A New Value for Cooperative Games on Intersection-Closed Systems

Martin \v{C}ern\'y

REVIEW 3 major objections 5 minor 13 references

A New Value for Cooperative Games on Intersection-Closed Systems

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper introduces a unique fair-payoff rule for intersection-closed incomplete cooperative games and shows it equals the average Shapley value over all positive completions.

desk verdict The UD-value is a real, axiomatically grounded allocation rule for incomplete cooperative games, but the paper's expected-Shapley interpretation quietly depends on a prior choice, and Examples 2, 3, and Table 1 contain fixable errors. read the letter →

arxiv 2501.05169 v1 pith:WYFGF3JA submitted 2025-01-09 cs.GT

Martin Černý This is my paper · ORCID

classification cs.GT MSC 91A12

keywords cooperativegamesincompleteShapleyvalueintersection-closedsetsystemsuniform-dividendpositiveextensionsaxiomaticcharacterizationallocationrules

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 the uniform-dividend value (UD-value), a payoff-allocation rule for cooperative games in which only some coalitions' worths are known. Its central claim is that when the known coalitions form an intersection-closed system containing the grand coalition, the rule is uniquely determined and equals the expected Shapley value across every positive completion of the partial data. This matters because incomplete games otherwise leave the fair split ambiguous; the UD-value offers a canonical choice that does not force a pessimistic guess about missing coalition values. The paper also characterizes the rule axiomatically and compares it with two existing values, finding that it typically behaves more like the zero-surplus R-value than like the IC-value.

What carries the argument

The engine is the closure operator $c_K(T)=\bigcap\{S\in K: T\subseteq S\}$. Two coalitions are indistinguishable when they have the same closure, and the UD-value forces their dividends $\delta^K_v(S)$ to be equal; the known coalition values then become linear equations in these dividends. When $K$ is intersection-closed, $c_K(T)\in K$ for every $T$, so the equations can be solved by ascending inclusion-minimal known coalitions, yielding uniqueness. For the expected-value theorem, the set of positive extensions decomposes into simplices indexed by indistinguishability classes, each simplex distributing a fixed total surplus $\Delta_v(S)$ among the coalitions in $C(S)$, and averaging the vertices is exactly the uniform dividend rule.

What would settle it

Enumerate every positive extension of a small intersection-closed incomplete game, for instance the 3-player example in the introduction with $v(\{1,2\})$ unspecified, compute the uniform average of the Shapley payoff vectors, and compare it with the vector produced by Definition 2; any mismatch would refute Theorem 3.

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Extended reading notes

Core claim

On a $P_n$-extendable intersection-closed incomplete game $(N,K,v)$, the UD-value $\Phi^K(v)$ is the unique allocation rule obtained by requiring indistinguishable coalitions—those with the same closure under $K$—to receive equal dividends. Theorem 3 states that $\Phi^K(v)=\mathbb{E}_{w\sim P_n(v)}[\phi(w)]$, where the expectation is uniform over all positive games agreeing with the known values. Thus the rule is exactly the average Shapley value over all nonnegative-surplus completions of the incomplete data. The paper further proves uniqueness (Proposition 2) and gives two axiomatic characterizations (Propositions 7 and 8) modelled on Shapley's axioms.

Load-bearing premise

The load-bearing premise is that every positive extension is equally likely, so the uniform average is the right way to summarize the incomplete data; change that prior and the expected-Shapley interpretation changes.

Editorial extensions

If this is right

For any intersection-closed incomplete game with $N\in K$, the UD-value is a single well-defined payoff vector, so no arbitrary choice of missing coalition values is needed before allocating the grand coalition's worth.
For every $P_n$-extendable such game, the UD-value is literally the average of the Shapley values of all positive completions of the data, giving it a neutral, data-only interpretation.
The R-value and the UD-value are both Shapley values of positive extensions, whereas the IC-value is in general only monotone, which the experiments confirm by showing the first two values lie closer together.
Any rule on intersection-closed incomplete games satisfying Efficiency, Additivity, IC-null player, IC-equal treatment, and Equality must coincide with the UD-value (Proposition 7).
For non-intersection-closed systems the UD-value can fail to be unique, but the paper's sampling experiments indicate the unique case becomes common as the number of players grows.

Reading between the lines

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

Editorial inference: with a non-uniform prior over missing coalition values, the same construction yields a prior-weighted expected Shapley value; Proposition 2's uniqueness would survive, but Theorem 3's equality would not.
Editorial inference: nothing restricts the equal-dividends idea to Shapley; applying the same indistinguishability classes to other complete-game solution concepts would produce a family of analogous values for incomplete games.
Editorial inference: the large share of unique cases outside intersection-closed systems points to an open structural question—characterize the set systems for which the dividend-to-payoff map is injective on the solution space of Definition 2.

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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 core contribution is worth separating from the sloppiness around it. The UD-value is defined by a linear system that equalizes dividends among indistinguishable coalitions, and Proposition 2 convincingly shows that this system has a unique solution when the known-coalition set is intersection-closed and contains N. The two axiomatic characterizations in Section 4.2 are plausible and follow the Shapley template with weakened null-player and equal-treatment axioms. None of that is trivial, and the connection to the average value in [7] is acknowledged honestly rather than hidden.

The main theorem, however, is softer than the headline. Theorem 3 says the UD-value equals the expected Shapley value over all positive extensions, and the paper justifies this by saying every extension is equally likely. But the extension set is an infinite union of simplices; "equally likely" is not a probability measure on that set. The proof implicitly picks the uniform (Lebesgue) distribution on each simplex, which is exactly what makes the average the centroid. Change the prior over surplus vectors and the expectation changes while the UD-value does not. So the expected-Shapley reading is a modeling assumption, not an unassailable interpretation. The uniqueness result survives, but the paper should either state the uniform-prior assumption explicitly or rephrase the claim as an average over the polytope's vertices. The stress-test note lands.

The examples have real errors. In Example 2, the stated value UD^{K-1}_3(v_{-1})=1 is actually the R-value; the UD-value is 1.5, and with that correction the balanced-contributions violation in the example disappears. In Example 3, the coalition P={1,2} is not a coalition of partners under the paper's own definition, because S={2,3} has S\P={3} not in K and v({2,3})=1, violating condition 2. Table 1's count of set systems for n=6 is wrong by nearly nineteen orders of magnitude: 2^62 is about 4.6e18, not 8.5e37. These are fixable, but they currently undermine the comparative and empirical sections. The numerical experiments also lack code or data, and the sampling procedure for random intersection-closed systems is not described in enough detail to reproduce.

For a reader in cooperative game theory, the UD-value is a genuine addition: it gives a unique, axiomatically characterized rule on intersection-closed systems and a centroid interpretation that can be made precise. The citation pattern looks normal, and the self-citation to [7] is for context, not load-bearing. I would send this to a referee, but I would ask the author to correct the examples and table, clarify the prior in Theorem 3, and make the experiments reproducible. As it stands, the central theoretical claim is probably correct, but the paper needs careful revision before it is publishable.

Referee Report

3 major / 5 minor

Summary. The paper introduces the uniform-dividend value (UD-value) for incomplete cooperative games, defined by equalizing the dividends of coalitions that have the same closure in the known set system K. It proves that the UD-value is unique when K is intersection-closed and contains N, derives two axiomatic characterizations, and states in Theorem 3 that the UD-value equals the expected Shapley value over all positive extensions. The paper also compares the UD-value with the R-value and the IC-value, gives examples intended to show violations of fairness-type axioms, and reports numerical experiments on uniqueness and on the distances between the three values.

Significance. The UD-value is a natural and clearly motivated allocation rule for incomplete cooperative games, and the paper's main structural results---uniqueness for intersection-closed systems, the two axiomatizations, and the explicit connection to averaging over extensions---are valuable. The proof of Proposition 2 is a clean triangular-system argument, and the axiomatic characterizations are genuine additions to the literature. The central interpretative claim in Theorem 3 is currently stated with an undefined probability measure, and two of the counterexamples in the axiomatic comparison are incorrect as written; these problems are local and repairable, so the core construction appears sound.

major comments (3)

[Section 3, Theorem 3 (Eq. (8))] The right-hand side E_{w∼Pn(v)}[ϕ(w)] is not well-defined as written because Pn(v) is an infinite set (a product of simplices by Proposition 4) and no probability measure on it is specified. The sentence 'Without additional assumptions on the incomplete game, every Pn-extension is equally likely' does not define a uniform measure on an infinite set, and the proof's step 'the average is given by an average of its vertices' implicitly selects the uniform Lebesgue measure on each simplex. This is a substantive modeling assumption: a different prior over the surplus vectors, such as a non-uniform Dirichlet distribution or a point mass at an extreme point, changes the expected Shapley value while leaving the UD-value unchanged. The theorem is correct only after explicitly defining the measure and should be restated as holding under that measure; otherwise the central interpretative claim that the UD-value is 'the' expected Shapley value over all positive extensions is not justified. The uniqueness and axiomatic results are unaffected by this issue.
[Section 4.2, Example 2 (Eq. (29))] The computed value UD^{K−1}_3(v−1)=1 is not the UD-value of the restricted game. On player set {2,3} with K−1={∅,{2},{2,3}} and v({2,3})=2, Definition 2 gives δ({3})=δ({2,3}) and the equation δ({2})+δ({3})+δ({2,3})=2, so δ({3})=δ({2,3})=1 and UD^{K−1}_3(v−1)=1+1/2=3/2, not 1. The value 1 is the R-value of this restricted game. With the correct value, the balanced-contributions equality in Eq. (29) holds for this instance, so the example does not demonstrate a violation by the UD-value. Please correct the computation or provide another counterexample.
[Section 4.2, Example 3] The coalition P={1,2} is not a coalition of partners under the definition given immediately above. For S={2,3}∈K, P\S={1}≠∅ and S\P={3}∉K, so condition 2 applies and requires v(T)=0 for every T∈K with T⊆S; however T={2,3} itself lies in K, is a subset of S, and has v({2,3})=1. Hence the Symmetric Partnership axiom is not applicable to this game, and the example does not establish a violation by the UD-value. If a different notion of partnership was intended, the definition needs to be changed accordingly.

minor comments (5)

[Section 3, Proof of Theorem 3] The proof heading reads 'Proof of Theorem 5' but should refer to Theorem 3.
[Section 4.2, Eq. (19)] In the IC-Equal treatment axiom, 'f_j^K(w)' should be 'f_j^K(v)'.
[Section 4.3] The paragraph on the distance from the Equal Division Rule refers to 'Figure 2' when presenting the ranking frequencies; the displayed figure is Figure 5.
[Proposition 4, Eq. (10)] In the displayed equation, the final summation has 'd_w(S)' as the summand; it should be 'd_w(T)'.
[Section 1] There is a typo in the sentence beginning 'THis means that...'; it should be 'This means that...'.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity; Theorem 3 is a conditional equivalence resting on an explicit (if under-specified) uniform-prior assumption, not on fitted parameters or self-citation.

full rationale

The derivation chain is self-contained. Definition 2 defines the UD-value by a linear system with the equal-dividend condition, and Proposition 2 proves uniqueness for intersection-closed K by triangular elimination; no external result is needed. Theorem 3 is proved directly: the set of Pn-extensions is a product of simplices (Proposition 4), and taking the arithmetic mean over each simplex gives constant expected dividends on each indistinguishability class, which is exactly the UD-dividend vector. This is a genuine equivalence, but it is conditional on the modeling assumption stated in Section 3: 'Without additional assumptions on the incomplete game, every Pn-extension is equally likely to represent the actual underlying game.' The paper does not derive that uniform prior from the axioms; it is an input. The theorem therefore interprets the UD-value as the expected Shapley value under that chosen prior rather than as a prediction forced by the definition. I found no fitted parameter renamed as a prediction and no load-bearing self-citation: references [5] and [7] are contextual and are not premises of the proofs. The only caveat is that the uniform distribution over the infinite set Pn(v) is not formally defined; this is a rigor/interpretation concern, not a circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The central definition is a system of linear equations with no fitted constants. The main interpretative claim rests on a uniform prior over positive extensions, which is stated but not derived. No extra physical or mathematical entities are introduced beyond the UD-value itself.

assumptions (3)

domain assumption Every Pn-extension is equally likely when defining the expected Shapley value
Stated in Section 3 before Definition 4; the equality in Theorem 3 is an expectation under this uniform prior. A different prior would break the interpretation.
domain assumption Positive games (nonnegative surpluses) are the only admissible underlying games
The paper restricts extensions to Pn, excluding games with negative dividends; this is a modeling choice for incomplete games (Section 3, Definition 4).
standard math Closure operator properties from lattice theory (Lemma 1, cited to [8])
Used in Proposition 2 and Proposition 4 to reduce sums over all subsets to sums over K via cK(T).

invented entities (1)

Uniform-dividend value (UD-value)
purpose: Allocation rule for incomplete cooperative games; assigns payoffs by equalizing dividends of coalitions with the same closure
Introduced in Definition 2; its properties are derived within the paper, so there is no independent falsifiable handle outside the paper itself.

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Cite this review

Pith. "Pith review of A New Value for Cooperative Games on Intersection-Closed Systems." pith.science (2026). https://pith.science/paper/WYFGF3JA

@misc{pith2026250105169,
  author       = {Pith},
  title        = {Pith review of: A New Value for Cooperative Games on Intersection-Closed Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WYFGF3JA}},
  note         = {Machine review of arXiv:2501.05169}
}

read the original abstract

We introduce a new allocation rule, the uniform-dividend value (UD-value), for cooperative games whose characteristic function is incomplete. The UD-value assigns payoffs by distributing the total surplus of each family of indistinguishable coalitions uniformly among them. Our primary focus is on set systems that are intersection-closed, for which we show the UD-value is uniquely determined and can be interpreted as the expected Shapley value over all positive (i.e., nonnegative-surplus) extensions of the incomplete game. We compare the UD-value to two existing allocation rules for intersection-closed games: the R-value, defined as the Shapley value of a game that sets surplus of absent coalition values to zero, and the IC-value, tailored specifically for intersection-closed systems. We provide axiomatic characterizations of the UD-value motivated by characterizations of the IC-value and discuss further properties such as fairness and balanced contributions. Further, our experiments suggest that the UD-value and the R-value typically lie closer to each other than either does to the IC-value. Beyond intersection-closed systems, we find that while the UD-value is not always unique, a surprisingly large fraction of non-intersection-closed set systems still yield a unique UD-value, making it a practical choice in broader scenarios of incomplete cooperative games.

Figures

Figures reproduced from arXiv: 2501.05169 by the authors.

**Figure 1.** Average ℓ1-norms of differences between the R-value, UD-value, and IC-value for all intersection-closed set systems with n = 3 players. Each set system is represented on the x-axis by its integer encoding, and the y-axis shows the average norm of the differences between the values, computed over 100 randomly generated games games with values selected uniformly from [0, 1]. all possible intersection-closed set system… view at source ↗

**Figure 2.** Frequencies of each ℓ1-norm difference being the smallest, second largest, or largest for all intersection-closed set systems with n = 3, 4, 5, 6 players. The y-axis shows the frequency of each rank, aggregated over sampled or exhaustively evaluated systems, highlighting consistent trends as n increases. 0.0-0.1 0.1-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.6-0.7 0.7-0.8 0.8-0.9 0.9-1.0 1.0-1.1 1.1-1.2 1.2+ 0 5 10 15 20… view at source ↗

**Figure 3.** Histograms of the average differences between value pairs for [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Average ℓ1-norms of distances between the R-value, UD-value, IC-value and the Equal Division Rule for 3 players. Each set system is represented on the x-axis by its integer encoding, while the y-axis shows the average distance between the values and ED, computed over 1…

**Figure 5.** Figure 5: Frequencies of each value’s distance from the ED rule being the closest, second [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]

**Figure 6.** Figure 6: Histograms of the average distances from the ED rule for [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]

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Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [7]

J. Bok, M. ˇCern´ y, 1-convex extensions of incomplete cooperative games and the average value, Theory and Decisions (2023). doi:10.1007/ s11238-023-09946-8

work page 2023

[2] [1]

R. B. Myerson, Graphs and cooperation in games, Mathematics of Op- erations Research 2 (3) (1977) 225–229. doi:10.1287/moor.2.3.225

doi:10.1287/moor.2.3.225 1977

[3] [2]

van den Brink, Games with a permission structure: A survey on generalizations and applications, Top 25 (1) (2017) 1–33

R. van den Brink, Games with a permission structure: A survey on generalizations and applications, Top 25 (1) (2017) 1–33

work page 2017

[4] [3]

Calvo, E

E. Calvo, E. Guti´ errez-L´ opez, The value in games with restricted coop- eration, Tech. Rep. 0115, University of Valencia, ERI-CES (2015)

work page 2015

[5] [4]

Masuya, M

S. Masuya, M. Inuiguchi, A fundamental study for partially defined cooperative games, Fuzzy Optimization Decision Making 15 (1) (2016) 281–306

work page 2016

[6] [5]

ˇCern´ y, J

M. ˇCern´ y, J. Bok, D. Hartman, M. Hlad ´ ık, Positivity and convexity in incomplete cooperative games, Annals of Operations Research (2024). doi:10.1007/s10479-024-06082-6

work page doi:10.1007/s10479-024-06082-6 2024

[7] [6]

B´ eal, I

S. B´ eal, I. Moyouwou, E. R´ emila, P. Solal, Cooperative games on in- tersection closed systems and the Shapley value, Mathematical Social Sciences 104 (2020) 15–22. doi:10.1016/j.mathsocsci.2020.01.005

work page doi:10.1016/j.mathsocsci.2020.01.005 2020

[8] [8]

B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, 2nd Edition, Cambridge University Press, Cambridge, UK, 2002

work page 2002