Approximately Efficient Multidimensional Bilateral Trade

1 Introduction

• The paper combines multidimensional Bayesian mechanism design, bilateral trade, and XOS combinatorial valuations, motivated by agreements involving many dimensions such as pay, hours, benefits, tariffs, and immigration.XOS valuations lie between submodular and subadditive valuations in generality.Source: §1 · pp. 3–4

1.2 Related Work

• In single-item bilateral trade, [DMSW22] proved a constant-factor first-best GFT approximation under arbitrary independent distributions, improving earlier approximation results for second-best GFT.Their mechanism was later refined to 1/3.15 by [Fei22].Source: §1 · p. 3, §1.2 · p. 5

1.1 Our Results

• A simple equal-probability randomization between seller-optimal and buyer-optimal mechanisms achieves at least 1/44 of first-best GFT while satisfying BIC, IIR, and ex-ante WBB.The result assumes one XOS buyer, one additive seller, and independent item values and costs.Source: §1.1 · p. 4
• The natural VCG-based buyers’ lawyer fails with a multidimensional seller because buyers impose positive externalities on one another, preventing budget balance.The paper instead restricts the lawyer to item pricing with probability 16/43.Source: §1.1 · p. 4
• With n XOS buyers and one additive seller, a simple BIC, IIR, and ex-ante WBB mechanism achieves an Ω(1) fraction of first-best GFT over independent items.The mechanism uses item pricing with probability 16/43 and otherwise permits an optimal seller-facing mechanism.Source: §1.1 · p. 4

1.2 Related Work

• Prior multidimensional GFT work [CGMZ21] studied one constrained-additive buyer and n single-dimensional sellers, achieving a log(n)^2 approximation to second-best GFT.The second-best benchmark is the maximum GFT achievable by any BIC, IIR, and WBB mechanism.Source: §1.2 · p. 5

2 Preliminaries

• The model has n buyers, m heterogeneous items, additive seller costs, XOS buyer valuations, and mutually independent buyer types, item types, and seller costs.Clause values may be arbitrarily correlated within each buyer-item pair, while independence is required across buyers and items.Source: §2 · p. 6
• XOS-over-independent-items valuations satisfy no externalities, monotonicity, and a finite maximum of additive item-level clauses, including additive, unit-demand, and constrained-additive models.The valuation class is defined following.Source: §2 · p. 6
• The framework defines feasible randomized allocations, expected GFT, first-best GFT, direct-revelation mechanisms, incentive compatibility, individual rationality, and budget-balance benchmarks.The second-best GFT is the maximum expected GFT achievable by a BIC, IIR, and ex-ante WBB mechanism.Source: §2 · p. 7

2.1 Unit-Demand Reduction to Single-Dimensional Markets

• For unit-demand buyers under downward-closed feasibility, the copies instance creates one single-dimensional pseudo-buyer per buyer-item pair, with matchings enforcing both feasibility and unit-demand constraints.Each pseudo-buyer (i, j) wants only item j, has value vij, and faces seller cost cj.Source: §2.1 · pp. 8–9
• The first-best unit-demand GFT equals the first-best GFT of the copies instance, and [BRTW26] yields 2Π*_B(Icopies) ≥ 1/3.15 GFT*_{F,ud}.The equivalence follows because each buyer can retain only the item maximizing vij − cj without reducing GFT.Source: §2.1 · p. 9

• A constant-factor first-best GFT approximation is obtained for multi-dimensional XOS buyers and an additive seller through decompositions and mechanism-design reductions.The proof combines a pointwise core-tail decomposition, seller-run entry fees, and reductions to single-dimensional copies instances.Source: §3 · p. 10
• Choosing τ as the unit-demand benchmark balances core utility concentration with tail control: the core satisfies bounded differences, while excess tail GFT is at most a constant factor of τ.For bilateral trade, τ is the one-trade benchmark GFT∗,ud F 1; for n buyers, it is the unit-demand matching benchmark GFT∗,ud F m.Source: §3 · p. 10
• Seller-profit entry fees approximate the core: a cost-price two-part tariff works for one buyer, while ASPE uses balanced prices and residual-core entry fees for multiple buyers.In the single-buyer case, concentration permits an entry fee equal to a constant fraction of expected core GFT; with multiple buyers, fees are calibrated sequentially to residual core GFT.Source: §3 · p. 11
• The cited results state that B(Icopies) suffice to cover the tail up to a constant factor, while 2Π∗ B(I) ≥ 1 3.15 GFT∗(I).Source: §3 · pp. 11–12
• Unrestricted buyer-profit maximization with VCG can violate ex-ante WBB because buyer profits are superadditive, so the construction restricts allocations to isolate utilities and ensure nonnegative VCG payments.The restricted mechanism randomly designates each item to at most one buyer, posts prices, and uses a half-capacity relaxation to preserve a constant fraction of optimal single-dimensional buyer profit.Source: §3 · pp. 12–13
• Combining core and tail profits yields the final constant-factor first-best GFT guarantee, using delegation for single-agent sides and an explicit DSIC, WBB, IIR tail auction for the buyers’ side.The advice auctions lower-bound optimal BIC auctions in single-agent settings, while the constructed buyer-side auction directly supplies the multi-buyer tail mechanism.Source: §3 · p. 13

4 The Multi-Dimensional Bilateral Trade

• A 1/44-fraction mechanism randomizes equally between seller-optimal and buyer-optimal mechanisms and is BIC, IIR, and ex-ante WBB.The setting has one XOS buyer and one additive seller with independent item values and costs.Source: §4 · p. 13
• Simple mechanisms randomizing between seller-side tariffs and buyer-side posted-price procurement menus achieve at least a 1/88 fraction of first-best GFT.The protocol delegates pricing power to the seller or buyer with equal probability and uses explicit candidate mechanisms rather than computationally complex optimal mechanisms.Source: §4 · pp. 13–14

4.1 The Core–Tail Decomposition

• The first-best GFT is bounded by the sum of core and tail GFT, using a threshold based on the optimal one-trade benchmark.Under the one-trade family, the XOS buyer reduces to a unit-demand buyer with independent item values.Source: §4.1 · pp. 15–16

4.2 Approximating the Core via an Entry Fee

• A cost-price two-part tariff charges an entry fee and prices each item at its realized cost, so seller profit equals the entry fee whenever the buyer enters.When expected core surplus µ is sufficiently large relative to threshold τ, setting the fee to µ/4 yields a constant expected-profit fraction of µ.Source: §4.2 · pp. 16–18

4.3 Approximating the Tail via the Unit-Demand Reduction

• The tail is upper bounded by the one-trade unit-demand benchmark, with E[GFTT(w)] ≤ 1/(1 − ln 2) GFT∗,ud_F1.The proof uses independence, a bound on the probability of simultaneous large surpluses, and a maximum-versus-sum inequality.Source: §4.3 · pp. 18–19, F 1 2GFT∗,ud F 1 · p. 19

F 1 2GFT∗,ud F 1

• Prophet inequalities produce order-oblivious posted-price mechanisms that approximate the seller’s one-trade profit benchmark and transfer to simultaneous menus for the original multidimensional buyer.The reduction uses a common virtual threshold and preserves the approximation guarantee through the equivalence result in [CHMS10].Source: F 1 2GFT∗,ud F 1 · p. 20

In the regular case one may take

• The seller-side construction is DSIC-B and ex-post IR, while the procurement analogue is DSIC-S and ex-post IR, with each side restricted to trading at most one item.For ironed distributions, boundary-price randomization matches the threshold rule’s acceptance probability.Source: In the regular case one may take · pp. 20–21

4.4 Putting Everything Together

• 1/44 of first-best expected GFT is achieved by randomizing equally between the seller-optimal and buyer-optimal mechanisms while satisfying BIC, IIR, and ex-ante WBB.The guarantee applies to an XOS buyer and additive seller over independent items.Source: §4.4 · pp. 22–23
• 1/88 of first-best expected GFT is achieved by a simple randomized protocol using the cost-price two-part tariff and simultaneous one-item posted-price menus.The protocol delegates pricing power to the seller or buyer with equal probability, avoiding computationally intractable optimal multidimensional mechanisms.Source: §4.4 · pp. 23–24

5 Multiple XOS Buyers

• Mfinal achieves a constant-factor approximation to first-best GFT while satisfying BIC, interim IR, and ex-ante WBB.It randomly delegates pricing power to the seller or uses the buyer-side mechanism Mcopy_B.Source: §5 · p. 24
• The main technical challenges are designing a weakly budget-balanced truthful buyer-side tail mechanism and replacing static core entry fees with ASPE fees depending on available items.VCG payments may violate weak budget balance, while sequential arrivals make continuation utilities depend on the remaining set; ASPE addresses this using anonymous prices and set-dependent entry fees δ_i(S).Source: §5 · p. 25

5.2 Proof Overview

• The proof overview approximates the core with an ASPE mechanism, the tail with copy-instance profit benchmarks and a new buyer-side mechanism, then aggregates both bounds.The resulting components follow the bilateral-trade blueprint while adapting implementation to multiple buyers.Source: §5 · p. 25, §5.2 · p. 27

5.1 The Core–Tail Decomposition

• The core–tail decomposition uses the unit-demand matching benchmark GFT*_{F_m} and threshold τ = 2 GFT*_{F_m}, yielding first-best GFT bounded by core plus tail.Under the matching constraint each buyer receives at most one item, so XOS valuations reduce to independent singleton values; the core and tail are then bounded separately.Source: §5.1 · p. 26

6 Approximating the Core via an ASPE Mechanism

• The mechanism adapts [CZ17]’s ASPE framework by representing the fixed-cost core objective as one-sided XOS welfare with capped net gains from trade.This representation enables anonymous sequential posted prices and entry fees in the original two-sided market.Source: §6 · p. 28
• Buyers are visited sequentially and face cost-plus-markup item prices, while inventory-dependent entry fees account for the remaining surplus after earlier buyers remove items.The markups are derived from supporting prices and generate seller profit when items sell; the entry fees extract residual surplus when inventory remains.Source: §6 · pp. 28–30
• The entry fee for buyer i and remaining set S is calibrated to the median auxiliary residual surplus, and the buyer accepts it with probability at least one half.Final truncation limits sensitivity to any single item, while pointwise attainability ensures the auxiliary surplus is feasible under the true cost-plus-markup menu.Source: §6 · p. 30
• ASPE(c) is buyer-DSIC and buyer ex-post IR, while its expected seller profit captures a constant fraction of µ(c) minus loss bounded by the fixed-cost unit-demand matching benchmark.The guarantee applies for every fixed seller cost vector c and therefore lower-bounds the seller’s optimal conditional profit.Source: §6 · p. 31
• Supporting prices retain half the welfare after the first truncation, and truncation and ASPE-analysis losses are charged to the fixed-cost unit-demand matching benchmark.This accounting combines item markups, entry-fee extraction, and the benchmark bound to prove the fixed-cost core approximation.Source: §6 · pp. 28, 31

7 Approximating the Tail via the Unit-Demand benchmark

• E[GFT_T(t, c)] ≤ 1/(1 − ln 2) GFT*_{F_m}^{ud}, reducing the tail component to the unit-demand matching benchmark.The reduction uses sparse, independent itemwise tail events to compare the expected sum of excesses with the expected largest excess, which is feasible under unit-demand constraints.Source: §7 · pp. 31–32

7.1 Approximating the Buyer-Side Copy Benchmark

• Randomized procurement menus approximate the buyer-side copy benchmark while remaining DSIC and ex-post IR for the seller, BIC and interim IR for buyers, and ex-post WBB.Menus designate each item to at most one buyer, post randomized prices, and use VCG selection; buyer separation ensures nonnegative VCG charges and no additional approximation loss.Source: §7.1 · pp. 33–34

7.1.1 The half-capacity relaxation

• The half-capacity relaxation preserves at least one half of the buyer-side copy benchmark by reducing each buyer’s total active probability from 1 to 1/2.This capacity reduction enables constant-probability implementation of an active item without competing active items for the same buyer.Source: §7.1 · p. 34, §7.1.1 · p. 35

7.1.2 Implementing the relaxation with procurement menus

• Procurement menus implement the relaxation by independently designating items to buyers and randomizing posted prices according to seller procurement-payment curves.The menu family is compact, utilities are continuous, and every fixed menu is DSIC and ex-post IR for the seller with ex-post strongly budget-balanced trade payments.Source: §7.1.2 · pp. 36–37
• The two-price procurement-menu construction exactly implements target allocation probabilities and expected procurement payments, with the only loss arising from simultaneous profitability; half-capacity limits this loss by ensuring at least 1/2 probability of unique profitability.When an item is uniquely profitable within its buyer’s cluster, Auction 1 necessarily allocates it.Source: §7.1.2 · p. 38

7.1.3 VCG over procurement menus

• VCG selection over procurement menus makes the buyer mechanism BIC and interim IR while preserving DSIC and ex-post IR for the seller and ex-post WBB.Buyer-exclusion invariance ensures that removing one buyer leaves other buyers’ trade opportunities unchanged; the VCG construction is based on this property.Source: §7.1.3 · pp. 40–42

7.2 Approximating the Seller-Side Copy Benchmark

• 6.75-approximation: Msvq_S is DSIC-B, strongly budget balanced, and achieves this guarantee for the seller’s optimal expected profit in the single-dimensional copies instance.The mechanism uses sequential posted prices based on allocation probabilities scaled by three; for general irregular distributions, it uses randomized posted prices from.Source: §7.2 · p. 43

8 Final Assembly

• Mfinal randomizes between seller pricing power with probability p and the buyer-side mechanism Mcopy_B with probability 1 − p.The protocol uses p = 27/43 to balance the two sides’ tail contributions, with ηS = 1/6.75 and ηB = 1/4.Source: §8 · pp. 44–45
• The mechanism inherits BIC, interim IR, and ex-ante WBB because its component mechanisms satisfy the relevant incentive, rationality, and budget properties and the randomization is report-independent.Mcopy_B is DSIC, buyer interim IR, seller ex-post IR, and ex-post WBB; the seller mechanism is BIC, interim IR, and ex-ante WBB.Source: §8 · p. 45
• Theorem 5.1 shows that Mfinal is BIC, interim IR, and ex-ante WBB, while achieving at least an α fraction of first-best GFT.The result applies under Assumption 7.3 to n XOS buyers and one additive seller over independent items.Source: §8 · pp. 45–46
• The proof splits into core-dominated and tail-dominated cases, using entry-fee profit in the former and seller- and buyer-side tail benchmarks in the latter.In both cases, benchmark comparisons lower-bound Mfinal’s expected profit and hence its expected GFT.Source: §8 · pp. 45–46

A Virtual Value Function

• Buyer virtual value is φi(bi) = bi − (1−Di(bi))/di(bi), with ironed virtual value ˜φi(bi) used when monotonicity requires ironing.The appendix defines virtual values from the buyer’s value distribution and uses the monotonized version when the original virtual value is not monotone.Source: A Virtual Value Function · pp. 50–51
• Seller virtual cost ψj(sj) is defined analogously from the seller’s cost distribution, with ironed virtual costs ˜ψj(sj) used when ψj is nonmonotone.These ironed quantities support the virtual-surplus formulation for seller-side profit.Source: A Virtual Value Function · pp. 50–51
• Myerson’s profit equivalence states that expected profit in any BIC and IR auction equals the corresponding expected virtual surplus.Thus, maximizing one side’s expected profit is equivalent to maximizing its corresponding virtual surplus.Source: A Virtual Value Function · p. 51

• The counterexample has two unit-demand buyers, an additive seller with two items, independent costs c1,c2 ~ U[0,1], and each buyer values only one item at 1.Source: B Missing Proof from Section 3 · p. 51
• Each buyer alone yields optimal expected seller profit Π∗({1}) = Π∗({2}) = 0.25 through a posted price of 0.5.Source: B Missing Proof from Section 3 · p. 51
• Representing both buyers with a bundle price P ∈ [1,2] gives a joint-profit lower bound because the seller accepts when c1 + c2 ≤ P.Source: B Missing Proof from Section 3 · p. 51
• The VCG transfer identities imply an ex-ante net budget of Π∗({1}) + Π∗({2}) − Π∗({1,2}); applying the bounds makes it strictly negative.Source: B Missing Proof from Section 3 · pp. 51–52

Proof of Lemma 4.9

• The proof of Lemma 4.9 derives a pointwise inequality by separating the maximum variable from the remaining sum with an indicator function.The argument checks zero, singleton-positive, and multiple-positive realizations.Source: Proof of Lemma 4.9 · p. 52
• The pointwise inequality is verified across all realizations, including cases with no positive variables, exactly one positive variable, or at least two positive variables.In the multiple-positive case, the right-hand side contains the full sum plus the maximum variable.Source: Proof of Lemma 4.9 · p. 52
• Taking expectations, independence factors each variable’s product with the event that another variable is positive, while a union bound controls that event’s probability.The resulting expression is rearranged into the final bound.Source: Proof of Lemma 4.9 · pp. 52–53

Proof of Proposition 4.13

• For every realized singleton-value vector, Proposition 4.13 constructs a common threshold and possibly randomized procurement-price vector for an order-oblivious posted-price mechanism.The mechanism guarantees a conditional expected buyer-profit bound stated in the proof.Source: Proof of Proposition 4.13 · p. 53

In the regular case one may take

• For independent seller costs, Samuel-Cahn’s prophet inequality supplies a threshold for stopping on the first positive virtual-surplus variable exceeding it.The variables are defined from singleton values and ironed virtual costs.Source: In the regular case one may take · p. 53
• The threshold is translated into procurement prices that implement the same acceptance rule for regular distributions and use boundary randomization for ironed distributions.Ironed distributions randomize on the threshold plateau to match the prophet-threshold acceptance probability.Source: In the regular case one may take · pp. 53–54
• Myerson’s procurement identity equates expected buyer profit with expected ironed virtual-cost surplus for the resulting posted-price mechanism.This transfers the prophet-inequality guarantee to buyer profit.Source: In the regular case one may take · p. 54
• The Proposition 4.13 reduction yields a 2-approximate order-oblivious mechanism and, in the warm-up instance, a DSIC and IR simultaneous procurement menu.The seller rejects all offers or selects the item maximizing qj(v) − cj.Source: In the regular case one may take · p. 54

D Missing Proofs from Section 6

• The induced valuations form a standard one-sided XOS instance over independent item coordinates, enabling application of the ASPE analysis in Section 7.2 of [CZ17].The benchmark allocation is identified with σC, while the corresponding cutoffs, truncated valuation, and residual-surplus function are χi, ηi, bvC_i, and eui.Source: D Missing Proofs from Section 6 · p. 54
• The reduction transfers the parts of [CZ17] that depend only on one-sided structure, including truncation, supporting-price, concentration, and entry-fee analyses.Lemmas 14–18 of [CZ17] control the first-truncation loss, while Lemmas 19–28 provide the supporting-price, concentration, and entry-fee analysis.Source: D Missing Proofs from Section 6 · p. 54
• The appendix’s additional tasks are to identify fixed-cost core GFT with induced-instance welfare, compare posted-price losses with the fixed-cost unit-demand matching benchmark, and implement entry fees in the original market.The original-market implementation uses prices cj + Qj.Source: D Missing Proofs from Section 6 · p. 54

D.1 Reduction to a One-Sided XOS Instance

• For every realization t, maximizing capped core contributions over feasible allocations yields the fixed-cost welfare identity.For each feasible allocation, buyer i’s capped core contribution is vC_i(ti, Si); maximizing gives the first identity, and σC supplies the welfare-maximizing allocation rule.Source: §D.1 · pp. 54–55
• The proof establishes the reduction by equating each feasible allocation’s capped core contribution with its induced one-sided valuation before maximizing.The resulting identities connect feasible bilateral allocations to the welfare benchmark defined by σC.Source: §D.1 · p. 55
• The baseline thresholds satisfy the two properties required by the [CZ17] analysis under the threshold-splitting convention.The first property follows from the definition of βij; summing across buyers and using ∑k qkj ≤ 1 establishes the aggregate bound, and dropping the ith term gives the second claim.Source: D Missing Proofs from Section 6 · p. 54, §D.1 · p. 55

D.2 Charging the First-Truncation Loss

• The favorite contribution is bounded by the optimal unit-demand matching GFT, while total contribution is at most favorite plus nonfavorite contribution.This follows pointwise from the favorite-item decomposition and then by taking expectations.Source: §D.2 · pp. 55–56
• The auxiliary benchmark PostRev is the supremum revenue of rationed sequential posted-price mechanisms for the induced unit-demand instance.Buyers are visited sequentially, prices may depend on buyer–item pairs, and each buyer can purchase at most one available item.Source: §D.2 · p. 56
• The Cai–Zhao first-truncation bound decomposes nonfavorite contribution into retained contribution and a high-singleton overlap term, bounding the latter by 8/3 times the RSPM benchmark.The reduction applies [CZ17] with b = 1/4, identifying its allocation, singleton values, cutoff, and truncated valuation with the constructions here.Source: §D.2 · p. 56
• Comparing the auxiliary revenue benchmark with GFT relies specifically on the paper’s fixed-cost reduction.This comparison is not asserted as a general property of the auxiliary benchmark.Source: §D.2 · p. 56
• The auxiliary posted-price revenue is at most the optimal unit-demand matching GFT, pointwise and therefore in expectation after taking the supremum.Every mechanism’s sold buyer–item pairs form a matching, whose revenue is bounded by the singleton GFT of that matching.Source: §D.2 · pp. 56–57

Supporting Prices and the Fixed-Cost ASPE

• The residual-surplus proxy is feasible for every buyer, type, and available set, providing the main bridge from the auxiliary one-sided instance to the original two-sided market.The proof uses an XOS clause and the fact that the proxy is obtained by deleting item coefficients.Source: §D.3 · p. 57
• 20/3: ASPE(c) satisfies the fixed-cost specialization of the Cai–Zhao seller-profit bound [CZ17].The specialization sets b = 1/4 and identifies the induced valuation, benchmark allocation, cutoff, truncated valuation, and residual-surplus function with the section’s corresponding objects.Source: §D.3 · p. 58
• The transfer applies because valuations are XOS over independent items, baseline-threshold properties hold, and supporting prices are exact with α = 1.These conditions verify the hypotheses of the supporting-price and entry-fee analysis in Lemmas 19–28 of [CZ17].Source: §D.3 · p. 58
• The market accounting carries over unchanged: entry is accepted with probability at least 1/2, availability is independent of the visited buyer’s type, and each sale yields seller margin Q_j.The pointwise utility dominance from Lemma D.7 supports the entry-fee argument, while exact supporting prices identify the seller’s margin.Source: §D.3 · pp. 57–58

D.4 Completing the Fixed-Cost Bound

• ASPE(c) is buyer-DSIC and buyer ex-post IR because fixed entry fees and item prices make truthful utility maximization dominant, while rejecting yields zero utility.Source: §D.4 · p. 58
• The fixed-cost profit proof combines Lemmas D.3–D.5 and then substitutes the resulting inequality into Lemma D.8, using Lemma D.5 again.Source: §D.4 · p. 58

E.1 Seller-Side Ironing Facts

• The seller revenue function’s lower convex envelope bR is continuous, convex, absolutely continuous, satisfies bR(0)=0, and has a nondecreasing measurable right derivative.Source: §E.1 · p. 59
• Every ironed point can be implemented with two quantiles a and b and mixing weight λ, including the trivial construction at q=0.Source: §E.1 · pp. 59–60
• For any nonincreasing allocation rule x with expected allocation q, the one-dimensional ironing inequality follows by shifting allocation mass toward the smallest quantiles against the nondecreasing derivative g.Source: §E.1 · p. 60

E.2 A Monotone Optimal-Matching Selector

• The selector chooses a maximum-weight matching by minimizing cardinality, then maximizing the specified secondary criterion, and finally applying fixed lexicographic tie-breaking.Source: §E.2 · p. 61
• This selector uses only strictly positive-weight edges, and each item’s selection indicator is nonincreasing in its cost, enabling the half-capacity relaxation argument.Source: §E.2 · p. 61

E.3 Proof of Lemma 7.7

• The procurement-menu family P is compact because its parameters lie in a bounded finite-dimensional box and its defining constraints are closed.This follows by viewing P as a closed subset of a compact box.Source: §E.3 · p. 62
• Buyer utility is continuous in menu parameters, using finitely many designation and price branches, almost-surely strict comparisons, bounded payoffs, and dominated convergence.Atomless independent costs make equality cases null, while Assumption 7.3 ensures continuous posted prices and uniformly bounded values.Source: §E.3 · pp. 62–63
• For each fixed procurement menu, buyer utility is continuous in type because induced trade probabilities and expected payments are type-independent, while each item value is continuous.The resulting utility map ti 7→Ui(ti, ρ) is therefore continuous.Source: §E.3 · p. 63
• The equivalent direct implementation is DSIC and ex-post individually rational for the seller because truthful reporting maximizes total utility across disjoint clusters and the empty selection is feasible.The seller selects in each cluster the item maximizing reported profit when nonnegative, with ties resolved by a fixed rule.Source: §E.3 · p. 63
• Trade payments are ex-post strongly budget balanced because each trading item’s designated buyer pays exactly the amount received by the seller.This equality holds for every realization of the menu, reports, and costs.Source: §E.3 · p. 63