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arxiv: 2505.21122 · v3 · pith:W4EFA5OVnew · submitted 2025-05-27 · 💻 cs.GT · econ.TH

Sequential Elimination and Union Shapley Value for Group Assessment in Coalitional Games

Piotr K\k{e}pczy\'nski , Oskar Skibski This is my paper

Pith reviewed 2026-05-22 02:03 UTC · model grok-4.3

classification 💻 cs.GT econ.TH
keywords coalitional gamesShapley valuesemivaluesgroup valuessequential eliminationinteraction indexunion shapley valuesynergy
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The pith

Sequential elimination yields order-independent group values for almost all semivalues in coalitional games.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors show that assessing groups by sequentially eliminating members one at a time produces the same group assessment no matter the sequence, for almost all semivalues. This order independence defines a new group value they call the Union Shapley Value. Using a class of weak consistent semivalues, they distinguish methods that evaluate a group's total contribution from those that isolate its synergistic effects. They prove that the Interaction Index is the synergistic counterpart to the merge approach of Marichal et al., and derive the Intersection Shapley Value as the corresponding synergistic form for their new Union value. The results position sequential elimination as one of the most natural ways to extend player values to groups.

Core claim

We show that sequential elimination in coalitional games does not depend on the order of players for almost all semivalues. In particular, we introduce the Union Shapley Value and investigate its axiomatic properties. Our analysis defines a class of group weak consistent semivalues satisfying a weak form of monotonicity, which clarifies differences between existing notions and reveals that group values either assess total worth or measure synergy, with the Interaction Index as the synergistic counterpart of the Marichal et al. value.

What carries the argument

The Union Shapley Value, defined via sequential elimination of group members where each removal's marginal contribution is aggregated, serving as an order-independent extension of the Shapley value to groups.

Load-bearing premise

That semivalues satisfy a weak form of monotonicity allowing consistent group extensions without order dependence.

What would settle it

A specific semivalue and coalitional game where two different orders of eliminating the same group members produce different aggregated assessments.

read the original abstract

Two straightforward methods to extend an assessment of individual elements to groups are to sum individual assessments or to treat the group as a single merged element and assess it accordingly. In this work, we analyze another natural approach based on sequential elimination: elements of the group are removed one by one, and their assessments are aggregated. We study this approach in the context of coalitional games and show that, for almost all semivalues, it does not depend on the order of players. In particular, we introduce a new group value, called the Union Shapley Value, and investigate its axiomatic properties. Our results build on a comprehensive analysis of group values in coalitional games. Specifically, we define a class of group (weak consistent) semivalues - a variant of semivalues satisfying a weak form of monotonicity. This framework allows us to clarify the differences between existing notions in the literature. We show that existing group values either assess the total worth of a group or measure its synergy. We distinguish these two approaches axiomatically and uncover a connection between the corresponding values. In particular, we show that the well-known Interaction Index is a synergistic counterpart of the value introduced by Marichal et al., which corresponds to the merge approach. The analysis also yields new synergistic group values associated with the Union Shapley Value, which we call the Intersection Shapley Value. Our results demonstrate that the sequential extension - and the Union Shapley value in particular - constitute one of the most natural extensions of player values to groups in coalitional games.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper examines extensions of individual player assessments to groups in coalitional games, contrasting summation and merging approaches with a sequential elimination method in which group members are removed one-by-one and their assessments aggregated. It claims that this sequential process yields an order-independent result for almost all semivalues, introduces the Union Shapley Value as the resulting group value, and develops its axiomatic properties. The authors define a class of weak-consistent group semivalues obeying a weak monotonicity condition, use this framework to distinguish merge versus synergy interpretations of existing group values, establish that the Interaction Index is the synergistic counterpart of Marichal et al.'s value, and introduce the Intersection Shapley Value as a new synergistic counterpart associated with the Union Shapley Value.

Significance. If the order-independence result and the axiomatic characterizations hold with the stated generality, the work supplies a principled and natural way to extend semivalues to groups, clarifies an important distinction between total-worth and synergy measures that has been implicit in the literature, and adds two new concrete group values (Union and Intersection Shapley Values) with explicit axiomatic grounding. The connection drawn between the Interaction Index and Marichal et al.'s value is a useful unification.

major comments (1)
  1. [Section 4] Section 4 (definition of weak-consistent semivalues and the weak monotonicity axiom): the central claim that sequential elimination is order-independent 'for almost all semivalues' is proved only after restricting attention to the subclass of semivalues that are weak-consistent and satisfy the newly introduced weak monotonicity condition. The manuscript neither supplies a measure on the space of weight vectors that define semivalues nor exhibits a concrete semivalue lying outside this subclass for which order dependence appears. Consequently the qualifier 'almost all' is effectively 'almost all inside the weakly monotonic consistent subclass,' which weakens the support for the Union Shapley Value as a general extension.
minor comments (1)
  1. [Section 3] Notation for the sequential-elimination operator could be introduced earlier and used consistently; the current presentation requires the reader to track several ad-hoc symbols across sections.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The main concern is the scope of the order-independence claim. We address it directly below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Section 4] Section 4 (definition of weak-consistent semivalues and the weak monotonicity axiom): the central claim that sequential elimination is order-independent 'for almost all semivalues' is proved only after restricting attention to the subclass of semivalues that are weak-consistent and satisfy the newly introduced weak monotonicity condition. The manuscript neither supplies a measure on the space of weight vectors that define semivalues nor exhibits a concrete semivalue lying outside this subclass for which order dependence appears. Consequently the qualifier 'almost all' is effectively 'almost all inside the weakly monotonic consistent subclass,' which weakens the support for the Union Shapley Value as a general extension.

    Authors: We agree that the order-independence theorem is established inside the class of weak-consistent semivalues obeying the weak monotonicity axiom introduced in Section 4. This class is not arbitrary: it is precisely the setting in which the sequential-elimination operator yields a well-defined group value that satisfies basic consistency and monotonicity requirements needed for the subsequent axiomatic analysis. All standard semivalues used in the literature (Shapley, Banzhaf, and their weighted variants with non-increasing weights) belong to this class. To meet the referee's request, we will add (i) an explicit parametrization of semivalues by their weight vectors, (ii) a short argument that the weak-monotonicity condition excludes only a lower-dimensional subset of the weight space, and (iii) a concrete counter-example of a semivalue with pathological weights that violates weak monotonicity and for which sequential elimination is order-dependent. These additions will clarify the precise scope of the 'almost all' qualifier without changing the main results. revision: yes

Circularity Check

0 steps flagged

Minor self-citation and new class definition; central claims retain independent axiomatic content

full rationale

The paper defines a new subclass of weak-consistent semivalues with a weak monotonicity condition specifically to support analysis of sequential elimination and to distinguish merge vs. synergy group values. Order-independence is then proven inside this subclass, with the qualifier 'almost all semivalues' implicitly restricted to it. The Union Shapley Value and Intersection Shapley Value are introduced via explicit definitions and axioms, with connections to Marichal et al. and the Interaction Index derived through direct comparison rather than reduction to prior results by construction. No fitted parameters are renamed as predictions, and no load-bearing step collapses to a self-citation chain. The derivations are self-contained against external benchmarks in cooperative game theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on standard axioms of cooperative game theory and the newly introduced definitions of group values and the weak consistency framework.

axioms (2)
  • standard math Semivalues satisfy efficiency, symmetry, and other standard properties in coalitional games
    The analysis is built on semivalues, which are standard in cooperative game theory.
  • domain assumption Weak form of monotonicity for group semivalues
    The paper introduces group (weak consistent) semivalues to enable the analysis of sequential elimination.
invented entities (2)
  • Union Shapley Value no independent evidence
    purpose: Group value based on sequential elimination
    Newly defined in this work as a natural extension of player values to groups.
  • Intersection Shapley Value no independent evidence
    purpose: Synergistic group value associated with Union Shapley Value
    Introduced as the synergistic counterpart to the Union Shapley Value.

pith-pipeline@v0.9.0 · 5816 in / 1551 out tokens · 92433 ms · 2026-05-22T02:03:48.523751+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Group Vitality Indices: Axioms and Algorithms

    cs.SI 2026-05 unverdicted novelty 6.0

    Every vitality index extends uniquely to groups using the group Shapley value, with a complete axiomatization and computational study of the resulting measures.

Reference graph

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