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Fair Allocation in Dynamic Mechanism Design

Alireza Fallah, Michael I. Jordan, Annie Ulichney

arXiv:2406.00147v3cs.GTcs.LGecon.TH

TL;DR

The paper asks how to maximize auction revenue while guaranteeing each group a minimum allocation over time. It characterizes fair mechanisms for static and dynamic settings, including allocation subsidies and dynamic payment transfers. It also gives approximation schemes and notes that ex post fairness for multiple groups may be infeasible.

  • Problem

    The paper asks how to maximize discounted seller revenue while ensuring each group receives a minimum discounted average allocation.

    Source: §1 · p. 2

  • Method

    It characterizes static and dynamic fair auctions, using recursive functions for dynamic allocations and payments.

    Source: §1 · p. 3, §4 · p. 12

  • Results

    The dynamic mechanism uses allocation subsidies, a participation bonus for winning groups, and an entry fee for participants, with subsidy levels depending on future utilities.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Approximation schemes provide efficient ways to find allocations that preserve seller utility while meeting relaxed fairness guarantees.

    Source: §7 · p. 22

    Under ex post fairness, imposing the constraint is infeasible unless the requirement applies to only one group.

    Source: §4 · p. 16

Abstract

from arXiv · show

We consider a dynamic mechanism design problem where an auctioneer sells an indivisible good to groups of buyers in every round, for a total of $T$ rounds. The auctioneer aims to maximize their discounted overall revenue while adhering to a fairness constraint that guarantees a minimum average allocation for each group. We begin by studying the static case ($T=1$) and establish that the optimal mechanism involves two types of subsidization: one that increases the overall probability of allocation to all buyers, and another that favors the groups which otherwise have a lower probability of winning the item. We then extend our results to the dynamic case by characterizing a set of recursive functions that determine the optimal allocation and payments in each round. Notably, our results establish that in the dynamic case, the seller, on the one hand, commits to a participation bonus to incentivize truth-telling, and on the other hand, charges an entry fee for every round. Moreover, the optimal allocation once more involves subsidization, which its extent depends on the difference in future utilities for both the seller and buyers when allocating the item to one group versus the others. Finally, we present an approximation scheme to solve the recursive equations and determine an approximately optimal and fair allocation efficiently.

1 Introduction

The paper studies revenue-maximizing auctions that guarantee each buyer group a minimum discounted allocation, using a proportional measure suited to settings where groups may value goods differently. It extends fair mechanism design from static auctions to dynamic settings and develops mechanisms and approximation schemes for these constraints.

1 Introduction

  • In the static case, fair allocation can both favor an under-allocated group and lower the reserve price to increase allocation overall.The reserve-price reduction may disproportionately benefit one group.Source: §1 · p. 2
  • For dynamic auctions, recursive functions characterize allocations and payments, with subsidy levels depending on buyers' and the seller's future utilities.The paper also provides an approximation scheme preserving seller utility while relaxing each group's fairness target to (1 − ε)(αi − ε), and a constant approximation scheme for δ near one.Source: §1 · p. 3
  • The dynamic mechanism pairs allocation subsidies with participation bonuses for winning groups and entry fees charged to all participants.The bonus equals the buyer's net future utility from not winning; the entry fee equals the expected bonus forgone by not participating.Source: §1 · p. 3
  • As the number of buyers grows, the entry fee's excess over the participation bonus decreases, so the fee offsets the bonus in large auctions.The participation bonus may exceed the entry fee.Source: §1 · p. 3
  • The results extend to more than two groups, and numerical experiments illustrate how fairness constraints affect seller and buyer utility.Source: §1 · p. 4

1.1 Related Work

  • Related work spans fair allocation and strategic mechanisms, while this paper combines dynamic allocation, multiple groups, and possibly asymmetric minimum allocation constraints.The closest cited setting studies a similar fairness notion in a single round for one group; this paper also introduces a second type of subsidization in the static case.Source: §1.1 · p. 4

2 Model

The model studies a seller allocating one indivisible item per round across two buyer groups, maximizing discounted expected utility subject to minimum discounted average allocation guarantees. It restricts attention to direct truthful mechanisms that satisfy individual rationality, under stated distributional assumptions.

2 Model

  • The seller allocates a single indivisible item in each of T rounds to one of two groups, each assumed to contain n buyers for simplicity.The analysis also applies to unequal group sizes.Source: §2 · p. 5
  • The model assumes buyers' values are independent within and between rounds, drawn from publicly known distributions, and regular in the main analysis.The regularity assumption can be relaxed using an ironing procedure.Source: §2 · pp. 5, 6, 8, 9
  • The mechanism specifies allocation and payment functions based on bids and public allocation history, and the seller's and buyers' utilities span the rounds.The seller discounts future utility by δ.Source: §2 · p. 6
  • The seller maximizes expected discounted utility while guaranteeing each group a minimum discounted average allocation αi.The group guarantees satisfy α1 + α2 ≤ 1, and feasibility is assessed using past allocations and expected allocations in remaining rounds.Source: §2 · pp. 6–7
  • The analysis seeks direct mechanisms that are ex post incentive compatible and individually rational, so truthful bidding is optimal and participation weakly beats skipping.Truthful reporting is evaluated against truthful reports by other buyers; the dynamic revelation argument applies under the paper's Markovian environment.Source: §2 · pp. 7–8

3 The Static Case

In the static case, the optimal fair allocation departs from the unconstrained rule only as needed to meet group targets, reallocating in ways that limit revenue loss. The fairness constraint can require both more overall allocations and a more balanced allocation between groups.

3 The Static Case

  • The fairness constraint controls both relative allocation probabilities and overall allocation rates, with the relevant adjustment depending on target levels and unconstrained allocations.Source: §3 · p. 10
  • When the item goes to a group, it is awarded to that group’s highest-value buyer.Source: §3 · p. 10
  • The group-level allocation depends on the two groups’ maximum values and uses shifted virtual-value thresholds to decide which group receives the item.The rule is specified by γ, η1, and η2; the theorem states it up to a measure-zero set.Source: §3 · pp. 10–11
  • The optimal fair mechanism reallocates the item to lower-value buyers in the cases that least affect expected seller revenue, with fairness constraints binding.The theorem’s parameters select the cost-minimizing reallocations needed to meet the target.Source: §3 · pp. 11–12
  • Fairness may require both increasing the overall probability of allocation and balancing allocation rates between groups.The overall increase is needed when unconstrained allocation probability is below the sum of the groups’ target levels.Source: §3 · p. 11
  • When γ > 0, group two’s allocation probability equals α2; when γ < 0, group one’s equals α1, and η2 > 0 also implies group two’s probability equals α2.Source: §3 · p. 11

4 The Dynamic Case

This section characterizes the dynamic mechanism through backward induction, relating fair allocations and payments to current values, future utilities, entry fees, and participation bonuses. It examines the seller’s revenue and net costs as buyer numbers grow, and presents exact and approximate methods for computing fair allocations with guarantees on fairness, seller utility, and computational cost.

4 The Dynamic Case

  • The analysis proceeds backward from the last round, updating residual minimum allocations and setting the seller’s and buyers’ interim utilities to −∞ when no feasible auction exists.The last round is treated as a static auction subject to its residual fairness requirements.Source: §4 · pp. 12–13
  • If only one group can still meet its residual allocation requirement, the item goes to that group’s highest-value buyer at the second-highest value, without a reserve price.The seller must allocate the item to satisfy fairness, so no reserve price is needed.Source: §4 · pp. 13–14
  • Increasing a group’s fairness requirement can reduce its buyers’ expected utility, as a stricter requirement may force allocation in an earlier round with lower values.The example describes two rounds with identical group value distributions, lower first-round values, and a higher first-group allocation requirement.Source: §4 · pp. 14–15
  • When allocation to either group remains feasible, the seller’s allocation rule balances groups’ maximum virtual values against differences in future seller and buyer utility.The allocation boundary is linear in the groups’ maximum virtual values; the seller’s future-utility difference shifts the threshold between groups.Source: §4 · p. 15
  • The dynamic payment rule includes a participation bonus to encourage truthful reporting and an entry fee charged to buyers.The bonus addresses the loss of future utility when a buyer’s group wins; the entry fee is charged to buyers who skip the current round.Source: §4 · pp. 15–16
  • Under ex post fairness, the recursive analysis remains, but the terminal static case must change; feasibility requires that only one group have a positive minimum allocation requirement.When a group has a positive requirement, the item must always go to that group.Source: §4 · p. 16

4.1 Do the entry fees cover the participation bonuses?

  • Under the stated assumption that buyers’ future utility rises when they do not win currently, the entry fee can be zero while the participation bonus is positive for one buyer, but entry fees offset bonuses as buyer numbers increase.The paper examines this net effect under Assumption 3, excluding the corner case in Remark 1.Source: §4.1 · p. 17
  • In expectation, total participation bonuses exceed total entry fees by O(1/n), so the seller pays at most a negligible amount as the auction grows.The bound’s constant depends on δ and the maximal buyer value.Source: §4.1 · pp. 17–18

4.2 An Approximation Scheme

  • When δ=1, the exact optimal allocation can be found with O(T²) oracle calls.This result holds under Assumptions 1, 2, and 4.Source: §4.2 · p. 18
  • For any ε∈[0,min_i α_i], x′ achieves group-i fairness ≥(1−ε)(α_i−ε) while giving the seller utility at least the optimum.The scheme computes an exact fair allocation over T0 = log(ε)/log(δ) rounds, then uses a standard second-price auction for the remaining rounds; the early allocation takes O(2^T0) oracle calls.Source: §4.2 · pp. 18–19
  • For fixed c<δ², x′′ guarantees group-i fairness ≥c·α_i and seller utility ≥c times the optimum, with polynomially many oracle calls.This scheme modifies the early-stopping approximation using discontinuous discounting and buckets of rounds with a common approximate discount factor.Source: §4.2 · p. 19

5 Extension to Multiple Groups

The paper extends its fairness-through-subsidization results to multiple groups and characterizes optimal allocations using group-specific subsidies. In the dynamic case, recursive allocation rules again favor otherwise under-allocated groups.

5 Extension to Multiple Groups

  • The multi-group dynamic result generalizes the static subsidization insight, with expected utilities evaluated using updated residual minimum allocations.The dynamic case is solved by backward induction.Source: §5 · pp. 19–20
  • For static allocations, the group-level decision depends only on each group’s maximum value, and the item goes to that group’s highest-value buyer.The allocation can be represented using nonnegative group-specific subsidies η_i; min_i η_i is the overall subsidy to society.Source: §5 · pp. 19–20
  • In the dynamic multi-group setting, the optimal fair allocation subsidizes groups that would otherwise be under-allocated.The allocation rule is characterized for residual minimum allocations when allocation remains feasible for any group.Source: §5 · p. 20

6 A Numerical Experiment

The numerical experiment examines how fairness constraints affect seller and buyer utilities for T = 2, 3, and 4. Its reported patterns distinguish the seller’s utility loss from the different utility responses of the two groups.

6 A Numerical Experiment

  • Seller utility decreases as fairness constraints rise, but only after the constraints bind.The experiment considers δ = 0.99, n = 1, and 10,000 iterations.Source: §6 · pp. 20–21
  • The experiment compares utility differences against the unconstrained allocation for T = 2, 3, and 4.The fairness constraints α1 and α2 range over a discretized grid from 0 to 0.5 in increments of 0.1; the supplied captions identify figures for T = 2 and T = 3.Source: §6 · pp. 20–21, Figure 2 · p. 21, Figure 3 · p. 21
  • For fixed α1, group one’s utility decreases as α2 rises, while group two’s utility remains relatively constant as α1 rises for fixed α2.The stated explanation is that satisfying group two’s constraint reallocates higher-value regions to group two, whereas reallocations to group one tend to come from lower-value regions that otherwise remain unallocated.Source: §6 · p. 21

7 Conclusion and Future Directions

The paper concludes that dynamic fair auctions can use subsidization and participation rewards alongside entry fees, and offers an efficient approximation scheme. It identifies interdependent values and the price of fairness as directions for future study.

7 Conclusion and Future Directions

  • The dynamic revenue-maximizing allocation subsidizes groups otherwise allocated the item too infrequently, while payments combine participation rewards with an entry fee each round.As the number of buyers grows, the entry fee covers the participation bonus, so the seller does not pay out of pocket.Source: §7 · pp. 21–22

A Extended Related Work

  • The related work spans group fairness in allocation, fair mechanism design, matching, ranking, assortment, and market-based redistribution.The paper distinguishes proportional group allocation from other notions such as group envy-freeness and democratic group fairness.Source: A Extended Related Work · p. 29

B Proofs

B.1 Proof of Theorem 1

The proof characterizes the optimal static fair allocation through affine boundaries in virtual values. It establishes the form of the allocation by comparing its loss with that of the unconstrained allocation.

B.1 Proof of Theorem 1

  • For a fixed set of allocated value profiles, an optimal fair allocation has an affine boundary in virtual values.Group one receives the item when ϕ1(v1) ≥ ϕ2(v2) + γ within the allocated set; the reverse inequality assigns it to group two.Source: §B.1 · pp. 30, 32
  • The proof shows that the optimal allocation’s region of deviation from the unconstrained rule matches the corresponding region for the constructed allocation, up to measure zero.Claim 1 lower-bounds the probability of the optimal allocation’s deviation within Q1 and states that equality requires the deviation set to lie within Q1.Source: §B.1 · pp. 30–32
  • The proof establishes that the group-specific thresholds satisfy η2 = η1 + γ in the optimal fair allocation.The argument rules out either strict inequality by constructing a change that lowers the seller’s loss.Source: §B.1 · p. 32

B.2 Proof of Proposition 2

The proof establishes tight group-allocation probabilities in the optimal fair allocation and distinguishes cases by the relationship between η2 and η1 + γ.

B.2 Proof of Proposition 2

  • Figures 5 and 6 depict the cases η2 < η1 + γ and η2 > η1 + γ, respectively.Source: Figure 5 · p. 33, Figure 6 · p. 34
  • For γ > 0, the optimal fair allocation satisfies P(G2) = α2.The proof argues that a modified allocation would increase seller utility if this equality failed.Source: §B.2 · p. 33
  • When η1 > 0, the optimal allocation also satisfies P(G1) = α1.A modified allocation with higher loss would contradict optimality.Source: §B.2 · pp. 33–34

B.3 Proof of Corollary 1

The proof derives group-level expected utilities and payments for the corollary, then treats a feasibility case as a static Vickrey auction.

B.3 Proof of Corollary 1

  • The expected payment expression is connected to the standard static payment derivation of Nisan et al. [2007, Lemma 13.11].Source: §B.3 · p. 35
  • The corollary proof applies Claim 3 in deriving expected seller and buyer utilities from group-level functions.Source: §B.3 · p. 35

B.4 Proof of Proposition 3

  • When feasibility requires allocating to a buyer in group i, the case reduces to a no-reserve Vickrey auction: the highest bidder wins and pays the second-highest bid.Source: §B.4 · p. 35
  • A buyer who does not receive the item gets only discounted expected utility from future rounds in this case.Source: §B.4 · p. 36

B.5 Proof of Theorem 2

The proof derives payments and allocation rules for the dynamic mechanism, characterizes a winning probability with one fewer group member, and extends the argument when nonallocation is feasible.

B.5 Proof of Theorem 2

  • Claim 4 characterizes buyer (i, k)'s expected payment to the seller at round t, which is then used to express seller utility.Source: §B.5 · p. 38
  • The round-t allocation is monotone: group 1 receives the item when its threshold condition holds, and otherwise group 2 receives it, yielding EPIC and IR allocation and payment rules.The allocation also determines the regions G_i and updates the interim utility functions.Source: §B.5 · p. 38

B.5.1 Characterization of ζt

  • The characterization of ζt concerns group i's probability of winning when it participates with one fewer buyer.Source: §B.5.1 · p. 39

B.5.2 Relaxing Assumption 2

  • The proof relaxes Assumption 2 to allow nonallocation to either group when feasible, then derives allocation and payment rules by similar steps.Source: §B.5.2 · p. 39

B.6 Proof of Proposition 4

The proof uses backward induction across two feasibility cases, then relaxes regularity by ironing virtual values while preserving the fairness-constrained revenue allocation.

B.6 Proof of Proposition 4

  • Backward induction from the terminal static round establishes the claimed property in both cases: allocation must go to a particular group, or either group may receive it.Source: §B.6 · pp. 40–41

B.7 Relaxation of Assumption 1 by Ironing

  • Ironing produces a monotone virtual-value function, yielding a threshold-form allocation that maximizes R subject to the fairness constraint.Source: §B.7 · pp. 41–42
  • For this allocation, E is zero, so maximizing R subject to fairness also maximizes R − E subject to fairness.Source: §B.7 · p. 42

B.8 Proof of Proposition 5

The proof constructs an approximation by running the exact fair allocation for a finite initial horizon and then switching to a standard second-price auction. It bounds the resulting fairness and seller-utility guarantees and the computation needed.

B.8 Proof of Proposition 5

  • The construction runs the recursive fair allocation at level α_i − ε for T0 rounds, then uses a standard second-price auction for the remaining T − T0 rounds.The initial horizon T0 is chosen later in the construction.Source: §B.8 · p. 42
  • Running x∗ only through T0 = log(ε) / log(δ) rounds preserves fairness at level α_i − ε over that initial horizon.The proof uses δ^T0 = ε and T0 ≤ T.Source: §B.8 · pp. 42–43
  • The approximation x′ satisfies fairness at level (1 − ε)(α_i − ε) over T rounds and guarantees seller utility at least that of the optimal fair allocation x∗.The seller-utility guarantee applies across both the initial fair-allocation interval and the remaining second-price-auction interval.Source: §B.8 · pp. 43–44
  • The recursive fair-allocation algorithm has complexity that increases exponentially with the number of rounds, while the approximation runs it for only T0 rounds.The supplied passage introduces a complexity bound but does not provide its expression.Source: §B.8 · pp. 42, 44

B.9 Proof of Proposition 6

Proposition 6 bounds the fairness and seller utility of an approximation using bucketed discounting and a finite fair-allocation horizon. The following proofs extend the allocation-boundary and payment arguments to multiple groups.

B.9 Proof of Proposition 6

  • x′′ guarantees fairness of at least (1 − ε)(1 − β)^2(α_i − ε) over T rounds and seller utility of at least (1 − β) times the optimal allocation’s utility.The guarantee holds under the stated assumptions, with δ < 1, ε ∈ (0, min_i(α_i)), and β > 1 − δ.Source: §B.9 · pp. 44–45
  • The x′′ construction approximates discounting as constant within each of T0/ℓ buckets, runs fair allocation for T0 rounds, then uses a standard second-price auction.Its initial fair-allocation level is (1 − β)(1 − α_i) under the approximated discontinuous discounting scheme.Source: §B.9 · p. 45

B.10 Proof of Theorem 3

  • For groups i and j, the allocation boundary is affine: group i wins if and only if ϕ_i(v_i) − ϕ_j(v_j) ≥ η_i,j; the boundary parameters add across groups.Under the stated nonnegative-parameter conditions, η_i,k = η_i,j + η_j,k.Source: §B.10 · p. 46

B.11 Proof of Theorem 4

  • The multi-group payment proof extends Theorem 2’s argument, with expected payment expressed using a group-wise value of c_i,k(·) that satisfies individual rationality with equality.The expected-payment derivation and seller-utility expressions are developed for each round and from time t onward.Source: §B.11 · pp. 46–47

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