{"id":"50d4d3a3-d778-4031-bbcb-31b2a6f1d6f5","arxiv_id":"2501.05169","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For incomplete cooperative games on intersection-closed set systems, the uniform-dividend value is uniquely defined and coincides with the expected Shapley value over all positive extensions.","lead":"A new rule, the uniform-dividend value, allocates payoffs in cooperative games where some coalition values are unknown, by splitting each ambiguous surplus equally among indistinguishable coalitions. For intersection-closed families of known coalitions, the rule is unique and equals the average Shapley value over all positive completions of the data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3's equality to the expected Shapley value rests on an undefined uniform prior over an infinite extension set; any other prior changes the expectation.","rationale":"The central theorem is mathematically sound once a measure on Pn(v) is specified; however, the paper never defines this measure. The informal 'equally likely' cannot define a distribution on an infinite set, and the proof's uniform-simplex centroid is a specific choice. This is load-bearing because the main interpretation of the UD-value as an expected Shapley value requires that choice. The uniqueness result and axiomatic characterizations are independent and appear correct. Peripheral errors (Examples 2–3, Table 1) are real but do not affect the core argument. The reader's conditional verdict should stand, with a request to state the measure explicitly and discuss sensitivity to the prior.","tokens_in":16224,"tokens_out":14053,"duration_ms":131431,"concrete_test":"Consider N={1,2}, K={∅,{1},{1,2}}, v({1})=0, v({1,2})=1. Compute the UD-value and compare it with the expected Shapley value under the distribution on Pn(v) that assigns probability 1 to the extension with d({2})=1, d({1,2})=0. The UD-value is (0.25, 0.75), while the point-mass expectation is (0, 1). If the two vectors differ, the equality in Theorem 3 is measure-dependent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem (Theorem 3) claims ΦK(v) = E_{w∼Pn(v)}[ϕ(w)], but Pn(v) is an infinite set (a product of simplices). The paper never defines a probability measure on Pn(v); the phrase 'every Pn-extension is equally likely' (Section 3) is not well-defined for an infinite set. The proof implicitly selects the uniform Lebesgue measure on each simplex, which yields the centroid dividends. This is an extra modeling assumption, not a consequence of the axioms or of 'no additional assumptions.' If a different prior over the surplus vectors is used—e.g., a point mass at one vertex, or a Dirichlet prior with non-uniform weights—the expected Shapley value changes, while the UD-value remains the same. Consequently, the central interpretative claim that the UD-value is 'the' expected Shapley value over all positive extensions is prior-dependent. The uniqueness result (Proposition 2) survives, but the equality in Theorem 3 is a statement about a particular, unstated measure.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":58,"tokens_out":17199,"duration_ms":218424,"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":[{"comment":"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":"Section 3, Theorem 3 (Eq. (8))"},{"comment":"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":"Section 4.2, Example 2 (Eq. (29))"},{"comment":"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.","section":"Section 4.2, Example 3"}],"minor_comments":[{"comment":"The proof heading reads 'Proof of Theorem 5' but should refer to Theorem 3.","section":"Section 3, Proof of Theorem 3"},{"comment":"In the IC-Equal treatment axiom, 'f_j^K(w)' should be 'f_j^K(v)'.","section":"Section 4.2, Eq. (19)"},{"comment":"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.","section":"Section 4.3"},{"comment":"In the displayed equation, the final summation has 'd_w(S)' as the summand; it should be 'd_w(T)'.","section":"Proposition 4, Eq. (10)"},{"comment":"There is a typo in the sentence beginning 'THis means that...'; it should be 'This means that...'.","section":"Section 1"}],"recommendation":"major_revision","confidential_remarks":"The central construction appears sound and the paper fits the journal's scope. The main revision burden is the undefined uniform distribution in Theorem 3 and the two incorrect counterexamples; all three issues are fixable without changing the core definition or the axiomatic framework."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nFor 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.","headline":"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.","tokens_in":16939,"tokens_out":6552,"would_cite":true,"duration_ms":60683,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A12"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["cooperative games","incomplete games","Shapley value","intersection-closed set systems","uniform-dividend value","positive extensions","axiomatic characterization","allocation rules"],"falsifier":"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.","tokens_in":15913,"feed_emoji":"⚖️","tokens_out":6643,"duration_ms":59576,"temperature":0.7,"pith_summary":"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.","feed_headline":"Unique payoff rule averages Shapley values over positive extensions","feed_subtitle":"Averaging every positive completion of partial data gives a unique fair split for intersection-closed games.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the IC-value for intersection-closed systems and supplies the two axiomatizations that the paper adapts to the UD-value.","marker":"[6]"},{"why":"Defines the average value over all admissible extensions, the object Theorem 3 connects to the UD-value.","marker":"[7]"},{"why":"Shapley's classic characterization provides the Efficiency, Additivity, null-player, and equal-treatment template used in Proposition 7.","marker":"[12]"},{"why":"Defines the R-value as the Shapley value of the zero-surplus extension, the main comparison baseline.","marker":"[3]"},{"why":"Introduces the partially defined cooperative game model that frames the incomplete-game setting.","marker":"[4]"},{"why":"Supplies positivity and extendability conditions for incomplete games used in Proposition 5.","marker":"[5]"},{"why":"Supplies the closure-operator facts used to rewrite the surplus equations in the uniqueness proof.","marker":"[8]"}],"fun_headline_variants":["New cooperative payoff: average Shapley over positive completions","Uniform-dividend value: unique fair split for partial games","Intersection-closed games get unique payoff via expected Shapley","UD-value averages Shapley values for incomplete games"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["New cooperative payoff: average Shapley over positive completions","Uniform-dividend value: unique fair split for partial games","Intersection-closed games get unique payoff via expected Shapley","UD-value averages Shapley values for incomplete games"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000501,"raw_usage":{"total_tokens":2442,"prompt_tokens":930,"completion_tokens":1512,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":1451}},"tokens_in":546,"tokens_out":1512,"duration_ms":11818,"temperature":1.0,"reasoning_tokens":1451,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:15:33.954219+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"B´ eal, I","cited_arxiv_id":null,"evidence_quote":"Introduces the IC-value for intersection-closed systems and supplies the two axiomatizations that the paper adapts to the UD-value."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the average value over all admissible extensions, the object Theorem 3 connects to the UD-value."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shapley's classic characterization provides the Efficiency, Additivity, null-player, and equal-treatment template used in Proposition 7."},{"cited_title":"Calvo, E","cited_arxiv_id":null,"evidence_quote":"Defines the R-value as the Shapley value of the zero-surplus extension, the main comparison baseline."},{"cited_title":"Masuya, M","cited_arxiv_id":null,"evidence_quote":"Introduces the partially defined cooperative game model that frames the incomplete-game setting."},{"cited_title":"ˇCern´ y, J","cited_arxiv_id":null,"evidence_quote":"Supplies positivity and extendability conditions for incomplete games used in Proposition 5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the closure-operator facts used to rewrite the surplus equations in the uniqueness proof."}],"review_version":1}