Truthful Mechanisms for Online Bipartite Matching

• The paper presents a systematic catalog of truthful mechanisms for online bipartite matching, emphasizing both deterministic and randomized strategies with exact and approximate truthfulness.
• It details algorithmic paradigms such as greedy selection, LP rounding, and secretary-based sampling that achieve competitive ratios like 2-approximation and e-competitiveness under varying models.
• The analysis extends to budgeted, ordinal, and generalized assignment variants, illustrating practical applications in dynamic, auction-based, and real-time allocation platforms.

A truthful mechanism for online bipartite matching is a mechanism where agents, represented as one side of a bipartite graph, have no incentive to misreport their private information—typically their incident edges, costs, or values—during the matching process. The literature has established a diverse set of results and methodological foundations for these mechanisms, involving both exact and approximate truthfulness, deterministic and randomized implementations, and varying assumptions about payments, edge weights, arrival models, and strategic constraints. This article systematically catalogs and analyzes these foundations with a focus on the formal guarantees, algorithmic paradigms, performance barriers, and broader applicability to mechanism design.

1. Foundational Problem Statements and Truthfulness Paradigms

In the online bipartite matching setting, the task is to instantaneously and irrevocably assign arriving agents (vertices on one side of the bipartite graph) to available resources or tasks (vertices on the other side) to maximize a welfare objective (e.g., total weight of the matching), subject to feasibility and often additional constraints such as budgets or capacities. Truthful mechanisms are designed such that agents cannot benefit by misreporting private inputs, such as their edge existence, declared interest, bids, or costs (1001.0436, Vaze et al., 2016, Vaze, 2016).

Mechanisms are characterized by:

• Edge revelation as private information, where each job or agent reports whether each possible edge exists (compatibility) (1001.0436).
• Classic models without payments—agents cannot be incentivized via monetary transfers, and incentive compatibility is enforced solely by algorithmic choices (tie-breaking, monotonic selection rules, critical values).
• Budgeted models with payments, where agents may misreport costs or bids, and critical payment rules aligned with Myerson's lemma ensure truthfulness (Vaze et al., 2016, Vaze, 2016).
• Ordinal (preference-only) models, where numerical utilities are hidden but ordinal rankings on the incident edges are reported (Anshelevich et al., 2016).

Truthfulness is further refined by considering deterministic and randomized mechanisms. Universally truthful mechanisms retain incentive compatibility regardless of randomization (Anshelevich et al., 2016).

2. Mechanism Design for Unweighted, Weighted, and Generalized Assignment Problems

Unweighted Bipartite Matching

For the unweighted problem (maximum cardinality matching), an optimal and strategyproof mechanism is achieved by computing any maximum matching and breaking ties in a predetermined, agent-independent order (e.g., lex order). Since tie-breaking does not rely on the agents’ private reports, no agent can benefit from misreporting, establishing strategyproofness and maximality without money (1001.0436).

Maximum Weight Bipartite Matching

For the max-weight case, with publicly known edge values but private edge existence, truthfulness imposes a worst-case 2-approximation barrier for deterministic mechanisms and an approximate barrier of 1.0938 for randomized mechanisms (Theorem 3.2) (1001.0436). The canonical solution is a greedy algorithm: sort all edges by public value, process in order, and include each as long as no endpoint is already matched. Since the edge order is pre-fixed and input-independent, reporting false incident edges cannot increase an agent's expected welfare.

Generalized Assignment and Knapsack-type Variants

For knapsack-like and generalized assignment problem (GAP) settings (public values and capacities, private edge existence):

• Fractional mechanism: Solve the LP relaxation, using lexicographic tie-breaking to maintain truthfulness in the fractional domain;
• Rounding: Via a method extending Lavi and Swamy, randomly round the fractional assignment, with the approximation ratio scaling with the integrality gap (e.g., 2 for GAP, 4 for certain knapsack variants) (1001.0436). This LP-based methodology decouples approximation from incentive compatibility, potentially generalizing to other packing and assignment polytopes.

3. Online, Budgeted, and Secretary-type Truthful Mechanisms

Online Arrival and Sampling-Based Algorithms

Mechanisms for online bipartite matching under adversarial or random arrivals employ sample-and-threshold paradigms inspired by the secretary problem:

• Two-phase design: Use the first fraction of arrivals as a sample to estimate thresholds (e.g., “buck-per-bang”), then use these to make irrevocable allocation decisions for future arrivals.
• Payment rules: Assign winners their critical prices, determined by the threshold and sample statistics, thereby aligning with Myerson's theorem and ensuring monotonicity (Vaze et al., 2016, Vaze, 2016).

A representative result is the 24β-competitive online mechanism for budgeted matching, with β the ratio of maximum to minimum utility (Vaze et al., 2016). In both the online knapsack and budgeted bipartite matching scenarios, critical payment mechanisms and monotonic selection rules ensure truthfulness, while approximation to the offline optimum is proportional to the sampling and acceptance probabilities derived from the secretary model.

Secretary Model and Online Weighted Matching

For online weighted bipartite matching with random arrivals (secretary model), an optimal e-competitive truthful mechanism is achieved using a variant of secretary algorithms:

1. Sampling phase: Observe the first k bidders, making no assignments.
2. Assignment phase: For subsequent arrivals, assign items based on the current offline optimum relative to the set of unassigned items and all observed bidders. Payments are set using a VCG-based formula computed before a bidder reveals valuations.
3. Truthfulness and independence: Fixed item prices, determined before seeing an agent’s report, eliminate the incentive to misrepresent, and the competitive ratio matches the secretary bound of e (Reiffenhäuser, 2021).

4. Ordinal Preference and Truthful Algorithms without Cardinal Utility

When only preference orderings are available, universally truthful approximation algorithms can be constructed using randomized serial dictatorship and greedy procedures on undominated edges (Anshelevich et al., 2016). For example:

• For max-weight matching, a mixed greedy-randomized mechanism achieves a 1.7638-approximation;
• For max k-matching, a randomized serial dictatorship achieves a 2-approximation;
• All such mechanisms are universally truthful, as agents cannot benefit by misreporting ordinal rankings due to the independence of the allocation order.

These principles extend naturally to online bipartite matching models with ordinal input, providing robust constant-factor truthful approximation guarantees in the absence of cardinal utility reports.

5. Efficiency, Barriers, and Performance Guarantees

The literature delineates strong and often tight performance barriers for truthful mechanisms:

Problem	Truthful Deterministic Ratio	Truthful Randomized Ratio	Notes
Max unweighted matching	1.0 (optimal)	1.0 (optimal)	Consistent tie-breaking suffices (1001.0436)
Max weight matching (no money)	2.0	>1.0938	Factor-2 lower bound tight (1001.0436)
Online budgeted matching	24β	–	Monotonic rules + sampling (Vaze et al., 2016)
Online weighted secretary	e (~2.718)	–	Sampling + VCG payments (Reiffenhäuser, 2021)
Ordinal max-weight matching	1.7638	1.7638	Greedy-randomized mix (Anshelevich et al., 2016)

These bounds are established using direct adversarial constructions (lower bounds), reduction to well-studied secretary and auction settings, critical tie-breaking, and integrality gaps for LP-based relaxations. In budgeted settings, the selection rule’s monotonicity and critical payments are essential for maintaining truthfulness, while competitive analysis ensures that expected welfare doesn't drop below the stated ratios.

6. Structural and Algorithmic Techniques

The construction and analysis of truthful mechanisms leverage several algorithmic strategies:

• Fixed, agent-independent ordering: Tie-breaking rules and edge-processing orders that do not depend on agent reports block manipulation (1001.0436).
• Fractional assignment and rounding: Defer allocation to a fractional LP solution and use randomized rounding, losing at most the integrality gap in approximation.
• Critical thresholds and monotonicity: Sample thresholds (“buck-per-bang”), monotonic accept/reject policies, and critical payments underpin truthful selection (Vaze et al., 2016).
• Randomized serial dictatorship and greedy selection: For ordinal settings, these methods are universally truthful due to their independence from reported orders.
• VCG and posted-price mechanisms: In auction and secretary variants, offline computation of VCG prices or item-thresholds that are independent of the current agent’s report is crucial (Reiffenhäuser, 2021).
• Martingale and dual-fitting analysis: Performance analyses use probabilistic arguments and primal-dual methods to certify welfare and truthfulness guarantees.

7. Broader Applicability and Extensions

The paradigm of decoupling approximation and truthfulness, as instantiated in these mechanisms, extends to:

• Other LP-based packing problems: Strategyproofness in fractional relaxations and randomized rounding generalize to scheduling, assignment, and clustering frameworks under suitable conditions on the underlying polytope.
• Dynamic, prediction-augmented, and semi-online models: If predictions or hybrids of adversarial/random arrivals are incorporated, the core properties of randomization and tie-breaking can be preserved to maintain truthfulness (Kumar et al., 2018).
• Online platforms without monetary transfers: Crowdsourcing, real-time task assignment, and ad allocation markets benefit from mechanisms that guarantee high efficiency and incentive compatibility without the burden of monetary incentives.

The field continues to pursue tight characterizations of the efficiency–truthfulness frontier, optimal tradeoffs for multi-parameter and group-strategyproof variants, and further generalizations to stochastic and partially observable online matching environments.

---

References

• (1001.0436) Truthful Assignment without Money
• (Vaze et al., 2016) Online Budgeted Truthful Matching
• (Vaze, 2016) Online Knapsack Problem and Budgeted Truthful Bipartite Matching
• (Anshelevich et al., 2016) Truthful Mechanisms for Matching and Clustering in an Ordinal World
• (Reiffenhäuser, 2021) An Optimal Truthful Mechanism for the Online Weighted Bipartite Matching Problem