law to favor small businesses (Pai and Vohra 2012). In spectrum auctions, sellers may set allocation guarantees to prevent bidders with low bids from starvation and thus prevent them from dropping out of future auctions (Wu et al. 2014). Fairness constraints can also induce less variation in payments, which is desirable when agents have budget constraints (Sinha and Anastasopoulos 2017).
In this paper, I study the two-agent feasible reduced-form problem when ex post allocation probabilities have type-contingent fairness constraints. My results can be used to study single item allocation problems such as ex post welfare maximizing auctions with two buyers.

Feasible reduced-form problem
A selling mechanism allocates goods via money transfer. For example, an indivisible item is often awarded to the highest bidder in a single-unit auction. When the bidders have private values, the ex post allocation rule to n bidders is the joint winning probability q = (q 1 , .., q n ) given the type profile, t = (t 1 , .., t n ), such that q i (t) ≥ 0, i = 1, ..., n, n j=1 q j (t) ≤ 1.
From prior work, such as that done by Myerson (1981), we understand that a Bayesian Nash equilibrium for quasi-linear utility bidders can be completely specified with the reduced form, which is a vector Q = (Q 1 , ..., Q n ) of the interim allocations: Interim allocations have much smaller dimensions than ex post allocations, so a reduced form is useful in mechanism design problems. However, a reduced form is implementable (or equivalently, is feasible) if and only if there exists a corresponding ex post allocation rule. Maskin and Riley (1984) first discussed and proved a special case of the feasible reduced-form problem in optimal selling mechanisms. Moreover, Matthews (1984) first proposed a conjecture on feasible reduced-form auctions: suppose all n agents draw their valuations (types) from the same type space, T . A symmetric interim allocation Q : T → [0, 1] is implementable with q i : T n → [0, 1] if and only if for every (1.1) Border (1991) first proved conjecture (1.1) for symmetric auctions in a single-unit environment. Later, Border (2007) revisited this problem and generalized (1.1) for asymmetric auctions in finite type spaces. Mierendorff (2011) then extended Border's (1991) proof to asymmetric auctions. Later, Hart and Reny (2015) showed that the symmetric Border's theorem is equivalent to a second-order stochastic dominance condition with reference to an "efficient auction." A fruitful method to generalize Border's theorem is to formulate the feasible reduced-form problem as a feasible circulation flow problem. Che et al. (2013) first used the circulation flow approach to study the reduced-form problem in multiunit auctions in which there are paramodular set constraints on the ex post allocation for any buyer subgroups. 2 Later, Li (2019) formulated a related circulation flow problem, providing the existence condition for the single-unit feasible reduced-form problem where the sum of ex post allocations across agents has a symmetric type-contingent lower bound. 3 A related body of literature has focused on the equivalence of Bayesian and dominant strategy incentive compatibility (BIC-DIC equivalence). By generalizing the geometric technique found in Border (1991), Manelli and Vincent (2010) proved that in the environment of single-unit, one-dimensional continuous private values and with linear utility, given any Bayesian incentive compatible (BIC) mechanism, there is a dominant strategy incentive compatible (DIC) ex post allocation with the same interim allocation. Gershkov et al. (2013) proved a different notion of BIC-DIC equivalence in social choice settings by showing that a DIC ex post allocation is one with the smallest weighted Euclidean norm among all ex post allocations associated with the same BIC mechanism. Kushnir and Liu (2019) further extended this BIC-DIC equivalence result to nonlinear utilities to accommodate applications in principal-agent models.

Contributions to the literature
The present paper investigates the feasible reduced-form problem for n = 2 when the ex post allocations are constrained by bounded integrable functions that depend 2 Che et al. (2013) investigated the feasible circulation flow problem where the demand nodes are the interim allocations and the supply nodes are the ex post allocations. The paramodular constraints on the ex post allocations are incorporated into the flow capacity constraints from the supply nodes. The researchers drew upon a result from Hassin (1982) regarding the existence of feasible circulation flow with paramodular constraints to obtain the corresponding existence condition for ex post allocations. The paramodular constraints studied by Che et al. (2013) can be examined using polymatroid optimization, which is solvable using a greedy procedure (see Vohra 2011, in particular, the discussion in Sect. 6.2.) 3 In her formulation, the demand node are the pairs (t i , i) ∈ (T , I ), and the supply nodes are vectors t ∈ T n , where the set of agents I = {1, ..., n} share the same type space, T . The flow capacity from a supply node is a symmetric type-contingent function, ρ(t). In the formulation, the constraints on the flow capacities in the network are also paramodular. Li (2019) obtained the existence condition of the reduced-form problem in which the sum of the symmetric ex post allocation is lower-bounded by ρ(t). on both agent types (see Theorems 2.1 and 2.2), interpreted as fairness constraints. Corollary 2.3 imposes realistic conditions on the constraints, making the conditions verifiable in polynomial time. Example 1 illustrates such a contribution.

Example 1 Suppose two agents have interim allocations
The types are uniformly distributed. We then ask whether there is an ex post allocation that satisfies When g 1 = g 2 = 1, the existing literature (e.g. Border 2007;Manelli and Vincent 2010;Che et al. 2013) suggests that the answer is positive. However, the existing studies provide no answer for when g 1 , g 2 ∈ L 1 ([0, 1] 2 ).
By Corollary 2.3, the answer is negative when g 1 = y, g 2 = x 1 3 . However, the answer is positive when Furthermore, Corollary 2.3 suggests that there is a DIC ex post allocation. More generally, we solve Problem 2.1 in a nonatomic probability space is the Borel σ −algebra on [0, 1] 2 (Theorem 2.1). When types are uniformly distributed, we solve Problem 2.2 with incentive compatibility constraints (Theorem 2.2 and Corollary 2.3). We then extend Corollary 2.3 to some special cases when the item is not necessarily sold (Corollaries 2.4 and 2.5).
Theorem 2.2 cannot be obtained from Theorem 2.1 combined with the incentive compatibility results found in Manelli and Vincent (2010), since the equivalence results do not apply to cases with type-contingent bounds. Nevertheless, when g 1 = g 2 = 1, our results become special cases of the existing results. For example, when g 1 = g 2 = 1, Theorem 2.1 follows from Che et al. (2013, Corollary 6). By combining Corollary 6 in Che et al. (2013) with Theorem 2 in Manelli and Vincent (2010) and setting n = 2, we obtain Theorem 2.2 with g 1 = g 2 = 1.
To prove the main results of Theorems 2.1 and 2.2, I reduce the problems to ones that are solvable by Lemmas 3.1 and 3.2, respectively. Although Lemma 3.2 follows from Theorem 4 and 7 in Gutmann et al. (1991), Gutmann et al.'s Theorem 4 relies on a result found in Kellerer (1961), but without a proof for it. For completeness, I include simplified proofs of the relevant results from Kellerer (1961) and Gutmann et al. (1991) in the present paper. 4 The remainder of this paper is structured as follows. Section 2 provides the main implementation results and their corollaries. Section 3 proves Theorems 2.1 and 2.2 based on Lemmas 3.1 and 3.2, respectively, and also proves Corollary 2.3. Section 4 proves Lemma 3.1. Finally, Sect. 5 proves Lemma 3.2.

Main results
To capture the most general fairness considerations, one might contemplate ex post allocations that are constrained by arbitrary bounded functions of both agents' valuations.

Primitives
In what follows, we consider a selling mechanism whereby there is one indivisible item to allocate to two agents. Each agent i = 1, 2 has a private value on [0, 1] distributed according to a nonatomic probability measure, μ i . The joint type space [0, 1] 2 is endowed with the product measure μ 1 × μ 2 .

Maximal auctions
In the main theorems, we consider the following case: Suppose the seller maximizes the following utilitarian welfare function where the two agents are given equal weights: The seller does not care about budget balance, and thus, she can always arrange an individual rational money transfer. In this case, q 1 + q 2 = 1 holds for all direct Footnote 4 continued while here, Lemma 3.2 is stated for any real numbers. Integers imply a corresponding result with rational components; perhaps one can also get arbitrary real numbers, but this does not seem to follow immediately (especially because-since there are many equalities in the assumption of the theorem-one needs to carefully approximate real numbers using rational ones to keep the constraints in the same direction). mechanisms that are incentive compatible and individual rational. 5 Following Hart and Reny (2015), an auction where q 1 + q 2 = 1 is called a maximal auction. We consider nonmaximal auctions (0 ≤ q 1 + q 2 ≤ 1) in Corollaries 2.4 and 2.5.

Ex post constraints
We require that

Main results
Problem 2.1 One is given two arbitrary, measurable functions Q 1 , Q 2 : [0, 1] → [0, 1]. The goal is to obtain the necessary and sufficient assumptions for Q 1 and Q 2 that would guarantee the existence of the measurable functions q 1 , q 2 defined on [0, 1] × [0, 1] with the following properties:

4)
and Notice that (2.4) and (2.5) imply the following constraint as well: The above constraint can be obtained simply by applying (2.5) to the set

Incentive compatibility with uniform types
A mechanism is incentive compatible if truth-telling is an equilibrium strategy. Results found in Myerson (1981) suggests the following definition. 6

Definition a mechanism is
(1) BIC if and only if the interim allocation is nondecreasing; while it is (2) DIC if and only if a bidder's ex post allocation is nondecreasing in her type.
Manelli and Vincent (2010) discussed the equivalence of BIC and DIC implementations without considering any ex post allocation constraints. We cannot directly apply such a result to Theorem 2.1 because we do not know if the ex post allocations still satisfy the ex post fairness constraints. I have a characterization for incentive compatible implementations when the types are uniformly distributed, i.e., when μ 1 = μ 2 = λ, the Lebesgue measure. However, I have no corresponding result for nonuniform types.

Problem 2.2 (Incentive Compatible Implementation) One is given two arbitrary
The goal is to obtain the necessary and sufficient assumptions for Q 1 and Q 2 that would guarantee the existence of the measurable functions q 1 , q 2 defined on [0, 1] × [0, 1] which satisfies conditions (i), (ii), (iii) in Problem 2.1, and (iv) The maps x → q 1 (x, t), and y → q 2 (s, y) are nondecreasing for a.e. s, t ∈ [0, 1].
The following theorem completely solves Problem 2.2.

Theorem 2.2 The pair (q 1 , q 2 ) that satisfies Problem 2.2 exists if and only if
To further simplify the condition, we consider the case when one's allocation upper bound would be a nonincreasing function of the other's type alone, and one's allocation lower bound would be a nondecreasing function of one's own type alone.

Corollary 2.3
The pair (q 1 , q 2 ) that satisfies Problem 2.3 exists if and only if (2.9) Condition 2.9 can be checked quickly. The corresponding algorithm used to test condition (2.9) for a discretized domain has polynomial complexity O(N 2 n). 7

Nonmaximal auctions
We provide some sufficient conditions for the generalization of Problem 2.3 below.

Problem 2.4 Problem 2.3 with (ii) replaced by
Notice that (ii ) holds if and only if 0 ≤ 1 0 Q 1 + Q 2 ≤ 1. If it happens that Q 1 +Q 2 ≤ 1, then the problem has a trivial solution: q 1 (x, y) = Q 1 (x) and q 2 (x, y) = Q 2 (y). (Of course, the integral inequality 1 0 (Q 1 + Q 2 ) ≤ 1 does not necessarily mean that the integrand is, at most, 1.) This demonstrates that if Q 1 , Q 2 are small enough, then the problem has a trivial positive solution; on the other hand, condition (2.9) requires Q 1 , Q 2 to be large enough. Thus, maximal auctions and nonmaximal auctions can correspond to two completely opposite scenarios. However, we can extend Corollary 2.3 to some special cases of nonmaximal auctions. (1)

Corollary 2.4 Problem 2.4 has a positive solution if
holds for all s, t ∈ [0, 1].

Proof of Theorem 2.1, Theorem 2.2, and Corollary 2.3
In this section, we reduce Theorem 2.1 to a problem solvable by Lemma 3.1, and we reduce Theorem 2.2 to a problem solvable by Lemma 3.2. We show Corollary 2.3 by further simplifying Theorem 2.2. The proof of Lemma 3.1 is found in Sect. 4, and the proof of Lemma 3.2 is found in Sect. 5.
First, we show that 1 0 Q 1 + Q 2 = 1 is necessary and sufficient for q 1 + q 2 = 1 for all (x, y) ∈ [0, 1] 2 . By integrating the inequality 0 ≤ q 1 + q 2 ≤ 1 over the domain [0, 1] 2 , one gets Since all functions are nonnegative, it follows that we must have (3.1) The same argument carries over to arbitrary nonatomic probability spaces

Lemma 3.1 The necessary and sufficient condition for the existence of such an f in (3.4) is
(3.5) Applying Lemma 3.1 to conditions 3.2 and 3.3, we obtain: (3.7) Simplifying the above, we obtain (2.5).

Proof of Theorem 2.2
The proof for Theorem 2.2 is similar to the proof for Theorem 2.1 but with the incentive compatibility requirements.

Proof of Lemma 3.1
In this section, I prove Lemma 3.1. First, I show the necessity of (3.5) in step I. Second, I demonstrate the sufficiency of (3.5) for the discrete case in step II. Finally, I pass the discrete result to the continuous functions in (3.5) in step III.

Step I. Proof of necessity
It is indeed easy to see that (3.5) is necessary for the existence of such an f in (3.4).

Step II. Proof of sufficiency for the discrete case
To understand why (3.5) is sufficient, I consider a discrete "version" of the problem here in step II. Later, in step III, I pass the solution for the discrete problem to a limit in order to obtain (3.5).
For step II, we obtain the sufficiency condition for the existence of a matrix with given row sums and column sums (marginals), which is dominated by another given matrix.
In what follows, I denote [n] = {1, ..., n} for any positive integer n ≥ 1.  In fact, one can easily see that Lemma 4.1 (3) is also a necessary condition for the existence of the matrix {s i j } n,m i, j=1 . The proof proceeds precisely in the same way as the proof in the continuous case in step I. However, this proof is not needed for the purposes of this step.
Next, I briefly summarize the induction argument. I start by reducing the values of each t i, j slightly, as long as the 2 n+m linear inequalities in hypothesis (3) are not violated. Here, the following two scenarios can occur. Case Without loss of generality, one can assume that Y 1 = {1, ..., n 1 }, Y 2 = {m 1 , ..., m} for some integers n 1 , m 1 , with 1 ≤ n 1 ≤ n and 1 ≤ m 1 ≤ m (see Fig. 1 for an illustration). Note that if the required s i, j exists, then i∈Y 1 In order to find s i j on the rectangle Y 1 × Y c 2 (and Y c 1 × Y 2 ), ideally, we would apply the induction assumption to the smaller rectangles; however, the obstacle is that (2) and (3) might not hold on Y 1 × Y c 2 (notice that the complement of a subset A ⊂ Y 1 in Y 1 differs from its complement in [n]). To solve this issue, we must modify the numbers p i , q j , t i j in a certain way so that the new numbers p i , q j , t i j will satisfy the induction assumption for these smaller rectangles. After this modification, one can then "glue" all the solutions for the four different rectangles into one solution on

Proof of Lemma 4.4 (i) t i, j ≥ 0 on [n]×[m] follows from the fact that t i j is nonnegative on [n] × [m]
. Therefore, (1) holds for t i j .
(ii) The following equalities show that (2) holds for and by separating the summations on the disjoint sets and rearranging them, one obtains the following: Equations (4.5) and (4.6) are the same as Lemma 4.1 (3) when the latter is restricted to the domains Y 1 ×Y c 2 and Y c 1 ×Y 2 , respectively, for the new variables Inductive step Let m + n > 2. Here, I show that if the required {s i j } m ,n i, j=1 exists for m + n < m + n, then the required {s i j } m,n i, j=1 also exists. Since |Y 1 | + |Y c 2 | = m 1 + n 1 < m + n and |Y c 1 | + |Y 2 | < m + n − m 1 − n 1 , then we can apply the induction hypothesis on the domain Combining the above with the result that

Step III. Proving (3.5) using the discrete case
We denote the intervals R i 1 = i−1 n , i n , R j 2 = j−1 n , j n and the rectangles R i j = R i 1 × R j 2 for i, j ∈ [n]. We take the functions f 1 , f 2 , g from (3.5) and define the following numbers: (4.7) Conditions (1), (2), and (3) in Lemma 4.1 hold for the numbers f i 1 , f j 2 , and g i j . Lemma 4.1 (1) holds because g i j ≥ 0. Lemma 4.1 (2) holds because To verify Lemma 4.1 (3), for any A, By (3.5), (3) holds. Therefore, by Lemma 4.1, there exists a matrix Next, I pass F i j to the weak * limit to find the sufficient condition for (3.5). Letting μ = μ 1 × μ 2 , I define the following functions: The convergence results in (4.8b), (4.8c), and (4.8d) are found by the Lebesgue differentiation theorem (see, e.g., Benedetto and Czaja 2010, Theorem 8.4.6). Next, I show that the weak * limit of F n exists and that it satisfies (3.4).
(1) We know that on [0, 1], 0 ≤ F n k ≤ G n k a.e., and that G n k → g a.e., G n k is bounded. Taking any measurable set U ⊆ [0, 1] 2 , |U | > 0. The following must then hold: . For the sake of contradiction, assume F > g + on U for some > 0. Therefore, It would then follow that U Fdμ ≤ U gdμ. This contradicts F > g + on U. ( By Fubini's theorem, since dμ = dμ 1 × dμ 2 , we can write the above as Therefore, the left-hand-side of (4.9) satisfies the following: since the limit can be taken inside of the integration by the Lebesgue dominated convergence theorem. By linearity of integration, (4.9) implies Hence, l(y) ≡ 0 on [0, 1], and thus, >0} (y) and repeat the same process. From this work, we conclude that f 2 (y) The proof is the same as in (2), with x, y reversed, and with f 2 replaced by f 1 .

Proof of Lemma 3.2
The proof for Lemma 3.2 uses several modifications of the proof for Lemma 3.1 to accommodate the incentive compatibility.

Step I. Sufficiency of 3.11
The proof for sufficiency is the same as that in the proof of Lemma 3.1, with μ 1 = μ 2 = λ, the Lebesgue measure.

Step II. Discrete case
For step II, we obtain the sufficiency condition for the existence of a matrix with given increasing row sums and column sums (marginals), which is dominated by another given matrix. Additionally, the entries of the desired matrix must increase monotonically with the row and column indexes, respectively. That is, for the matrix A = [a i, j ] n i, j=1 , we must have a i j ≤ a ik for all 1 ≤ j ≤ k ≤ n for any i = 1, ..., n, and a i j ≤ a k j for all 1 ≤ i ≤ k ≤ n for any j = 1, ..., n.
Lemma 4.1 gives us the condition for the existence of a dominated matrix for given marginals. When the marginals are both increasing, the following lemma shows that there exists a matrix that satisfies the required monotonicity property.
To summarize, by using Lemmas 4.1 and 5.1 , we completed the proof that a matrix {s i, j } (i, j)∈[n]×[m] whose column entries and row entries both increase exists if and only if { p i , q j , t i j } satisfies the hypothesis in Lemma 4.1, as well as the additional condition that p i , q i is nondecreasing.

Step III. Proving (3.11) using the discrete case
The proof of (3.11) is almost the same as that for (3.5) with a few modifications. First, we take μ 1 = μ 2 = λ, the Lebesgue measure. When defining the numbers g i j , f i 1 , f j 2 , we take the functions f 1 , f 2 , g from (3.11) instead of from (3.5). Thus f i 1 , f j 2 are increasing since f 1 , f 2 are increasing. Conditions (1), (2), and (3) in Lemma 4.1 hold for the numbers f i 1 , f j 2 , and g i j by the same argument as that in the previous subsection. By Lemma 5.1, without loss of generality, one can assume that F i, j will increase in both i and j since f i 1 , f j 2 are increasing.
By Lemma 4.5, the weak * limit of F n exists, and it satisfies (3.10) (1) (2) (3). Furthermore, as f 1 = 1 0 F increases in x and f 2 = 1 0 F increases in y, it is clear that F must increase in both x, y independently. 8 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.