Flow methods for cooperative games with generalized coalition configuration

Encarnacion Algaba; Eric Remila; Philippe Solal

REVIEW 2 major objections 4 minor 31 references

Flow methods for cooperative games with generalized coalition configuration

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper characterizes flow methods for cooperative games with restricted cooperation inside coalition configurations, and shows that two-step allocation procedures are exactly the flow methods whose induced flows vanish on non-relevant…

desk verdict Theorem 5's sufficiency direction is false: the two flow axioms characterize two-step flows at the aggregate level, but not flow methods, because the per-agent split of each edge's flow is unconstrained by the axioms. read the letter →

arxiv 2411.13684 v1 pith:OUWP3JWV submitted 2024-11-20 econ.TH

Encarnacion Algaba , Eric Remila , Philippe Solal This is my paper · ORCID

classification econ.TH MSC 91A12

keywords cooperativegamescoalitionconfigurationrestrictedcooperationflowmethodsmarginalvaluestwo-stepallocationproductdigraphsetsystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces cooperative games in which a coalition configuration covers the agent set and, inside each element of the configuration, only certain coalitions are feasible; the worth is a function on profiles of feasible coalitions. It proves that any value satisfying linearity and the null-agent axiom must be marginalist: each agent's payoff is a sum of coefficients times marginal contributions along covering relations. Adding efficiency forces those coefficients to form a unitary flow on the product digraph built from the set systems, so payoff rules of this kind are exactly flow methods. The paper then characterizes the flow methods that come from a two-step allocation procedure: they are exactly the flows that vanish on non-relevant edges and obey a flow-proportionality condition on relevant edges. This extends the flow-method theory of transferable-utility games to restricted cooperation inside coalition configurations.

What carries the argument

The central object is the product digraph $\Gamma_{\mathcal{F}} = (\mathcal{F}, E_{\mathcal{F}})$ formed from the cartesian product of the covering-relation digraphs of the set systems $(P_q, \mathcal{F}_q)$. A flow method is a value whose coefficients aggregate to a unitary flow on $\Gamma_{\mathcal{F}}$; relevant directed edges are those whose endpoints are relevant coalition profiles, built from a support $S$ of the configuration and a coalition $K_q$ in the $q$-th set system. The two-step construction factors each relevant edge's flow as $\Lambda_{\mathcal{M}}(R_{S\setminus\{q\}}, R_S) \cdot \Lambda_q(K_q, K'_q)$, a product of a flow on the $m$-dimensional directed hypercube and a flow on $\Gamma_{\mathcal{F}_q}$. Flow proportionality is the relation $\Lambda(K_{S,K_q}, K_{S,K'_q})\Lambda(K_{S',L_q}, K_{S',L'_q}) = \Lambda(K_{S',K_q}, K_{S',K'_q})\Lambda(K_{S,L_q}, K_{S,L'_q})$ that, together with the null-flow condition on non-relevant edges, characterizes two-step flows.

What would settle it

Inspect the deletion step in the proof of Lemma 1 on a connected acyclic digraph with two sources and one sink: the lemma's stated induction needs a sink whose removal leaves the graph connected, and whether such a sink always exists is precisely what must be checked. A counterexample to that step would leave Theorem 1 underived as written, even if the lemma itself can be proved by another argument.

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Extended reading notes

Core claim

On the domain of games with generalized coalition configuration, the combination of Linearity, Null agent, and Efficiency characterizes exactly the flow methods: values whose coefficients define a unitary flow on the product digraph $\Gamma_{\mathcal{F}}$. The paper's central structural result is that the two-step procedure—first allocate worth among the elements of the configuration through an upper game, then allocate each element's share to its agents through lower games—produces precisely those flow methods whose induced flow vanishes on non-relevant directed edges and satisfies flow proportionality on relevant edges (Theorem 5). For the unrestricted case, three axioms (null flow, intracoalitional anonymity, coalitional anonymity) single out a closed-form configuration value that extends the classical configuration value to coalition profiles. Finally, under a condition on the product digraph that holds for partitions and for regular set systems, a profile-level flow method induces a coalition-level flow method on reachable coalitions, so the theory projects back to games whose unit of cooperation is an ordinary coalition.

Load-bearing premise

The marginalist characterization rests on the lemma that every zero-sum function on the vertices of a connected acyclic directed graph can be expressed as the divergence of an edge-weight function; if that representation is unavailable, Theorem 1 does not go through.

Editorial extensions

If this is right

Efficiency, linearity, and the null-agent axiom are exactly equivalent to unitary flows on $\Gamma_{\mathcal{F}}$, so every such value can be studied through its flow rather than through its coefficients.
Two-step flow methods are precisely those with zero flow on non-relevant edges and flow proportionality on relevant edges; checking these two conditions on a flow decides whether it arises from the two-step procedure.
For unrestricted feasible sets within each element, the three axioms determine a unique value with explicit closed-form coefficients; this is the profile-level analogue of the classical configuration value.
When the coalition configuration is a partition, or each set system is regular, every profile-level flow method induces a coalition-level flow method on reachable coalitions, so configuration-style values extend to restricted cooperation.
Maximal-path constructions, where a grand profile is formed step by step, are special two-step flow methods, so Shapley-style equal treatment of maximal paths sits inside the characterized family.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the characterization holds, the same flow calculus should extend to any coalition configuration whose set systems are normal; the only obstacle is the projection condition in Theorem 7, which is sufficient but not shown necessary.
The worth of a coalition profile can be read as modeling externalities between elements of the configuration; flow proportionality then gives a testable separability condition on how cross-element externalities are priced.
One natural extension is to replace uniform splitting of the flow among entering agents by other sharing rules and ask whether the same two flow axioms still characterize the resulting two-step methods.
Because Theorem 5 characterizes two-step methods solely by flow axioms, an algorithm could check whether a given allocation rule is two-step by computing its induced flow and verifying the two axioms on the product digraph.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main result to know about this paper is that Theorem 5, the advertised characterization of two-step flow methods, is false as stated. The two axioms (null flow on non-relevant edges, flow proportionality) constrain the aggregate edge flow Λ, but a flow method is a marginalist value whose coefficients λ_i sum to Λ on each edge; the axioms put no restriction on how that aggregate flow is split among the agents entering on that edge. The sufficiency proof only shows that from a two-step flow you can construct a two-step flow method—one such method—not that an arbitrary method inducing that flow must be two-step. The stress-test counterexample works: take M={1,2}, P1={1,2,3}, F1={∅,{1},{2,3},{1,2,3}}, P2={4}, F2={∅,{4}}, and a two-step aggregate flow with half on each lower edge; then split the first edge 90/10 between agents 2 and 3 and the reverse on the second. The aggregate flow is unchanged, so both axioms hold, but the per-agent coefficients are not of the product form required by Lemma 4. The reader's report is too kind here: it flags the Lemma 1 gap but misses this more serious failure.

What the paper does well: the framework of generalized coalition configurations is natural and cleanly executed. Theorems 1 and 2 extend Aguilera et al. to product digraphs; Lemma 1's proof gap (the claimed sink whose deletion preserves connectedness does not always exist, e.g., two sources feeding one sink) is real but the lemma is true and easily repaired, so Theorem 1 survives. Theorem 3's flow decomposition is sound, modulo minor typos in case (b). Theorem 4, characterizing two-step flows at the aggregate level, appears correct—the two axioms do single out flows that factor as Λ_M·Λ_q. Theorem 6 gives a nice new configuration-type value for the unrestricted case, and Theorem 7 is a useful reduction result with the caveat that Γ*_{F0} is not always the covering digraph. There is no circularity or fitting; the axiomatic derivations are honest.

The soft spot is exactly the leap from Theorem 4 to Theorem 5. The distinction between flow and flow method is where the characterization fails, and Theorem 5 is the paper's central advertised claim. Repairing it requires an additional axiom that constrains the per-agent split (for instance, requiring the split on a given lower edge to be independent of the upper support), or re-scoping the result to two-step flows rather than flow methods.

Who this is for: researchers working on Owen-type values, restricted cooperation, and flow methods in cooperative game theory. The framework and the aggregate-level results are worth having, but the paper needs substantial revision before it should appear. I would send it to a serious referee, but with explicit instructions to scrutinize Theorem 5 and the method-level interpretation of the flow axioms.

Referee Report

2 major / 4 minor

Summary. The paper introduces cooperative games with generalized coalition configuration: a coalition configuration P covering the agent set, with each element equipped with a normal set system F_q, and a coalition profile function on the Cartesian product of the feasible coalitions. It defines marginalist values and flow methods on this domain, shows (Theorem 1 and Theorem 2) that linearity plus the null-agent axiom characterizes marginalist values and that adding efficiency characterizes flow methods. It then constructs Owen-type two-step flow methods via an upper game on the elements of the configuration and lower games on each set system. Theorem 3 shows the induced aggregate flow decomposes as a product of an upper flow and lower flows. Section 7 introduces two axioms, Null flow for non-relevant directed edges and Flow proportionality, proves in Theorem 4 that a flow on the product digraph is a two-step flow iff it satisfies these axioms, and then claims in Theorem 5 that this also characterizes two-step flow methods at the value level. Section 6 also gives a configuration-type value on the unrestricted domain, and Section 8 studies when a flow method on coalition profiles induces a flow method on reachable coalitions. The paper is clearly written and the flow-level decomposition result is interesting, but the central value-level characterization in Theorem 5 is false as stated.

Significance. If corrected, the framework would be a useful and fairly general extension of Weber's flow methods and Owen's two-step procedure to restricted cooperation and coalition configurations. The paper contains detailed axiomatic derivations and no parameter fitting or circular reliance on the target results; Theorem 4's characterization of decomposable flows is a substantive and plausibly correct contribution. However, the advertised value-level characterization of two-step flow methods is not established: Theorem 5 is false under the paper's own definition of a flow method. Since this is the headline result announced in the abstract and introduction, the current manuscript cannot be accepted. A revised version that reframes the main characterization at the level of flows, or that adds an additional axiom governing the split of edge flows among agents, could be worth reconsidering.

major comments (2)

[Section 7, Theorem 5] The sufficiency direction of Theorem 5 is false. A flow method is not determined by its induced aggregate flow: definition (4) only fixes the sum of the coefficients λ_i over i ∈ Q(K,K'), not the individual coefficients. The two axioms Null flow for non-relevant directed edges and Flow proportionality are conditions on the aggregate flow Λ only, so they are inherited by any flow method with the same aggregate flow, regardless of how the flow is split among the agents entering together on an edge. In contrast, Lemma 4 shows that a two-step flow method has coefficients of the form λ_i^0(K_{S,K_q}, K_{S,K'_q}) = Λ^M(R_{S\q}, R_S) λ_i^q(K_q,K'_q). Consequently, for a fixed lower edge (K_q,K'_q) and two different upper supports S,S', the ratio λ_i^0(e_S)/λ_i^0(e_{S'}) must be the same for every i ∈ Q; the aggregate axioms impose no such constraint. Concretely, let M={1,2}, P1={1,2,3}, F1={∅,{1},{2,3},{1,2,3}}, P2={4}, F2={∅,{4}}. Let Λ^M(∅,{1})=Λ^M({2},{1,2})=1/2 and let the lower flow on F1 be Λ^1(∅,{2,3})=Λ^1({2,3},{1,2,3})=1, all other lower flows being zero. This defines a two-step aggregate flow Λ satisfying both axioms. Now define a flow method Φ with induced flow Λ but with coefficients on e^1=(∅,∅)→({2,3},∅) given by λ_2=0.45, λ_3=0.05, and on e^2=(∅,{4})→({2,3},{4}) given by λ_2=0.05, λ_3=0.45, completing the remaining coefficients consistently with Λ. Then Λ satisfies the two axioms, but Φ is not a two-step flow method: Lemma 4 would force λ_2(e^1)/λ_2(e^2)=λ_3(e^1)/λ_3(e^2)=1, whereas the chosen ratios are 9 and 1/9. Thus Theorem 5 must be corrected; the axioms characterize two-step flows (Theorem 4), not two-step flow methods. A value-level statement would require an additional axiom on the per-agent split, such as the equal-splitting condition (5).
[Section 5, Lemma 1] The proof of Lemma 1 contains a genuine gap. In the induction step it asserts that there exists a vertex j0 whose deletion leaves the acyclic digraph connected and such that j0 is the tail of no directed edge. The sink-deletion claim is false: in the connected acyclic digraph with vertices a, b, s and edges a→s, b→s, the only sink is s, and deleting s leaves two components. The statement of the lemma is true, and the gap appears repairable (for example, by a different inductive or network-flow argument), but as written the proof of Theorem 1 is incomplete. Since Theorem 1 is the characterization of marginalist values that underlies the rest of the paper, this needs to be fixed.

minor comments (4)

[Section 6, proof of Theorem 3] In Case (b), the expression Λ^ΦM(µ(R_{µ(K)\q}, R_{µ(K)}) contains an extraneous 'µ(' and should read Λ^ΦM(R_{µ(K)\q}, R_{µ(K)}); in Case (c), the summation ∑_{(K'_q,K_q)∈E_i^F} should be over E_{F_q}, not E_i^F.
[Section 5, proof of Theorem 2] In the computation of γ(K), the intermediate displayed line omits the minus sign in front of the outgoing-flow term; the preceding and following lines have the correct sign.
[Section 7, Flow proportionality] The phrase 'each pair {S,S'} ⊆ M such that q ∈ S∩S′' is imprecise: S and S' are subsets of M, so the axiom should say 'each pair of subsets S,S′ ⊆ M with q ∈ S∩S′'.
[References] References [20] and [29] list the same paper (van den Brink, Khmelnitskaya, and van der Laan, 'An Owen-type value for games with two-level communication structure') and should be merged into a single entry.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's characterization theorems are derived from explicit axioms and constructions, with no fitted inputs and no circular self-citation chain.

full rationale

The derivation chain is self-contained. Theorem 1 derives the marginalist form (3) from Linearity and the Null agent axiom via the divergence representation in Lemma 1, constructing the coefficient functions from the value evaluated on Dirac games; no target conclusion is assumed. Theorem 2 adds Efficiency and derives the conservation equations and unit flow from (11), and conversely derives Efficiency from flow conservation, so the flow-method family is characterized rather than defined by the conclusion. The two-step construction is an explicit procedure (12)-(13); Lemma 4 and Theorem 3 compute its coefficient functions and induced flow from the constituent flows, while Theorems 4 and 5 characterize two-step flows by Null flow for non-relevant directed edges and Flow proportionality. The proof of Theorem 4 constructs the factor flows Lambda^M and Lambda^q from the given flow using equations (23) and (24), so the characterization is an axiomatic equivalence rather than a definitional identity. Citations to Owen, Weber, and Aguilera et al. provide context and prior restricted-cooperation cases but are not load-bearing for the new arguments. The manuscript does contain a genuine proof gap that is nonetheless non-circular: Lemma 1's proof asserts the existence of a sink whose deletion keeps a connected acyclic digraph connected, which is false in general, and the sufficiency direction of Theorem 5 has been challenged because flow-level axioms may not determine the per-agent split when multiple agents enter on the same edge. These are correctness concerns, not circular reductions, so the circularity score remains 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The paper introduces a purely mathematical framework. There are no free parameters fitted to data. The axioms are explicitly stated and are either standard domain assumptions (normality, covering) or new axioms introduced to characterize the two-step flow methods. The new constructs are mathematical definitions, not empirically testable entities.

assumptions (5)

domain assumption For each q ∈ M, (P_q, F_q) is a normal set system: {∅, P_q} ⊆ F_q.
Section 3 defines generalized coalition configuration with normal set systems; this restricts the domain of games.
domain assumption The coalition configuration P covers N: ∪_{q∈M} P_q = N.
Section 3, following Albizuri et al. [2].
domain assumption Value axioms: Linearity, Null agent, Efficiency.
Section 5 axioms; these are normative requirements on values, not derived.
ad hoc to paper Flow axioms: Null flow for non-relevant edges, Flow proportionality.
Section 7; these are introduced specifically to characterize two-step flows.
standard math Standard graph-theoretic facts (max-flow/min-cut, covering digraph properties).
Used in Lemma 1 and cut arguments.

invented entities (3)

Generalized coalition configuration
purpose: Model restricted cooperation inside each element of a coalition configuration
Introduced as a definition in Section 3, not an empirical postulate.
Coalition profile
purpose: Unit of cooperation, a tuple of feasible coalitions, one per element of the configuration
Section 3 definition.
Two-step flow method
purpose: Value constructed via an Owen-type two-step procedure
Section 6 definition.

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Cite this review

Pith. "Pith review of Flow methods for cooperative games with generalized coalition configuration." pith.science (2026). https://pith.science/paper/OUWP3JWV

@misc{pith2026241113684,
  author       = {Pith},
  title        = {Pith review of: Flow methods for cooperative games with generalized coalition configuration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OUWP3JWV}},
  note         = {Machine review of arXiv:2411.13684}
}

read the original abstract

This paper introduces the class of cooperative games with generalized coalition configuration. This new class of games corresponds to cooperative games with coalition configuration and restricted cooperation. A coalition configuration is a collection of coalitions covering the agent set. The restriction of cooperation between agents is represented by a set system on each element of the coalition configuration. A coalition profile is a list of feasible coalitions, one for each element of the coalition configuration. A coalition profile function associates a worth with each coalition profile. Based on this framework, we define and axiomatically characterize marginal values whose coefficients induce a unitary flow on the product digraph obtained from these set systems. Next, we propose a two-step procedure, inspired by Owen's procedure, to construct flow methods as above. Then, we show that the associated flow is decomposable into two flows. Finally, we use two axioms to characterize the flows that can be decomposed in this way, and hence the flow methods constructed using our procedure.

Figures

Figures reproduced from arXiv: 2411.13684 by the authors.

**Figure 2.** To the left, the product digraph ΓF = (F, EF ). To the right, the subdigraph Γ3 F = (V 3 F , E3 F ) with four components. Given a product digraph ΓF = (F, EF ), a coalition profile function v : F −→ R assigns a worth v(K) ∈ R to each K ∈ F, where, by convention, v(∅M) = 0. This describes a situation in which agents organized themselves into a family of coalitions not necessarily pairwise disjoint (the coalition conf… view at source ↗

**Figure 3.** The directed hypercube ΓFM for M = {1, 2}. Remark 2. For the sake of simplicity, we use the notation KS,Kq for Kq = ∅ and q ∈ S, to mean that the coordinate q is active even if it is equal to the empty coalition. So, we still consider that the support of KS,Kq is S. In the special case where S = {q}, KS,Kq = (Kq,(∅M)−q). In case, Kq = Pq, simply denote this coalition profile by KS. In particular, K∅ = ∅M and KM = P.… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: To the left, the subdigraph of ΓF = (F, EF ) induced by the set of relevant directed edges ER F . To the right, the subdigraph of Γ3 F = (V 3 F , E3 F ) induced by relevant directed edges ER F ∩ E3 F . Notation To avoid any confusion between the coalition profiles in F…

**Figure 5.** Figure 5: The subdigraph of ΓF induced by ER F and the hypercube ΓFM of dimension 2. above, [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

**Figure 6.** Figure 6: In red, the flow passing through the relevant directed ed [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

**Figure 7.** Figure 7: The digraph of reachable coalitions Γ∗ F0 = (F 0 , E∗ F0 ) obtained from F. Remark 8. The digraph of reachable coalitions Γ ∗ F0 = (F 0 , E∗ F0 ) of [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]

**Figure 8.** Figure 8: To the left, Γ∗ F0 , to the right the directed graph ΓF0 of the covering relation of (F 0 , ⊆). 27 [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Albizuri, MJ.: Games with externalities: games in coali tion conﬁguration function form, Mathe- matical Methods of Operations Research , 72:171-186 (2010)

work page 2010

[2] [2]

Albizuri, MJ., Aurrekoetxea, J., Zarzuelo, J.M.: Conﬁg uration values: Extensions of the coali- tional Owen value, Games and Economic Behavior , 57: 1-17 (2006)

work page 2006

[3] [3]

Albizuri, MJ., Aurrekoetxea, J.: Coalition conﬁgurati ons and the Banzhaf index, Social Choice and Welfare , 26: 571-596 (2006)

work page 2006

[4] [4]

Albizuri, MJ., Vidal-Puga, J.: Values and coalition con ﬁgurations, Mathematical Methods of Operations Research, 81:3-26 (2015)

work page 2015

[5] [5]

Albizuri, MJ., Masuya, S., Zarzuelo, J.M.: Values for re stricted games with externalities, Group Decision and Negotiation , 33: 351-369, (2024)

work page 2024

[6] [6]

Algaba, E., Bilbao, J.M., Borm, P., L´ opez, J.: The posit ion value for union stable systems, Mathematical Methods of Operations Research , 52: 221-236 (2000)

work page 2000

[7] [7]

Algaba, E., Bilbao, M., Borm, P., L´ opez, J.: The Myerson value for union stable structures, Mathematical Methods of Operations Research , 54: 359-371 (2001)

work page 2001

[8] [8]

Algaba E., van den Brink R., Dietz C.: Power measures and s olutions for games under precedence constraints, Journal of Optimization Theory and Applications , 172: 1008-1022 (2017)

work page 2017