| paper | Outsourcing Adjudication to Strategic Jurors |
| authors | Ioannis Caragiannis, Nikolaj Schwartzbach |
| venue | IJCAI 2023 |
| filed under | voting · manipulation |
| judged by | gpt-5.6-luna / xhigh (triple__luna__xhigh__c2r1) |
| judge confidence | medium |
| authors would recognise it | yes |
Theorem 7
statement extracted from the paper’s text layer
Given a rational society \(\mu=(\rho,1-\rho)\) of well-informed and misinformed juror types, panel size \(k\), target \(x\in(1/2,1)\), and effort data \(\lambda=f^{-1}(x)\) and \(d=f'(\lambda)\), a panel is sampled from \(\mu\). Choose payment values \(p(1/k),\ldots,p(1)\) minimizing \(k[xA(p)+(1-x)B(p)]\), where \(J\sim\operatorname{Bin}(k-1,x)\), \(A(p)=\sum_{j=0}^{k-1}\Pr[J=j]p((j+1)/k)\), and \(B(p)=\sum_{j=0}^{k-1}\Pr[J=j]p((k-j)/k)\), subject to \(xA(p)+(1-x)B(p)\ge\lambda\), \(d(A(p)-B(p))=1\), and \(p((j+2)/k)-p((j+1)/k)+p((k-j)/k)-p((k-j-1)/k)\ge0\) for \(0\le j\le k-2\).
A high-multiplicity eligible-juror society with type masses \(\mu=(\rho,1-\rho)\), from which a finite panel of \(k\) jurors is randomly selected; the decision variable is the panel payment schedule \(p(j/k)\), and the objective is minimum expected total payment supporting the target equilibrium.
The continuous mass describes an eligibility pool rather than the finite panel that actually votes, and because every selected juror votes for \(T\) with probability \(x\), \(\rho\) disappears from \(J\sim\operatorname{Bin}(k-1,x)\) and from the LP.
fatal: False
The mirror covers the payment-optimization result of Section 4 and Theorem 7. It leaves the general equilibrium characterizations, equilibrium multiplicity, best-response dynamics, stability, and simulation results outside the stated problem.
The strongest positive case is narrow but genuine. The lead anchor is Theorem 7, proved in this paper, which states that the authors’ payment-selection problem can be solved in polynomial time by linear programming. It is a natural Class A candidate: the continuous population removes irrelevant juror identities while preserving the strategic payment mechanism.
The appropriate scenario is a large Web3 pool of eligible jurors from which a finite panel is randomly selected for each dispute. A juror type records the complete relevant information: her effort-to-signal function, effort cost, and payment environment. Take two types, \(w\) and \(m\), with population masses \(\mu_w=\rho\) and \(\mu_m=1-\rho\). Type \(w\) has effort function \(f\), while type \(m\) has effort function \(1-f\), exactly as in Section 4 of the paper. The population may contain millions of anonymous jurors but only two relevant types, so this is a plausible high-multiplicity regime: \(N\gg k\) jurors, \(\tau=2\) types, and a panel of \(k\) selected jurors.
The finite panel is important. It preserves the paper’s strategic feature that a juror’s vote changes the fraction of panel votes agreeing with her, while the eligible society itself is represented by the continuous mass vector \(\mu\). This is a continuous population mirror, not a claim that an atomless population must vote as one indivisible panel.
Call the resulting problem Continuous Juror-Pool Payment Design. An instance consists of a rational mass \(\rho\), a panel size \(k\), a target vote probability \(x\in(1/2,1)\), and an effort function \(f\), represented through the values \(\lambda=f^{-1}(x)\) and \(d=f'(\lambda)\). A panel of \(k\) jurors is sampled from the population. Each juror chooses effort \(\lambda_i\), receives a signal for the ground-truth alternative \(T\) with the probability prescribed by her type’s effort function, and then either follows or reverses that signal according to \(\beta_i\). The panel outcome is majority vote. A payment schedule is \(p(j/k)\) for \(j=1,\ldots,k\), where a juror receives \(p(r/k)\) when \(r\) panel members, including herself, cast the same vote.
The decision variable is the payment schedule \(p\). The target equilibrium is that type \(w\) uses \((\lambda,1)\), type \(m\) uses \((\lambda,0)\), and hence every selected juror votes for \(T\) with probability \(x\). The objective is to minimize the expected total payment to the panel, subject to individual rationality and the target strategy profile being an equilibrium. As in the paper, one may additionally require the finite-panel version of Lemma 4’s inequalities, ensuring that the payment schedule has the desired simple-equilibrium structure.
Precisely, let \(J\) be the number of other panel members voting for \(T\). Under the target profile, \(J\sim\operatorname{Bin}(k-1,x)\). Define \(z_j=\Pr[J=j]\), \(A(p)=\sum_{j=0}^{k-1}z_jp((j+1)/k)\), and \(B(p)=\sum_{j=0}^{k-1}z_jp((k-j)/k)\). Here \(A(p)\) is the expected payment from voting \(T\), and \(B(p)\) the expected payment from voting \(F\). A schedule is feasible when
\(xA(p)+(1-x)B(p)\ge\lambda\),
\(d(A(p)-B(p))=1\),
and, for every \(j=0,\ldots,k-2\),
\(p((j+2)/k)-p((j+1)/k)+p((k-j)/k)-p((k-j-1)/k)\ge0\).
The objective is to minimize \(k\bigl(xA(p)+(1-x)B(p)\bigr)\). A solution is an explicitly encoded schedule \(p(1/k),\ldots,p(1)\) satisfying these constraints, together with the induced target equilibrium. If desired, the grid values can be extended linearly to a function on \([0,1]\); the continuity being used here is in the population mass \(\mu\), not in the outcome space or in an artificial infinite-dimensional payment variable.
I expect this problem to be tractable, in Class A. Every quantity \(A(p)\), \(B(p)\), the individual-rationality condition, the equilibrium equation, and the Lemma 4 restrictions are linear in the \(k\) payment variables. Thus the continuous pool size \(N\) disappears from the optimization, and the problem is solved in time polynomial in \(k\) and the encoding length of the rational data, exactly paralleling Theorem 7. For a finite pool with \(N\mu_t\) jurors, the continuous solution also supplies a natural high-multiplicity approximation: sampling without replacement approaches the binomial model as \(N\) grows, while the payment schedule remains independent of the identities of the \(N\) jurors.
The mirror is plausible because it preserves the authors’ actual question: anonymous agents are paid according to agreement with the panel majority, choose effort strategically, may invert their signals, and are indifferent to the final adjudication outcome. The only reinterpretation is that the paper’s \(n\) jurors are a randomly appointed panel drawn from a much larger continuous pool. That is especially credible in the Web3 setting motivating the paper, where a large eligible population and random juror appointment are natural protocol components. The type mass \(\rho\) means the fraction of the pool having each signal-quality profile, not the fraction of named individuals manually assigned to a category.
This mirror deliberately covers only the paper’s Section 4 payment-optimization result, anchored on Theorem 7. It does not claim to solve the paper’s equilibrium-selection problem. In particular, Lemma 3, also proved here, says that a good equilibrium has a corresponding bad equilibrium with the same probability. The continuous problem therefore asks for a payment schedule supporting and minimizing the cost of a good equilibrium, just as Theorem 7 does; it does not silently upgrade existence into uniqueness, stability, or guaranteed correct adjudication.
The further questions are worthwhile: can the payment schedule be made stable under the paper’s best-response dynamics; can \(\rho\) and the effort functions be uncertain; can one optimize accuracy and payment jointly; and does the LP remain tractable with \(\tau>2\) heterogeneous juror types? A particularly natural extension replaces the two-point mass vector by a finite distribution over training backgrounds, expertise levels, and effort functions.
The weakest point is that the mirror retains a finite panel \(k\). If one insists that the entire atomless society votes directly, the binomial fluctuation and finite-juror pivot effects disappear, and the paper’s payment-design problem changes substantially. I would concede that objection. The positive case rests on the authors’ own institutional setting, however: adjudication is already outsourced to a selected set of jurors. Continuizing the large eligible pool while keeping the selected panel finite is therefore not a tractability-motivated distortion of their question; it is the high-multiplicity version of the Web3 scenario they actually study.
The proponent has found a plausible high-multiplicity story, but it does not yield a continuous mirror of Theorem 7. The continuous parameter \( \rho \) belongs to an eligibility pool, not to the strategic population that actually adjudicates the dispute.
Under the proposed target equilibrium, a well-informed juror follows her signal with effort \( \lambda \), so she votes \(T\) with probability \(f(\lambda)=x\). A misinformed juror reverses her signal, so she also votes \(T\) with probability \(1-f(\lambda)=x\). Consequently, for a selected panel of size \(k\), the number of other \(T\)-votes is still \(J\sim\operatorname{Bin}(k-1,x)\), independently of \( \rho \) and of the size of the eligible pool. The quantities \(A(p)\), \(B(p)\), the individual-rationality constraint, the equilibrium equation, and the Lemma 4 inequalities are exactly Theorem 7’s LP with \(n=k\).
Thus the claim that the continuum “removes irrelevant identities” is misleading. Those identities were already irrelevant to Theorem 7. The theorem’s LP is driven by the finite panel size, the \(k\) possible payment fractions, and the binomial distribution of the other panel votes. Increasing the number of eligible but unselected jurors changes none of these. The pool is a sampling frame or prior over finite games, not the society on which the paper’s outcome and incentives operate. This is a sensible institutional interpretation of the Web3 setting, but the continuous mass is computationally inert.
The stronger alternative is to make the entire jury atomless. That destroys the finite object Theorem 7 optimizes. If \(q\) is the aggregate fraction voting \(T\), then an individual juror has no effect on \(q\). Her expected payment depends only on \(p(q)\) if she votes \(T\), and on \(p(1-q)\) if she votes \(F\). The binomial expectations become the two point evaluations \(p(q)\) and \(p(1-q)\); the finite payment grid and the inequalities controlling changes in vote share by \(1/k\) disappear. Majority outcome uncertainty and finite-panel feedback also disappear. What remains is a mean-field equilibrium problem, not the continuous relaxation of the paper’s payment-selection theorem.
One can preserve the finite strategic interaction by introducing positive-mass voting cells, finite panels, or a continuum of independent juries. But then those cells or juries are the actual agents, and the continuous distribution merely mixes over separate finite games. Alternatively, one can make \( \rho \) matter by allowing type-dependent target vote probabilities, heterogeneous effort functions, or endogenous panel composition. That could motivate a new mechanism-design problem, but it abandons the complementary two-type symmetry that makes Theorem 7’s LP the stated result. It is an extension of the paper, not its high-multiplicity mirror.
Lemma 3 does not repair this gap. Its paired good and bad equilibria apply to each finite strategic game and remain present after conditioning on any pool composition; the proposed continuum adds no population-level structure relevant to equilibrium selection.
The honest concession is that a large anonymous juror pool is a credible high-multiplicity scenario. What it does not provide is an active continuous society for this computational result. The finite-panel version leaves the mass parameter decorative; the atomless version removes the finite-panel mechanism; and intermediate versions are mixtures of finite games or new models. On the programme’s criterion, Theorem 7 therefore supplies no worthwhile continuous 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.