Private Blotto: Viewpoint Competition with Polarized Agents

The strict computational-anchor requirement fails: the numbered results characterize Nash stability and misallocated effort but assert no complexity, algorithmic, approximation, or parameterized result. The high-multiplicity clone-limit is an author-recognizable extension, though atomic tie and parity information is not represented by mass alone. Theorem 3 cannot repair the missing anchor because it continuizes individual effort rather than the population.

The objection that survived

The same limiting mass allocation can arise from clone sequences with different O(1) occupancy, tie, and parity data, so mass alone does not determine one-agent stability.

What the mirror covers

The proposed mirror targets only Theorem 1's median-outcome existence characterization; it leaves Theorems 2–5 and Lemmas 1–11 otherwise uncovered, and Theorem 3 is out of scope.

Open questions for a prover

The case FOR (proponent)

The strongest honest positive case is a qualified one. Strictly under the ChoCo anchor rule, this paper has no qualifying computational-complexity result: Theorems 1–5 and Lemmas 1–11 are structural existence and characterization results, not statements that a decision problem is in P, NP-hard, FPT, or similar. They are proved in this paper, but no running-time claim is made. Thus the paper does not support an unconditional positive verdict.

If structural characterizations are admitted as computational anchors, my lead would be Theorem 1, proved here. It states that, under the median outcome and sufficiently large empty-item cost, a stable arrangement exists exactly when \((N_a,N_b)\) lies outside the median-critical region.

The corresponding mirror should be called High-Multiplicity Median Private Blotto.

A society consists of two complete types, A and B, with rational masses \(\mu_A,\mu_B\), \(\mu_A+\mu_B=1\). Type includes bias, bandwidth, access, and any other feature used by the game. The natural regime is a large pool of users annotating a comparatively small batch of posts: millions of users, two or a small fixed number of viewpoint/access types, and \(M\) fixed or much smaller than the population. This is plausible for Community Notes-like annotation or decentralized political activism.

To preserve the paper’s one-agent action, I would not define an ordinary atomless Nash equilibrium. Instead, the continuous solution is a limit of exact high-multiplicity games. An instance contains \(M\), rational \(\mu_A,\mu_B\), biases \(\beta_A,\beta_B\), and \(c\geq \frac12|\beta_A-\beta_B|\). For every admissible population scale \(q\), there are \(q\mu_A\) type-A and \(q\mu_B\) type-B agents. Each agent still chooses exactly one item. Let \(a_i,b_i\) be the numbers assigned to item \(i\), and let \(x_{A,i}=a_i/q\), \(x_{B,i}=b_i/q\) be the corresponding masses.

The continuous problem is:

Given \(M,\mu_A,\mu_B,\beta_A,\beta_B,c\), determine whether there exists a sequence \(q_r\to\infty\) and exact pure Nash-stable arrangements \((a^{(r)},b^{(r)})\) whose normalized allocations converge to a mass allocation \(x\). If so, output such an \(x\) together with a generating construction; otherwise certify that no such limiting stable allocation exists.

The median on each item is computed from the finite clone arrangement: it is \(\beta_A\) when \(a_i>b_i\), \(\beta_B\) when \(b_i>a_i\), and their midpoint in a tie. Stability is also checked exactly as in the paper: moving one individual of either type from any occupied source item to any target item must not reduce that individual’s cost. The objective is therefore to find a limiting stable allocation, not to fractionalize an individual’s effort or introduce coordination.

This is author-recognizable: same items, same two polarized types, same one-item action, same decentralized deviations, and the population is changed only from an explicit list of agents to rational type masses. The high-multiplicity regime is not artificial here; it is exactly the setting of many nearly interchangeable users grouped by viewpoint and platform-relevant attributes.

I would expect this mirror to be Class A. Theorem 1 gives a bounded exceptional region in \((N_a,N_b)\), while any fixed positive masses eventually satisfy \(N_a+N_b\geq 2M+1\). Lemmas 3 and 4 are constructive, so they suggest an efficient procedure for generating stable clone arrangements and hence a limiting mass allocation. The continuous limit therefore removes the bounded small-population obstruction without changing the individual action. Further questions include the corresponding mean-outcome problem, whether instability persists at fixed type proportions, weighted item importance, and what changes with a fixed finite number of types larger than two.

The weakest point is serious. If one removes the clone-scale interpretation and uses an ordinary atomless game, an individual has zero effect on item outcomes, so unilateral Nash deviations become vacuous. If one instead allows positive-mass deviations, the model becomes coalitional and contradicts the paper’s decentralized premise. The clone-limit formulation avoids both defects, but it is an extension rather than a literal atomless Nash game.

Theorem 3 is tempting as an anchor because it gives the proportional fractional arrangement, but it explicitly allows players to allocate effort fractionally. That is action/outcome continuization, which ChoCo excludes from the verdict. Theorem 2 is also a warning: its instability is driven by integer granularity, so a fully atomless mirror may erase precisely the phenomenon the paper studies.

So my honest submission would be: one plausible, author-recognizable high-multiplicity mirror, led by Theorem 1 and likely tractable, but not a clean qualifying case under the strict computational-anchor rule.

The case AGAINST (opponent, writing after the proponent)

The negative case is decisive under ChoCo’s strict anchor rule: this paper contains no qualifying computational result. Theorems 1–5 and Lemmas 1–11 characterize existence and structure of pure Nash equilibria, but make no claim about complexity, algorithms, approximation, parameterization, or computational hardness. Theorem 1 is therefore not a computational anchor in the programme’s sense.

Even if structural results are admitted, the proposed mirror of Theorem 1 does not produce a meaningful continuous problem. Fix \(M\) and a population proportion \((\mu_A,\mu_B)\), and scale the electorate by \(q\). For every sufficiently large \(q\), \(N_A+N_B=q\ge 2M+1\), so Lemma 3 already guarantees a stable arrangement. The median-critical region is bounded in the raw counts and disappears entirely from the normalized population simplex. Thus the proposed continuous input has no nontrivial existence boundary: every fixed mass distribution eventually lies in the guaranteed-stability regime.

This is not merely the objection that the answer might be boring. The equilibrium property itself is not a property of the limiting mass allocation. A unilateral deviation changes an item’s load by one agent and can flip its median at an exact tie. After normalization, that deviation has size \(1/q\), which vanishes. Two clone sequences can converge to the same mass allocation while differing by \(O(1)\) agents at an item, with different medians and different deviation incentives. The limiting \(x\) therefore does not contain enough information to determine stability.

There are only unattractive ways around this. An ordinary atomless game makes each individual’s effect on medians, means, and item occupancy zero, so unilateral deviations become vacuous. Allowing positive-mass deviations turns the model into a coalitional game, contrary to the paper’s defining decentralized premise. The proposed sequence-based formulation preserves the finite game only by retaining the entire hidden integer-scale construction—\(q\), parity, and \(O(1)\) residual counts. At that point the continuous society is not the game’s state; it is merely a label attached to a sequence of discrete instances.

Scaling \(M\) with \(q\) does not repair the mirror of Theorem 1. To retain the critical region one needs \(M=\Omega(q)\), so each item receives only \(O(1)\) agents. The relevant phenomenon then remains atomic occupancy and item-level identity, not high-multiplicity population structure. Replacing the growing set of items by a continuum makes individual-item medians and emptiness ill-defined; enriching types with item-specific preferences creates a different game rather than a continuization of this paper’s theorem.

Theorem 3 is explicitly about fractional effort, hence outcome/action continuity rather than population continuization. Theorem 2 is a warning rather than an anchor: its persistent instability is caused by integer allocation. That phenomenon disappears in the atomless limit, while retaining it requires precisely the finite-agent microstructure the proposed mirror is supposed to eliminate.

The paper’s crowdsourcing setting does make many nearly interchangeable users plausible, so “there is no high multiplicity” would be an inferior objection. The problem is more fundamental: Private Blotto’s contribution is an atomic pure-equilibrium phenomenon, and its key distinctions are encoded in finite deviations and per-item occupancies. No distribution over voter or agent types alone preserves those distinctions. The paper may motivate an asymptotic game-theory study, but it offers no worthwhile ChoCo continuous-computational mirror.

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.