Answer: Universal EVS convergence is false. On inputs admitting the specified EVS limit, evaluation is polynomial-time; the limit is admissible and has the policy-determined minimum objective \(J=0\).
Theorem 4 — class: algorithmic
PDF text-layer excerpt — equations and symbol order may be approximateGiven any solvable v, there is Eπ∼U(Π(N))[SDEVS(v, π)] ≤Eπ∼U(Π(N))[SDM(v, π)] where M is an arbitrary anonymous mechanism satisfying SF, OIR, I4EA and MOS. We further argue that no mechanism can optimize SD for every single (v, π). Although critical players may initially be symmetric when the marginal player joins, this symmetry can be disrupted as additional players arrive. For example, in Example 4, players A, B, and C are symmetric in the local order [A, B, C], but this symmetry no longer holds in the order [A, B, C, D].
statement extracted from the paper’s text layer; consult the paper for the exact wording
Given a finite set of complete contribution roles \(T\), rational role masses \(\mu\), and a succinct monotone Boolean value functional \(V:[0,1]^T\to\{0,1\}\) on arrived role masses, define finite replicas by \(v_n(S)=V((|S\cap N_t|/n)_{t\in T})\), with \(|N_t|/n\to\mu_t\). Under uniform arrival orders and tagged-agent delay deviations, does EVS converge to a measurable per-capita policy that minimizes the limit of \(n\,\mathbb{E}_{\pi}[\sum_i(\phi_i(v_n,\pi)-\mathrm{SV}_i(v_n))^2]\) among anonymous policies satisfying the limiting SF, OIR, I4EA, and MOS conditions, and can that policy be computed from the chosen encoding of \(V\)?
A society is given by its support \(T\): the list of its \(\tau\) complete contribution roles with positive binary-rational masses \(\mu_t\), where \(\sum_t\mu_t=1\); \(\tau\) is the support size, not the number of possible roles. Roles determine all contributions, with no additional player attributes.
The input value functional \(V:[0,1]^T\to\{0,1\}\) is an AND/OR circuit of size \(m\), whose leaves are rational affine threshold tests with nonnegative coefficients and explicitly specified strict or weak comparisons. Assume \(V(0)=0\) and \(V(\mu)=1\); let \(L\) be the total input bit length.
For every positive integer \(n\) with \(n\mu_t\in\mathbb Z\), form \(N=\bigsqcup_tN_t\), with \(|N_t|=n\mu_t\), and \(v_n(S)=V((|S\cap N_t|/n)_t)\). Promise that every such replica is solvable: no winning coalition \(S\) has a unique indispensable player \(i\) with \(v_n(\{i\})=0\), where indispensable means \(v_n(S\setminus\{i\})=0\).
An online policy uses only the ordered arrived coalition and its restricted game, assigns nonnegative cumulative shares summing to its value at every prefix, and is anonymous under relabelings preserving the game. SF means expected shares under uniform orders equal Shapley values; OIR means cumulative shares never decrease; I4EA means moving one player later while fixing everyone else's relative order cannot increase her final share; MOS means an earlier symmetric player receives a weakly larger share. These requirements apply also to restricted games; symmetry means equal marginal effects on every coalition excluding the two players.
Define EVS as follows. At the first winning prefix \(P\), let \(i\) be its last arrival and \(C=\{j\in P:v_n(P\setminus\{j\})=0\}\). Let \(\widehat P=Q\cup\{i\}\), where \(Q\) is the shortest prefix preceding \(i\) for which \(v_n(Q\cup\{i\})=1\), and put \(r=|\{j\in\widehat P:v_n(\widehat P\setminus\{j\})=0\}|\). Give \(i\) share \(1/r\), each other member of \(C\) share \((1-1/r)/(|C|-1)\), and everyone else zero; if \(|C|=1\), give \(i\) one. Shares are zero before winning and frozen afterward.
Generate uniform orders using independent marks \(U_i\sim\mathrm{Uniform}[0,1]\); ties have probability zero. A per-capita policy is a nonnegative measurable field \(q_t(u,s)\) for \(0\leq u\leq s\leq1\), describing cumulative payment density at cutoff \(s\) to role \(t\) arriving at \(u\).
Define convergence to \(q\) by \(\mathbb E[n^{-1}\sum_{i:U_i\leq s}(n\phi_i(v_n|_{A_s},\pi|_{A_s})-q_{t(i)}(U_i,s))^2]\to0\) for every fixed \(s\in[0,1]\), where \(A_s=\{i:U_i\leq s\}\). A limiting admissible policy is such a limit of anonymous online policies satisfying SF, OIR, I4EA and MOS exactly on solvable finite games; this specifies the limiting axioms, including tagged-player delay deviations.
For an admissible policy with witnessing sequence \(\phi^{(n)}\), use the objective \(J=\lim_{n\to\infty}n\mathbb E_\pi[\sum_i(\phi_i^{(n)}(v_n,\pi)-\mathrm{SV}_i(v_n))^2]\), when it exists; all limits run through the exact replicas above. Computational evaluation means approximating \(\int_a^b q_t(u,1)\,du\) to additive error \(\varepsilon\), for rational \(0\leq a<b\leq1\) and \(\varepsilon>0\).
Does EVS converge to an admissible measurable per-capita policy whose objective limit exists, depends only on that policy, and is minimal among admissible policies with existing objective limits?
On inputs admitting such an EVS limit, is its computational evaluation polynomial-time in \(m,\tau,L\), the query bit length and \(\log(1/\varepsilon)\), or is it NP-hard?
For finite solvable monotone Boolean games, EVS satisfies SF, OIR, I4EA and MOS and minimizes expected squared Shapley distance among anonymous mechanisms satisfying these axioms. [Ge, Zhang and Zhao](https://arxiv.org/html/2511.11023v1).
Finite monotone Boolean games admit SF, OIR and I4EA simultaneously exactly on the solvable domain. [Ge et al.](https://aamas.csc.liv.ac.uk/Proceedings/aamas2024/pdfs/p651.pdf). No result settling the continuous problem specified here was located.
Universal convergence is false. On inputs admitting the specified EVS limit, evaluation is polynomial-time. In fact, those inputs admit a precise characterization: near full participation, the circuit must reduce to unanimity on one nonempty set of roles. For every such input, EVS is admissible and its objective is \(J=0\), hence minimal.
I use the intended monotone interpretation of the leaves: each tests \(\mathbf a\cdot x\ge b\) or \(\mathbf a\cdot x>b\), with \(\mathbf a\ge0\). An affine constant is absorbed into \(b\).
The requirement of convergence at every fixed cutoff, including the cutoff at which the population crosses a threshold, is essential to this result.
Fix a cutoff \(s\). Write \(\Phi_{n,i}(s)\) for player \(i\)’s cumulative share in the restricted game at \(s\), extending it by zero for unarrived players.
Suppose a sequence converges to \(q\) as specified. Then, for every role \(t\),
\[ \text{[Equation omitted to respect the requested delimiter format.]} \]
More explicitly, the following convergence holds in mean square:
\(R_{n,t}(s):=\sum_{i\in N_t}\Phi_{n,i}(s)\ \longrightarrow\ \mu_t\int_0^s q_t(u,s)\,du.\)
Here and below, all limits run through exact replicas.
To prove this, first observe that \(q_t(\cdot,s)\) is square-integrable on \([0,s]\). Indeed, every finite share lies between zero and one, so \(n\Phi_{n,i}(s)\) is bounded. If \(q_t(\cdot,s)\) had infinite squared integral, the expected squared approximation error would be infinite for every replica.
Define \(f_t(u)=q_t(u,s)\mathbf1_{\{u\le s\}}\) and
\(\widehat R_{n,t}(s)=n^{-1}\sum_{i\in N_t}f_t(U_i).\)
Cauchy–Schwarz gives
\(\mathbb E|R_{n,t}(s)-\widehat R_{n,t}(s)|^2\le \mu_t\,\mathbb E\!\left[n^{-1}\sum_{i:U_i\le s}(n\Phi_{n,i}(s)-q_{t(i)}(U_i,s))^2\right]\longrightarrow0.\)
Independence of the marks gives
\(\mathbb E\widehat R_{n,t}(s)=\mu_t\int_0^s q_t(u,s)\,du,\qquad \operatorname{Var}(\widehat R_{n,t}(s))=\frac{\mu_t}{n}\operatorname{Var}(f_t(U))\longrightarrow0.\)
This proves the assertion.
Summing over roles and using efficiency, we obtain a necessary condition:
At every fixed \(s\), the random value \(v_n(A_s)\) must converge in mean square to a deterministic constant.
In particular, convergence is impossible if the probability that \(A_s\) wins has a limit strictly between zero and one.
There is also a general positive conclusion, independent of the classification below.
For any admissible policy, put \(g_t(u)=q_t(u,1)\). Mean-square convergence implies
\(n\mathbb E\sum_i(\Phi_{n,i}(1))^2\longrightarrow\sum_t\mu_t\int_0^1g_t(u)^2\,du.\)
For justification, regard \(n\Phi_{n,i}(1)\) and \(g_{t(i)}(U_i)\) as vectors with squared norm \(\mathbb E[n^{-1}\sum_i(\cdot)^2]\). Their difference tends to zero in norm, while the latter vector has squared norm exactly \(\sum_t\mu_t\int_0^1g_t^2\).
Players within a role are symmetric, so write their common Shapley value as \(\mathrm{SV}_{n,t}\). SF and the role-total convergence above imply
\(n\mathrm{SV}_{n,t}\longrightarrow\int_0^1g_t(u)\,du.\)
Expanding the squared distance and using SF therefore proves
\(J(q)=\sum_t\mu_t\left[\int_0^1g_t(u)^2\,du-\left(\int_0^1g_t(u)\,du\right)^2\right].\)
Thus every admissible policy has an existing, finite objective limit, and that limit depends only on its full-participation field, up to equality almost everywhere.
Take one role, with mass \(\mu_1=1\), and let
\(V(x)=\mathbf1_{\{x\ge1/2\}}.\)
The replica is the symmetric threshold game with winning threshold \(k_n=\lceil n/2\rceil\).
Every replica satisfies the solvability promise. A winning coalition of size \(k_n\) has all its members indispensable; a larger winning coalition has none. Unique indispensability occurs only when \(k_n=1\), in which case that player’s singleton wins.
EVS gives each of the first \(k_n\) arrivals share \(1/k_n\), and everyone else zero.
At the fixed cutoff \(s=1/2\),
\(v_n(A_{1/2})=\mathbf1_{\{\operatorname{Binomial}(n,1/2)\ge\lceil n/2\rceil\}}.\)
The central limit theorem gives
\(\mathbb P(v_n(A_{1/2})=1)\longrightarrow1/2.\)
The necessary condition from Section 1 fails. Hence EVS cannot converge to a field of the specified kind.
This failure persists despite an existing finite objective limit. Symmetry gives \(\mathrm{SV}_i=1/n\), and direct calculation yields
\(n\mathbb E\sum_i(\phi_i^{\mathrm{EVS}}-\mathrm{SV}_i)^2=\frac{n}{k_n}-1\longrightarrow1.\)
So finite optimality and existence of the numerical objective limit do not imply the requested policy convergence.
For each threshold leaf, define
\(A=\mathbf a\cdot\mu,\qquad D=\{t:a_t>0\}.\)
Construct a monotone Boolean circuit \(H\) on subsets of roles by retaining the original AND/OR gates and replacing its leaves as follows.
| Original leaf | Condition | Replacement |
|---|---|---|
| \(\mathbf a\cdot x\ge b\) | \(b<A\) | Constant \(1\) |
| \(\mathbf a\cdot x\ge b\) | \(b=A\) | Test \(D\subseteq S\) |
| \(\mathbf a\cdot x\ge b\) | \(b>A\) | Constant \(0\) |
| \(\mathbf a\cdot x>b\) | \(b<A\) | Constant \(1\) |
| \(\mathbf a\cdot x>b\) | \(b\ge A\) | Constant \(0\) |
An empty conjunction is constant \(1\). Since \(V(\mu)=1\), we have \(H(T)=1\).
For a feasible vector \(0\le x\le\mu\), define its completed roles by
\(F(x)=\{t:x_t=\mu_t\}.\)
Two properties will be used.
Global domination: \(V(x)\le H(F(x))\).
Indeed, a weak saturated leaf satisfies
\(\mathbf a\cdot x\ge\mathbf a\cdot\mu\quad\Longleftrightarrow\quad x_t=\mu_t\text{ for every }t\in D,\)
because all terms \(a_t(\mu_t-x_t)\) are nonnegative. Every other replacement either preserves the leaf or increases it to constant \(1\). AND/OR gates preserve this inequality.
Local equality: there exists \(\delta\in(0,1)\) such that
\(V(x)=H(F(x))\quad\text{whenever}\quad (1-\delta)\mu_t\le x_t\le\mu_t\text{ for every }t.\)
For every leaf with \(b<A\) and \(A>0\), choose \(\delta\) small enough that \((1-\delta)A>b\). There are finitely many leaves, so one positive \(\delta\) works simultaneously. Leaves with \(A=0\) are constant. Saturated and impossible leaves already have the asserted exact behavior.
Define the deterministic ray threshold
\(c=\inf\{s\in[0,1]:V(s\mu)=1\}.\)
Monotonicity and the endpoint assumptions imply \(c\in[0,1]\).
Suppose \(0<c<1\), and write
\(X_n(c)=(|A_c\cap N_t|/n)_t.\)
The coordinates are independent scaled binomial variables. The multivariate central limit theorem gives
\(\sqrt n(X_n(c)-c\mu)\ \Rightarrow\ Z,\)
where \(Z\) is a centered Gaussian vector with positive diagonal covariance
\(\operatorname{diag}(\mu_t c(1-c))_{t\in T}.\)
A nonconstant threshold leaf with \(b\ne cA\) has an asymptotically fixed truth value. An active leaf with \(b=cA\) asymptotically tests the sign of \(\mathbf a\cdot Z\). Strict and weak comparisons have the same limit because this nonzero Gaussian linear form equals zero with probability zero.
Substitute these limiting tests into the circuit, obtaining a Boolean function \(G(Z)\). Its discontinuities lie in finitely many Gaussian-null hyperplanes, so
\(\mathbb P(V(X_n(c))=1)\longrightarrow p:=\mathbb P(G(Z)=1).\)
All active tests are false on the strictly negative orthant and true on the strictly positive orthant. The resulting circuit values are respectively zero and one: they agree with \(V((c-\eta)\mu)\) and \(V((c+\eta)\mu)\) for sufficiently small positive \(\eta\).
The Gaussian has positive density everywhere. Consequently, both orthants have positive probability, and \(0<p<1\).
Section 1 therefore rules out convergence for any efficient policy to a specified field at this cutoff.
Suppose \(c=0\). Then \(V(s\mu)=1\) for every \(s>0\).
For all sufficiently large replicas,
\(V((1/n)_{t\in T})=1.\)
To verify this, compare leaf truth values near the origin. Positive thresholds remain false for sufficiently large \(n\); negative thresholds remain true; weak zero thresholds are true; and strict zero thresholds with nonzero coefficients are true when every coordinate is positive. These are exactly the leaf values on a sufficiently small positive multiple of \(\mu\).
Consider the event that the first \(\tau\) arrivals contain exactly one player from every role. Its probability is
\(\frac{\tau!\prod_t(n\mu_t)}{n(n-1)\cdots(n-\tau+1)}\longrightarrow\tau!\prod_t\mu_t>0.\)
On this event, the first winning prefix has at most \(\tau\) players. EVS’s final shares are supported on that prefix and sum to one, so Cauchy–Schwarz gives
\(\sum_i(\phi_i^{\mathrm{EVS}})^2\ge1/\tau.\)
It follows that \(n\mathbb E\sum_i(\phi_i^{\mathrm{EVS}})^2\) diverges. Mean-square convergence at full participation would make this quantity bounded, as proved in Section 1. This is a contradiction.
Therefore, every input admitting the specified EVS limit has \(c=1\). By local equality along the ray, this also implies \(H(\varnothing)=0\).
Theorem. Within the stated solvability promise, EVS admits the specified limit if and only if \(H\) has exactly one inclusion-minimal winning role set \(B\), and \(B\ne\varnothing\).
Choose \(\delta\) from Section 3 and put \(s_0=1-\delta/2\). Let \(G_n\) be the event that, for every role \(t\),
\((1-3\delta/4)\mu_t\le |A_{s_0}\cap N_t|/n\le(1-\delta/4)\mu_t.\)
For each role, the count is binomial with \(n\mu_t\) trials and success probability \(s_0\). Hoeffding’s inequality and a union bound give
\(\mathbb P(G_n^c)\le2\tau\exp(-n\delta^2\min_t\mu_t/8).\)
For sufficiently large \(n\), on \(G_n\):
Since \(H(\varnothing)=0\), all those latter coalitions lose. Thus both the first winning prefix and EVS’s minimal critical prefix occur after \(s_0\). Their values, and the values after deleting any one member, are determined exactly by \(H\) on completed roles.
For each role, let \(M_{n,t}=\max_{i\in N_t}U_i\), its completion time. Independence across roles and
\(\mathbb P(n(1-M_{n,t})>x)=(1-x/n)^{n\mu_t}\longrightarrow e^{-\mu_t x}\)
show that the scaled completion gaps converge jointly to independent exponential variables with rates \(\mu_t\).
For any nonempty proper role set \(B\), the event
\(E_{n,B}=\{\max_{t\in B}M_{n,t}<\min_{t\notin B}M_{n,t}\}\)
therefore has a limiting probability strictly greater than zero. The comparison boundary has probability zero in the exponential limit, and the required strict ordering holds on a nonempty open rectangle.
Now let \(B\) be an inclusion-minimal winning set of \(H\). On \(E_{n,B}\cap G_n\), precisely the roles in \(B\) are complete at the first winning prefix. Every proper subset of \(B\) loses. Hence its indispensable players are exactly all players in those roles.
The minimal critical prefix has the same completed-role set: it cannot win with a proper subset of \(B\), and no outside role has completed yet. Its indispensable players are again exactly all players in \(B\).
Writing \(\mu(B)=\sum_{t\in B}\mu_t\), EVS therefore gives exactly \(1/(n\mu(B))\) to each player in these roles and zero to everyone else.
Suppose \(H\) has two distinct minimal winning sets \(B,B'\). They are incomparable, so choose \(t\in B\setminus B'\). On events of probabilities bounded away from zero, the final role total \(R_{n,t}(1)\) equals respectively
\(\mu_t/\mu(B)\quad\text{and}\quad0.\)
Intersecting with \(G_n\) preserves those positive probability bounds. These distinct values prevent convergence in mean square to a deterministic constant, contrary to Section 1.
Together with Section 4, this proves necessity.
Suppose \(H\) has a unique minimal winning set \(B\ne\varnothing\). Monotonicity then implies
\(H(S)=1\quad\Longleftrightarrow\quad B\subseteq S.\)
Indeed, every winning set contains a minimal winning subset, which must be \(B\); every superset of \(B\) wins.
Global domination shows that every winning coalition in every replica contains all players belonging to roles in \(B\).
On \(G_n\), the first winning prefix and minimal critical prefix consequently have the same indispensable players: exactly all players in roles \(B\). EVS gives them the uniform allocation
\(\psi_{n,i}=\mathbf1_{\{t(i)\in B\}}/(n\mu(B)).\)
Thus \(\phi^{\mathrm{EVS}}=\psi_n\) on \(G_n\). Everywhere, nonnegative final shares sum to one, and hence
\(\sum_i(\phi_i^{\mathrm{EVS}}-\psi_{n,i})^2\le2+2/(n\mu(B)).\)
It follows that
\(n\mathbb E\sum_i(\phi_i^{\mathrm{EVS}}-\psi_{n,i})^2 \le(2n+2/\mu(B))\mathbb P(G_n^c)\longrightarrow0.\)
Define the field, on \(0\le u\le s\le1\), by
\(q_t(u,s)=\begin{cases}1/\mu(B),&s=1\text{ and }t\in B,\\0,&\text{otherwise}.\end{cases}\)
This is nonnegative and measurable. The preceding estimate proves convergence at \(s=1\).
For every fixed \(s<1\), winning requires all \(n\mu(B)\) players in those roles to have arrived. Therefore,
\(\mathbb P(v_n(A_s)=1)\le s^{n\mu(B)}.\)
Since the squared sum of nonnegative shares is at most their total value,
\(\mathbb E\!\left[n^{-1}\sum_{i:U_i\le s}(n\Phi_{n,i}^{\mathrm{EVS}}(s))^2\right] \le n s^{n\mu(B)}\longrightarrow0.\)
This proves convergence at every cutoff and completes the characterization.
The replicas are monotone Boolean games and are solvable by promise. Every restriction is also solvable: a forbidden winning coalition in a restriction would be the same forbidden coalition in the original game.
The finite EVS theorem consequently supplies SF, OIR, I4EA and MOS on every relevant game and restriction. EVS is anonymous because its construction uses only the ordered restricted game. Thus EVS itself provides the required finite witnesses, including the task’s tagged-player delay requirements. See [Ge, Zhang and Zhao, Theorem 2](https://arxiv.org/html/2511.11023v1).
For the field proved above, \(q_t(u,1)\) is constant in \(u\) for every role. The objective formula in Section 1 gives \(J=0\).
Every competing objective is a limit of nonnegative squared distances and is therefore nonnegative. EVS is consequently minimal among admissible policies. Its objective depends only on the policy, and the same independence holds for every admissible policy by Section 1.
The characterization does not require enumeration of winning role sets.
Construct \(H\) by exact rational comparisons and define
\(B_*=\{t\in T:H(T\setminus\{t\})=0\}.\)
A role belongs to \(B_*\) exactly when every winning role set contains it. Indeed, a winning set omitting \(t\) exists exactly when \(T\setminus\{t\}\) wins, by monotonicity.
Consequently:
Thus the admitting-input criterion is precisely
\(B_*\ne\varnothing\quad\text{and}\quad H(B_*)=1.\)
Each saturated leaf becomes a conjunction of at most \(\tau\) role variables. Constructing \(H\), evaluating it on the \(\tau\) sets \(T\setminus\{t\}\), and evaluating it on \(B_*\) all take polynomial time. The rational sums and comparisons have polynomial bit complexity in the stated input length.
On an admitting input, the requested integral is exactly
\(\int_a^bq_t(u,1)\,du= \begin{cases}(b-a)/\mu(B_*),&t\in B_*,\\0,&t\notin B_*.\end{cases}\)
This rational number has polynomial encoding length and can be computed exactly. Hence evaluation is polynomial-time in the requested parameters, with no approximation-dependent numerical work.
⟨structured verdict parsed — rendered above⟩
The argument is sound in substance. It establishes failure of universal convergence, characterizes the inputs admitting an EVS limit, and proves polynomial-time evaluation on those inputs. I found one minor omission concerning the range of a probability formula, and no blocking issue.
The prover explicitly interprets each leaf as a lower threshold, either \(\mathbf a\cdot x\ge b\) or \(\mathbf a\cdot x>b\), with \(\mathbf a\ge0\). This is the natural monotone interpretation intended by the task. Absorbing the affine constant into \(b\) is valid.
The proof correctly works on the feasible box \(0\le x\le\mu\), although the circuit is defined on the larger box \([0,1]^T\). Every replica coalition produces a vector in the feasible box.
The square-integrability argument is valid. For each fixed replica, efficiency and nonnegativity bound every share by one, so the scaled share is bounded by \(n\). Since a tagged player’s mark is uniform, a non-square-integrable role slice would force infinite expected squared approximation error. Positive role masses ensure every role is represented in each exact replica.
The Cauchy–Schwarz bound has the correct normalization: the number of players in role \(t\) is \(n\mu_t\), giving the factor \(\mu_t\).
Independence gives the stated mean and variance of \(\widehat R_{n,t}(s)\). Consequently, the claimed mean-square convergence of each role total follows.
Summing finitely many role totals and using efficiency proves the necessary deterministic-value condition. In particular, a Boolean value whose winning probability converges to a number strictly between zero and one cannot satisfy that condition.
The norm argument correctly shows convergence of the scaled second moment of the allocation. Its limiting norm is finite by the square-integrability already established.
Players within each role are symmetric because roles determine all contributions. SF therefore identifies the expected role total with \(n\mu_t\mathrm{SV}_{n,t}\), yielding the asserted limit of \(n\mathrm{SV}_{n,t}\).
Expanding the objective and using SF gives the stated formula
\[
\text{}
\]
equivalently, in the requested inline notation,
\(J(q)=\sum_t\mu_t\left[\int_0^1g_t(u)^2\,du-\left(\int_0^1g_t(u)\,du\right)^2\right]\).
Both terms converge as claimed. Thus the conclusion that every admissible policy has a finite objective limit determined by its full-participation field is justified. This conclusion uses the strong mean-square convergence specified in the task, rather than merely weak convergence of allocations.
The one-role threshold example satisfies all input assumptions.
The solvability check is correct. At a winning coalition of exactly \(k_n\) players, all members are indispensable; above that size, none is indispensable. When \(k_n=1\), an indispensable player’s singleton wins, so the forbidden configuration still does not occur.
The EVS allocation is correctly identified as uniform over the first \(k_n\) arrivals: both relevant critical-player counts equal \(k_n\).
At cutoff \(s=1/2\), the binomial central limit theorem gives limiting winning probability \(1/2\), for both even and odd replica sizes. The deterministic-value condition therefore fails.
The objective calculation also checks:
\(n\mathbb E\sum_i(\phi_i^{\mathrm{EVS}}-\mathrm{SV}_i)^2=n/k_n-1\to1\).
This is a valid nondegenerate counterexample and directly answers the universal convergence question negatively.
Every entry of the replacement table is correct on the feasible box.
For a weak saturated leaf, equality with its maximum feasible value requires completion of every role having a positive coefficient. Positivity of the masses and nonnegativity of the coefficients justify this equivalence.
Global domination follows because each replacement either agrees with the original leaf on feasible vectors or raises its truth value, and AND/OR gates preserve coordinatewise Boolean inequalities.
Local equality is also justified. A sufficiently small common positive \(\delta\) makes every sub-saturated leaf true throughout the specified neighborhood. Leaves with zero coefficient vector are constant, and saturated or impossible leaves already have exact replacements. Finiteness of the circuit permits one common \(\delta\).
For \(0<c<1\), the stated multivariate central limit theorem has the correct covariance. The coordinates are independent binomial counts, and every diagonal variance is positive.
Nonactive leaves stabilize. Each active nonconstant leaf has a nonzero Gaussian linear form, so its boundary has probability zero. The finite union of these boundaries remains Gaussian-null, justifying convergence of the circuit’s winning probability.
The negative and positive orthants produce the circuit values immediately below and above \(c\). Monotonicity and the definition of \(c\) make these values zero and one. Both orthants have positive Gaussian probability, so the limiting winning probability lies strictly between zero and one. The exclusion follows.
For \(c=0\), the comparison of leaf values near the origin correctly establishes that one player from every role forms a winning coalition for sufficiently large replicas.
The probability of obtaining all roles among the first \(\tau\) arrivals is calculated correctly and has a positive limit. On this event, EVS distributes value over at most \(\tau\) players, so its squared-share sum is at least \(1/\tau\). The resulting divergence contradicts the bounded second moment required by convergence at full participation.
These arguments exhaust the cases other than \(c=1\). Local equality then correctly implies \(H(\varnothing)=0\).
The Hoeffding bound is correct. Each count deviates from its mean by at most \(n\mu_t\delta/4\), producing exponent \(n\mu_t\delta^2/8\); the union bound gives the displayed estimate.
The three asserted consequences hold for sufficiently large \(n\). In particular, the margin between the lower count bound and the local-equality boundary accommodates deletion of one player. The upper bound leaves at least two unarrived players per role.
Monotonicity ensures that losing at \(A_{s_0}\cup\{i\}\) also excludes any shorter critical prefix. Thus the first winning prefix and minimal critical prefix are both in the region where \(H\) controls their values and deletion tests.
For multiple minimal winning sets, the completion-time limit and independence correctly give positive limiting probabilities for the required role orderings, subject to the minor range omission identified below. Distinct minimal winning sets are incomparable and therefore proper subsets of \(T\), so the ordering events used in this argument are covered.
On each such event, the first winning prefix and minimal critical prefix complete exactly the chosen minimal winning set. Minimality makes precisely the players in those roles indispensable. The resulting uniform EVS allocation is correct.
Choosing a role in the difference of two minimal winning sets yields two distinct final role totals, each occurring with probability bounded away from zero. This contradicts deterministic role-total convergence.
For a unique nonempty minimal winning set, the equivalence \(H(S)=1\) if and only if \(B\subseteq S\) is valid. Global domination therefore forces every winning coalition to contain every player from the roles in \(B\).
On the regularity event, both critical sets are exactly those players. The comparison allocation and its squared-error bound are correct. Multiplication by \(n\) is still dominated by the exponentially small exceptional-event probability.
For every fixed \(s<1\), winning requires all \(n\mu(B)\) required players to arrive. The probability bound \(s^{n\mu(B)}\) and the resulting mean-square error estimate are correct, including \(s=0\). The separately defined field is measurable and satisfies the prescribed convergence at all cutoffs.
Restriction preserves solvability: any forbidden coalition in a restriction would have the same values and indispensability relations in the original game.
The cited finite theorem applies. [Theorem 2](https://arxiv.org/html/2511.11023v1) establishes SF, OIR, I4EA and MOS for WVS on solvable monotone Boolean games with weakly decreasing weights. The paper defines EVS using positive constant weights, which satisfy that hypothesis and yield the task’s allocation formula.
The explicit EVS construction is invariant under relabeling and uses only the ordered restricted game. It therefore supplies anonymous online witnesses. Applying the theorem on restrictions covers the task’s restricted-game requirements and player-delay comparisons.
The full-participation field is constant in arrival time within each role, so the verified objective formula gives \(J=0\). Nonnegativity of every competing objective establishes minimality without requiring an additional continuous optimization theorem.
The identification of \(B_*\) is correct: a role is required by every winning set precisely when deleting it from \(T\) makes \(H\) lose.
The test \(B_*\ne\varnothing\) and \(H(B_*)=1\) is equivalent to existence of a unique nonempty minimal winning set.
Constructing the saturation circuit requires rational arithmetic of polynomial bit complexity. Its role conjunctions have at most \(\tau\) variables each. The stated number of circuit evaluations is polynomial; enumeration of minimal winning sets is unnecessary.
The integral formula follows from the established field. Its rational numerator and denominator have polynomial encoding length, so exact computation suffices for every requested additive accuracy. The conditional polynomial-time evaluation claim is established.
Where: Section 5, completion-time survival formula.
Issue: The identity
\(\mathbb P(n(1-M_{n,t})>x)=(1-x/n)^{n\mu_t}\)
omits its domain. It is valid for \(0\le x\le n\). For negative \(x\), the left side is one while the displayed right side exceeds one.
Minimum clarification: State that the identity is used for \(0\le x\le n\), and that the exponential limit is taken for each fixed \(x\ge0\). This is an expositional correction; the subsequent convergence and ordering-event arguments are unaffected.
⟨structured verdict parsed — rendered above⟩
Tier: machine-proved, adversarially reviewed by an independent model. No human has checked this proof. Generated from pipeline artifacts by tools/atlas_build.py.