| paper | d4-journals-mss-2013-0031 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | yes |
other→P discrete result → its continuous mirror
Answer: Polynomial-time solvable in \(m,\tau,d,L\); every promised input admits an exact rational lottery with at most \(\tau\) cutoffs. · discrete side: other · adjudication: proved
full proof & adversarial review →
Proposition 4
statement extracted from the paper’s text layer
Given a finite type set \(T\), rational masses \(\mu_t\) summing to one, and an explicit rational, differentiable, monotone, convex production function \(V:[0,\mu]\to[0,1]\) with \(V(0)=0\) and \(V(\mu)=1\), compute an exact lottery \(\Pi\) over cutoffs \(q\in(0,1]\). Rank the population mass by \(h_t=\int_0^1 \partial_t V(\lambda\mu)\,d\lambda\); for each cutoff, let the top \(q\) mass decide by simple majority and share the unit payoff equally across selected mass, using the finite-clone limit convention. Require the expected aggregate payoff assigned to each type \(t\) to equal \(\mu_t h_t\).
Types \(t\) have masses \(\mu_t\) and coalition production is given by \(V(x)\). The economic allocation density is \(h_t=\int_0^1 \partial_t V(\lambda\mu)\,d\lambda\). The decision variable is a probability vector over ranked mass cutoffs; its constraints require expected type allocations to equal \(\mu_t h_t\). There is no separate optimization objective beyond finding a lottery satisfying those equalities.
The mirror’s tractability depends on adding an explicit coalition-production function \(V\), so its computational result is specific to that input representation and nearly follows from a formula.
fatal: False
The mirror covers Proposition 4’s zero-distortion random-selection construction and its probability formula. It leaves the weighted-voting distortion results, outside-option analysis, average-game result, and ECB rotation calculations untouched.
There is no named computational-complexity result in this paper: no proposition classifies a problem as tractable, hard, or parameterized tractable. Its numbered propositions are mathematical results about voting power and distortion. So none is an eligible anchor under the stated rule.
The strongest result to build a *candidate* mirror around is Proposition 4, proved in the paper. It says that for any economic game, rank players by decreasing Shapley value, and let \(\upsilon_m\) be the simple-majority game among the first \(m\) players. Choose \(\upsilon_m\) with probability \(m(\phi_m-\phi_{m+1})\) for \(m<n\), and choose \(\upsilon_n\) with probability \(n\phi_n\). The resulting random-selection voting rule gives every player an expected Shapley–Shubik index equal to their economic Shapley value, so expected political distortion is zero. This is an explicit construction, but the paper does not state a complexity bound for computing it.
A plausible population regime is a very large producer cooperative or infrastructure consortium: tens of thousands of member firms or households, but only a few dozen complete economic-role types. Members of a type have the same contribution technology, payoff rights, and voting role. The type masses are fractions of the membership; the number of agents can be enormous while the support size \(\tau\) stays modest. This fits the paper’s settings of common projects and joint ventures better than its ECB example, whose governors are a small set of distinct actors.
The candidate mirror is Continuum Random-Committee Replication. Its instance is a type set \(T\), rational masses \(\mu_t\) summing to one, and an explicit rational coalition-production function \(V(x)\) on \(0\le x\le\mu\), with \(V(0)=0\), \(V(\mu)=1\), and the monotonicity and convexity conditions needed for nonnegative economic allocations. Here \(x_t\) is the mass of type \(t\) in a coalition. Define the type’s per-unit economic allocation by the Aumann–Shapley limit
\[
h_t=\int_0^1 \partial_t V(\lambda\mu)\,d\lambda,
\]
so its aggregate allocation is \(\mu_t h_t\), and \(\sum_t\mu_t h_t=1\).
Sort all population mass by decreasing \(h_t\), producing a nonincreasing step function \(h(r)\) over ranks \(r\in[0,1]\). The question is: compute an exact probability distribution \(\Pi\) over cutoffs \(q\in(0,1]\) such that selecting the top \(q\) mass, deciding by simple majority, and equally sharing the unit payoff among the selected mass yields expected aggregate payoff \(\mu_t h_t\) for every type. A solution is the distribution \(\Pi\), not just a yes/no answer. For a committee cutoff \(q\), the per-unit payoff is \(1/q\) for selected members and zero for the rest; thus the expected allocation density at rank \(r\) is
\[
g(r)=\int_{[r,1]}\frac{1}{q}\,\Pi(dq).
\]
In this explicit input model, I expect the problem to be tractable. At each rank boundary \(b\) where \(h\) drops, assign probability \(b(h(b^-)-h(b^+))\) to cutoff \(b\), and assign probability \(h(1^-)\) to cutoff \(1\). This gives \(g(r)=h(r)\) almost everywhere. Computing the Aumann–Shapley values from an explicit rational polynomial \(V\), sorting the types, and outputting these at most \(\tau\) rational probabilities takes polynomial time in the representation size, \(\tau\), and bit length. That is a proposed complexity result for this mirror, not a claim made by the paper.
The connection to Proposition 4 is unusually direct: it preserves the paper’s comparison between economic Shapley allocation and political Shapley–Shubik allocation, and its random committees are nested prefixes ranked by economic importance. Population mass replaces counts of indistinguishable agents; the probability rule is the mass-limit counterpart of the paper’s formula. Random selection alone would only continuize the *mechanism*, which is outside the programme’s scope. Here the society itself is continuous, and the committee lottery is part of the mirrored mechanism.
The weakest point is that a continuum electorate has no ordinary individual pivotal voter: each individual’s Shapley–Shubik power vanishes. The mirror therefore defines political allocations as limits of finite clone electorates, where symmetric simple-majority committees divide the Shapley–Shubik payoff equally among their members. It also replaces the paper’s arbitrary finite characteristic function with a smooth type-mass production function. Those are substantial co-modifications, though they follow naturally from the high-multiplicity regime and preserve the paper’s central question. A useful next question would be whether the same construction remains tractable when \(V\) has a compact implicit representation, or when committee sizes and membership must satisfy additional institutional constraints.
The strongest objection is upstream: this paper states no computational-complexity result. Propositions 1–5 and Corollary 6 characterize power and distortion; Proposition 4 even gives the random-selection probabilities explicitly once the Shapley values are known. Turning that construction into “compute the probabilities” creates a new algorithmic question, rather than continuizing a computational problem the paper studies.
The best proposed mirror, for a large cooperative with a few economic-role types, is plausible. But under the proponent’s explicit polynomial input \(V\), its answer is the direct continuum limit of Proposition 4: compute the Aumann–Shapley values, sort them, and assign the cutoff probabilities from their drops. That is a useful identity, perhaps, but it supplies no substantial complexity landscape or pricing problem. If \(V\) is instead given only implicitly, the difficulty shifts to evaluating coalition worths and computing Shapley-type allocations. The society’s mass distribution no longer identifies the input: one must also choose and encode a function over coalition compositions. Any complexity result would then depend on that added representation, not on the paper’s voting mechanism.
The ECB application itself is an especially poor high-multiplicity instance. It has a small set of named states with distinct economic roles. Replicating them into masses changes the object being studied; preserving the paper’s arbitrary coalition-specific game instead leaves no compact type-level input. The proponent’s cooperative example gets around this by imposing a structured mass-production function, but that is precisely the extra model that makes the construction immediate.
This does not establish that no worthwhile mirror exists. A continuum limit theorem for a large cooperative could be sensible, and the authors might recognize it. The negative case is strongest if confined to ChoCo’s computational aim: the paper offers no computational result to mirror, and its best candidate either has an explicit formula or gets its complexity from a separately imposed representation of coalition production. I cannot honestly rule out every scenario; the weakness is that a plausible high-multiplicity cooperative remains available.
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.