Trust in Shapley: A Cooperative Quest for Global Trust in P2P Network

The paper contains no numbered result asserting computational hardness, tractability, parameterized complexity, or approximation. The proposed typed \(q\)-matrix construction is a plausible new high-multiplicity computation for \(u_G\), but it is not a continuous analogue of a qualifying computational result and omits the distinctive external term \(w_G\). Therefore bit (a) fails.

The objection that survived

The \((\mu,q)\) construction does not encode the coalition-specific outsider minima of \(w_G\), so it is not a full-game mirror; the proponent already concedes this and restricts the question to \(u_G\).

What the mirror covers

The proposed mirror covers only the internal trust game \(u_G\) and Proposition 3.6's Shapley formula; it leaves the full game \(v_G\), external game \(w_G\), and the paper's structural propositions and experiments outside.

Open questions for a prover

The case FOR (proponent)

Strictly speaking, this paper has no qualifying named computational anchor. Proposition 3.3 proves monotonicity and superadditivity, Proposition 3.4 gives a core allocation, and Proposition 3.6 gives a closed-form Shapley value for the internal trust game. None states that a computational problem is in \(P\), NP-hard, FPT, or similar. The sampling discussion in Section 5 is neither a numbered result nor a formal complexity theorem. Thus, under the programme’s literal anchor rule, the paper cannot support a formal positive verdict.

The strongest honest salvage is the following conditional lead, anchored structurally by Proposition 3.6, proved in this paper.

Define Normalized Type-Internal Trust-Shapley\(_\infty\). An instance consists of finitely many peer types \(R=\{1,\ldots,\tau\}\), rational masses \(\mu_r>0\) with \(\sum_r\mu_r=1\), and a rational row-stochastic matrix \(q\), where \(q_{rs}\) is the fraction of a type-\(r\) peer’s normalized trust directed toward type \(s\). Peers of the same type have identical trust shares toward every type. The output is the exact type-level global-trust vector \(z\), or equivalently the ranking of types by aggregate trust.

For a denominator-clearing integer \(N\), create \(N\mu_r\) clones of type \(r\). Give a type-\(r\) clone trust \(q_{rs}/(N\mu_s)\) to each type-\(s\) clone when \(r\ne s\), and distribute \(q_{rr}\) uniformly among the other type-\(r\) clones. This is a genuine high-multiplicity regime: \(N\) named peers are represented by \(\tau\) complete trust types.

Proposition 3.6 then gives every type-\(r\) clone the same internal-game Shapley value, namely \(z_r=\frac12\left(1+q_{rr}+\frac{1}{\mu_r}\sum_{s\ne r}\mu_s q_{sr}\right)\). The aggregate trust mass of the type is \(\Phi_r=\mu_r z_r\). Hence the continuous problem is solvable exactly in \(O(\tau^2)\) rational arithmetic, so its expected classification is Class A. This is not merely replacing a named peer by a weight: clearing denominators reconstructs the finite clone graph and preserves the paper’s internal trust game and Shapley semantics.

The mirror is recognizable to the authors because it retains their players, directed local trust, normalized transaction-derived trust, and Shapley-based global reputation. Its scope is deliberately limited to the internal trust component covered by Proposition 3.6. It does not claim to continuize the entire external-minimum game.

That limitation is also the weakest point. The paper’s distinctive contribution is the external trust term in Definition 3.1, while the proposed mirror handles only the part for which Proposition 3.6 already supplies a formula. Extending the full game requires deciding what the minimum over outsiders means for a divisible coalition and how the external term should scale under cloning; with dense normalized graphs it becomes lower-order, so simply dropping it would not be faithful. The atomless Shapley value also has to be defined as a normalized finite-clone limit rather than assumed to exist automatically.

Natural follow-up questions are whether the full external-trust Shapley vector has a polynomial-time type formulation, whether sparse typed trust graphs preserve the external term, and how much adversarial mass is required to change the type ranking. These are worthwhile ChoCo-style questions, but they are new computational extensions rather than results already anchored by this paper.

The case AGAINST (opponent, writing after the proponent)

The proponent concedes the decisive point: this paper has no qualifying computational anchor. Proposition 3.3 proves game-theoretic properties, Proposition 3.4 exhibits a core allocation, and Proposition 3.6 gives an algebraic Shapley formula. None states the complexity or approximability of a computational problem. The sampling paragraph is neither a numbered result nor a formal algorithmic theorem. Thus the proposed task is manufactured after the fact, rather than being a continuous version of a computational result in the paper.

Even granting that broader standard, the proposed mirror preserves only the paper’s least distinctive component. Proposition 3.6 concerns the internal game \(u_G\), whose Shapley value is already just half the sum of incoming and outgoing edge weights. The paper’s distinctive construction is the external term \(w_G\), involving a minimum over the particular outsiders of a coalition. The proposed \(q\)-matrix and masses specify only aggregate type-to-type trust. They do not determine those coalition-specific minima. The clone construction resolves this by imposing a highly special complete, block-uniform graph; that is a new exchangeability assumption, not a consequence of the paper’s model.

The failure is structural. In the paper, trust is a relation between named peers, and the external term depends on which individual peers remain outside a coalition. If that relational topology is included in the type, then different peers generally have different types and the high-multiplicity compression disappears. If it is omitted, two graphs with the same type masses and aggregate matrix \(q\) can have different external games and different Shapley values. Preserving the missing information would require an additional matching, coupling, or graphon-like object, making the problem a network-continuum re-modelling rather than a population mirror.

There is also a genuine continuum degeneration under the proponent’s own clone scaling. Since each source distributes unit trust over \(N\mu_s\) targets, an individual edge has weight \(O(1/N)\). The internal game has total value \(O(N)\), whereas the external minimum contributes only \(O(1)\): each of \(O(N)\) targets receives a minimum edge weight of order \(O(1/N)\). Consequently the external contribution vanishes in the per-peer Shapley limit. The natural continuum limit therefore deletes precisely the paper’s novel external-trust mechanism. Multiplying the external term by \(N\), or replacing individual minima by minima over type averages, could prevent that disappearance, but either change alters Definition 3.1 and its trust normalization.

A sparse repeated-motif construction could retain an \(O(N)\) external term, but then the essential peer-to-peer matching must be supplied in addition to the type distribution. That is potentially interesting network theory, yet it is not a faithful mirror of a named result here. The internal-only construction is valid as a symmetric special case, but it is merely Proposition 3.6 rewritten with rational multiplicities; the full model either loses its distinctive term or requires a new relational model.

So the negative case is strong under ChoCo’s stated source rule and reasonably strong on fidelity. It is not an impossibility claim about all future graphon or mean-field trust models: such a successor could be worthwhile. It would, however, be a new computational programme, not a continuous mirror supported by this paper.

The adversarial triple: the proponent anchors on up to three named results; the opponent sees that case and must defeat every anchor; the judge decides which case convinced it. These are the pipeline’s own outputs, generated by tools/triple_run.py — no human edited them. The paper’s own text is not reproduced here beyond the quoted statement above.