Winner Determination¶

An equivalent problem in the context of combinatorial auctions is called the winner determination problem.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12897
Domain group: Applied Sciences & Engineering
Origin domain: Computer Science & Software Engineering
Subdomains: Combinatorial Auctions, Algorithmic Mechanism Design → Computer Science & Software Engineering

Core Idea¶

Winner Determination is treated here as the recurring social sciences, humanities, and arts identity summarized by this source-grounded definition: An equivalent problem in the context of combinatorial auctions is called the winner determination problem.

The welfare maximization problem is an optimization problem studied in economics and computer science. Its goal is to partition a set of items among agents with different utility functions, such that the welfare – defined as the sum of the agents' utilities – is as high as possible. In other words, the goal is to find an item allocation satisfying the utilitarian rule.

An equivalent problem in the context of combinatorial auctions is called the winner determination problem. In this context, each agent submits a list of bids on sets of items, and the goal is to determine what bid or bids should win, such that the sum of the winning bids is maximum. It is usually assumed that the utility functions are monotone set functions, that is, Z_1\supseteq Z_2 implies u_i(Z_1) \geq u_i(Z_2).

For Winner Determination, the abstraction is narrower than the article's general subject matter: a positive case must preserve An equivalent problem in the context of combinatorial auctions is called the winner determination problem. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in social sciences, humanities, and arts, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — In general, computing the value of a fractional bundle might require 2 m calls to a value oracle; however, it can be computed approximately with high probability by random sampling.
Constitutive relation — The welfare maximization problem has many variants, depending on the type of allowed utility functions, the way by which the algorithm can access the utility functions, and whether there are additional constraints on the allowed allocations.
Operating condition — When all agents are additive, welfare maximization can be done by a simple polynomial-time algorithm: give each item j to an agent for whom v_{i,j} is maximum (breaking ties arbitrarily).
Recognition evidence — Welfare maximization with additive utilities under heterogeneous matroid constraints can be done in polynomial time, by reduction to the weighted matroid intersection problem.
Admissible variation — The maximum welfare can be approximated by the following polynomial-time greedy algorithm.
Characteristic consequence — The utility of k increases by his marginal utility of g, which at most v by the greedy selection.
Failure boundary — The marginal utility of the remaining bundle of i increases by at most v.

What It Is Not¶

Not the whole field of social sciences, humanities, and arts. The node requires the specific identity stated by An equivalent problem in the context of combinatorial auctions is called the winner determination problem.
Not an over-broad reading. In general, there may be a different matroid for each agent, and the allocation must give each agent i a subset X i that is an independent set of their own matroid.
Not an over-broad reading. In general, computing the value of a fractional bundle might require 2 m calls to a value oracle; however, it can be computed approximately with high probability by random sampling.
Not an over-broad reading. For general monotone submodular functions, Buchbinder and Feldman described a different local search based algorithm that is deterministic and guarantees (1-1/e-\epsilon) -approximation in polynomial time for any constant \epsilon > 0.
Not automatically Maximum satisfiability problem. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Winner Determination applies literally inside social sciences, humanities, and arts wherever the source-defined carrier and relation can be established. Its documented habitats include:

Definitions. Each agent i in N has a utility function u_i: 2^M \to \mathbb{R}.
Definitions. The function assigns a real value to every possible subset of items.
Definitions. It is usually assumed that the utility functions are monotone set functions, that is, Z_1\supseteq Z_2 implies u_i(Z_1) \geq u_i(Z_2).
Definitions. The welfare maximization problem has many variants, depending on the type of allowed utility functions, the way by which the algorithm can access the utility functions, and whether there are additional constraints on the allowed allocations.
Submodular agents. A submodular agent has a utility function that is a submodular set function.
Algorithms using a value oracle. Dobzinski and Schapira present a polytime n/(2n-1) -approximation algorithm, and an (1-1/e)≈0.632-approximation algorithm for the special case in which the agents' utilities are set-coverage functions.

Outside social sciences, humanities, and arts, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Winner Determination names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is An equivalent problem in the context of combinatorial auctions is called the winner determination problem. The strongest recognition evidence in the frozen account is: Welfare maximization with additive utilities under heterogeneous matroid constraints can be done in polynomial time, by reduction to the weighted matroid intersection problem. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification In general, there may be a different matroid for each agent, and the allocation must give each agent i a subset X i that is an independent set of their own matroid. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Winner Determination compresses multiple social sciences, humanities, and arts details into a stable diagnostic relation. The source shows both the central mechanism—the welfare maximization problem has many variants, depending on the type of allowed utility functions, the way by which the algorithm can access the utility functions, and whether there are additional constraints on the allowed allocations.—and the practical consequence—the utility of k increases by his marginal utility of g, which at most v by the greedy selection. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the social sciences, humanities, and arts entities to which the claim applies.
State the relation. Use the source-grounded identity: An equivalent problem in the context of combinatorial auctions is called the winner determination problem.
Check operation and conditions. When all agents are additive, welfare maximization can be done by a simple polynomial-time algorithm: give each item j to an agent for whom v_{i,j} is maximum (breaking ties arbitrarily).
Demand recognition evidence. Welfare maximization with additive utilities under heterogeneous matroid constraints can be done in polynomial time, by reduction to the weighted matroid intersection problem.
Test variation. Change an implementation or setting while preserving the maximum welfare can be approximated by the following polynomial-time greedy algorithm.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Winner Determination transfers literally when a new case preserves the same carrier type, relation, and recognition test. Each agent i in N has a utility function u_i: 2^M \to \mathbb{R}. The function assigns a real value to every possible subset of items.

Beyond the home domain. Transfer the broader Optimization relation when the social sciences, humanities, and arts-specific differentia cannot be filled. Retain the name Winner Determination only when the same carrier, operation, and rejection conditions are present literally rather than metaphorically.

Examples¶

Canonical¶

In cases when fractional bundles can be evaluated efficiently (e.g. when utility functions are set-coverage functions), the algorithm can be made deterministic. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → An equivalent problem in the context of combinatorial auctions is called the winner determination problem; recognition evidence → Welfare maximization with additive utilities under heterogeneous matroid constraints can be done in polynomial time, by reduction to the weighted matroid intersection problem

Applied / In Practice¶

One may want to maximize the welfare among all allocations that are fair, for example, envy-free up to one item (EF1), proportional up to one item (PROP1), or equitable up to one item (EQ1). The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Fairness constraints; invariant → An equivalent problem in the context of combinatorial auctions is called the winner determination problem; boundary → the case exits the class when in general, there may be a different matroid for each agent, and the allocation must give each agent i a subset X i that is an independent set of their own matroid

Structural Tensions¶

T1 — Stable identity versus admissible variation. In general, there may be a different matroid for each agent, and the allocation must give each agent i a subset X i that is an independent set of their own matroid. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. In general, computing the value of a fractional bundle might require 2 m calls to a value oracle; however, it can be computed approximately with high probability by random sampling. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. For general monotone submodular functions, Buchbinder and Feldman described a different local search based algorithm that is deterministic and guarantees (1-1/e-\epsilon) -approximation in polynomial time for any constant \epsilon > 0. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. However, there are algorithms based on state space search that work very well in practice. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. In general, computing the value of a fractional bundle might require 2 m calls to a value oracle; however, it can be computed approximately with high probability by random sampling. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Winner Determination literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. The welfare maximization problem has many variants, depending on the type of allowed utility functions, the way by which the algorithm can access the utility functions, and whether there are additional constraints on the allowed allocations. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Winner Determination distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Winner Determination is mixed or framed-leaning. Its structural side is the repeatable organization summarized by An equivalent problem in the context of combinatorial auctions is called the winner determination problem. Its framed side is the social sciences, humanities, and arts vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: When all agents are additive, welfare maximization can be done by a simple polynomial-time algorithm: give each item j to an agent for whom v_{i,j} is maximum (breaking ties arbitrarily). Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. An equivalent problem in the context of combinatorial auctions is called the winner determination problem. The reviewed portable genus is Optimization; the candidate preserves that parent relation across admissible variants. The source-grounded carrier and relation are expressed by these conditions: In general, computing the value of a fractional bundle might require 2 m calls to a value oracle; however, it can be computed approximately with high probability by random sampling. The welfare maximization problem has many variants, depending on the type of allowed utility functions, the way by which the algorithm can access the utility functions, and whether there are additional constraints on the allowed allocations. The recognition and variation tests add: When all agents are additive, welfare maximization can be done by a simple polynomial-time algorithm: give each item j to an agent for whom v{i,j} is maximum (breaking ties arbitrarily). Welfare maximization with additive utilities under heterogeneous matroid constraints can be done in polynomial time, by reduction to the weighted matroid intersection problem.

What is domain-bound. social sciences, humanities, and arts fixes the carrier, technical vocabulary, admissible evidence, and exceptions that distinguish Winner Determination from other Optimization instances. Its documented habitat includes the condition that Each agent i in N has a utility function ui: 2^M \to \mathbb{R}. A second source-grounded application condition is that The function assigns a real value to every possible subset of items. Those details determine what the words denote, what observations warrant classification, and which apparent similarities are false positives.

Why the node remains domain-specific. Removing the social sciences, humanities, and arts differentia leaves the parent rather than the candidate. The edge records that reduction without claiming that every topical neighbor is hierarchical. The final collapse test is source-specific: The maximum welfare can be approximated by the following polynomial-time greedy algorithm. If that condition or the defining relation is absent, the case may instantiate Optimization, but it is not Winner Determination.

This entry is a kind of Optimization.

Immediate parent — Optimization (subsumption). Winner Determination is a domain-specific kind of Optimization. Winner Determination is a strict kind of Optimization: An equivalent problem in the context of combinatorial auctions is called the winner determination problem. The parent supplies the necessary broader identity—Finds best solution under constraints.—while the candidate adds its domain carrier, relation, and rejection conditions.
Other nearby abstractions. Retrieval neighbors remain comparison surfaces only; no additional parent is asserted without a necessary-genus or structural-prerequisite test.

Relationships to Other Abstractions¶

Current abstraction Winner Determination Domain-specific

Parents (1) — more general patterns this builds on

Winner Determination is a kind of Optimization Prime

Winner Determination is a strict kind of Optimization: An equivalent problem in the context of combinatorial auctions is called the winner determination problem.

Hierarchy path (1) — routes to 1 parentless root

Winner Determination → Optimization

Neighborhood in Abstraction Space¶

Winner Determination sits in a sparse region of the domain-specific corpus (62^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish An equivalent problem in the context of combinatorial auctions is called the winner determination problem?
Maximum satisfiability problem. The optimization problem of assigning Boolean variables to maximize the number or total weight of satisfied clauses in a conjunctive normal form formula. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Optimization. Finds best solution under constraints. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Allocation. Assign a limited supply across competing claimants under a feasibility constraint, independent of which criterion fills in the rule. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Winner Determination remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside social sciences, humanities, and arts lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Welfare_maximization (revision 1365865501).
Preserved source candidate: https://www.sciencedirect.com/science/article/pii/S0377221722007822
Preserved source candidate: https://www.sciencedirect.com/science/article/pii/S0004370222001606
Preserved source candidate: https://www.sciencedirect.com/science/article/pii/S0165489622000798
Preserved source candidate: https://doi.org/10.1145/501158.501161
Preserved source candidate: https://doi.org/10.1007/s00453-007-9105-7
Preserved source candidate: https://doi.org/10.1145/1386790.1386805
Preserved source candidate: https://doi.org/10.1007/BFb0121195
Preserved source candidate: https://dl.acm.org/doi/10.5555/1109557.1109675

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.