# Method explorer: sources, definitions, and scope Version: 2026-10-20, English dark-interface revision. This website is an educational companion to the manuscript. No experiment outcomes, checkpoints, or accuracy estimates are loaded. All numerical examples below run locally in the browser and are explicitly labeled as teaching models. ## Narrative and source mapping The presentation follows population construction → equilibrium reasoning → neural mechanism learning → voluntary participation → constrained allocation. The method-clarity and claim/evidence checks from the research-paper-writing skill informed the separation of these concepts. Repository-relative source paths used in the read-only review: | Website component | Source | Interpretation | | --- | --- | --- | | State law, responses, consistency | `main_final2/main_m_tdsc.tex`, MFG section, approximately lines 1299–1530 | A client policy and population law must be mutually consistent; R5 makes type evolution exogenous. | | Scope of the neural mechanism | Same source, approximately lines 1330 and 1638 | The deployed network is not an HJB–FP/MFE numerical solver. | | Inputs and separate encoders | Same source, AHN modules, approximately lines 1690–1785 | Full bids, repeated mean bids, and B; two tanh MLPs; single-item allocation has an abstain logit. | | Allocation and payment | Same source, `eq:alloc-net`, `eq:eps-out`, `eq:pay-net` | Allocation scales the reported capacity; sigmoid payment is followed by the total-budget projection. | | Deviation search and training | Same source, approximately lines 1806–1916 | Recompute population context after a deviation; searched gains do not certify a global maximum. | | Aggregation weights | Same source, P2 formulation and online FL algorithm | Use alpha_i = epsilon_i / sum(epsilon), not data-weighted epsilon. | | Allocation surrogate | `main_final2/revisions/p2_mechanism_design_20261018/exp/p2_reference.py`, `j_kappa`, approximately lines 107–114 | J_kappa = kappa * squared weight mismatch + active_count / total_privacy^2; empty procurement has infinite objective. | Line numbers refer to the source inspected for this website revision; named equations and functions are the stable references. The manuscript's evaluated configuration disables optional critic/alignment terms. No claim is made that the P2 development objective or a population-only architecture is the final adopted mechanism. ## 1. Empirical population teaching example - N is selected from 10, 50, 100, 500, and 1,000. - Baseline scalar cost reports are deterministic: b_j = 0.5 + 2.4 * ((7j) mod 13) / 12, with zero-based j. - One report changes by Delta in [0, 3]. - The histogram and mean use all N reports; dots show at most 60. - phi(b) = b in this single-coordinate example, so the exact mean change is Delta/N. The manuscript uses a vector of mean bid coordinates. This demonstrates only the effect through the explicit population input. It is not a bound on the report's direct effect on its own allocation/payment, and it does not prove total regret is O(1/N). The state law, report law, and feature mean are separate objects. The original AHN retains full-profile inputs. ## 2. Fixed-point teaching game This is a separate game, not the paper's auction or an empirical result. - Three equally likely exogenous types: s in {0.2, 0.4, 0.6}. - Action a in [0, 1]; coupling lambda in [0, 0.6]. - Utility: u(s,a,m) = -(a - (s + lambda*m))^2. - Best response: BR(s,m) = clip(s + lambda*m, 0, 1). - Start with m_0 = 0 and policy a_0(s) = s. - At step k, compute a_{k+1}(s) = BR(s,m_k), then m_{k+1} = 0.4*m_k + 0.6*mean(a_{k+1}). - Consistency residual: abs(mean(a_k) - m_k). - Best-response gap: max_s [u(s,BR(s,m_k),m_k) - u(s,a_k(s),m_k)]. The constrained optimum's utility is included even when clipping is active. - The dotted fixed point is computed by bisection of mean(BR(s,m)) - m. - Stop when consistency < 1e-5 and the best-response gap < 1e-8, or at 100 iterations. All permitted slider values converge before that limit. Clipping is 1-Lipschitz, so F(m) = mean(BR(s,m)) is lambda-Lipschitz. The relaxed iteration contracts by at most 0.4 + 0.6*lambda < 1. At lambda = 0.35, no action clips and m* = 0.4/0.65. At lambda = 0.6, the highest action saturates and m* = 8/9. These properties apply to this toy only. A finite-network equilibrium claim would need verified best-response, consistency, population-sampling, and representation errors for the actual utility. Equilibrium reporting need not be truthful reporting. The toy does not establish DSIC or an empirical regret certificate for MFG-RegretNet. ## 3. Neural architecture and incentive learning The diagram preserves separate allocation/payment encoders and the original allocation/payment heads. The one-item allocation is z_i = sigmoid(h_i1 - h_i0), epsilon_i = z_i * reported_capacity_i. Raw p_i = B*sigmoid(h_i); the projected offer is pbar_i = p_i / max(1, sum(p)/B). The clickable sequence explains a truthful reference, deviation search, context update, utility comparison, and mechanism-weight update. Its equations are explanatory; no numerical model training is performed. The displayed loss is schematic, with a procurement surrogate plus IR and regret penalties. True type and environment stay fixed during a unilateral report comparison. A full settlement audit must include local acceptance and the same ledger snapshot. Critic subtraction that changes with the report cannot silently replace true utility differences. Searching for deviations supplies a lower bound on the largest gain, not a global upper certificate. ## 4. Participation calculator One synthetic client has true capacity 0.80, previously used epsilon 0.40, and cumulative demo cap 1.00. Privacy cost is v*epsilon. It accepts exactly when: 1. epsilon <= 0.80; 2. 0.40 + epsilon <= 1.00; 3. payment >= v*epsilon. When accepted, realized utility is payment - v*epsilon. Otherwise payment, release, and realized utility are zero. Comparisons use 1e-9 floating-point tolerance. This is a deterministic participation illustration, not a deployment DP guarantee or a strategyproof payment mechanism. ## 5. Finite-grid allocation oracle Three clients have known unit costs [1,2,3], capacities [1,1,1], and target data shares pi = [0.5,0.3,0.2]. Each epsilon is in {0,0.25,0.5,0.75,1}, giving 125 candidates. The domain of B is [1,6] and that of kappa is [0,20]. Only nonempty candidates with sum(v_i*epsilon_i) <= B are feasible. Let S = sum(epsilon), alpha_i = epsilon_i/S, and I be the nonzero clients. The objective is J = kappa*sum((alpha_i-pi_i)^2) + |I|/S^2. For S = 0, J = infinity. The grid's smallest positive release (0.25) is above the P2 reference's minimum event threshold. The browser enumerates all candidates. Equal-objective ties prefer lower spending, then enumeration order. The comparator is the best feasible nonzero equal allocation on the same grid; if none fits, it is shown as infeasible. Payments at known cost only isolate the resource-allocation question. They are not proposed as an incentive-compatible payment rule. “Best” means best on this finite grid for this surrogate, not optimal real FL accuracy. The page does not claim any algorithm outperforms a baseline experimentally. ## Verification of this revision Browser checks use installed Playwright/Chromium, without a GPU or a package installation. The relevant acceptance checks are: - Dark color scheme, English static/dynamic text, navigation, no script or failed-resource errors, and no document-wide overflow at 360, 390, 768, 1024, 1440, and 1920 px. - Original workflow: all nine B = 10/30/100 and cap = 0.5/2/4 combinations; budget feasibility, accepted-client utility/capacity, refusal with zero payment/release, cumulative ledger over nine rounds, and export provenance. - Play/pause/resume, stepping, client selection, reset, and dialog keyboard close. - Exact Delta/N and independently recomputed means for every supported N. - Toy fixed points at lambda = 0, 0.35, and 0.6; numerical residuals; clipped actions; and run/pause/step/reset controls. - Participation acceptance, insufficient compensation, capacity refusal, ledger refusal, and acceptance exactly at the ledger boundary. - Independent enumeration of the allocation objective across 15 B/kappa pairs; equal-allocation comparison and rejection of empty procurement. - HTTP 200 for both Tailscale-hosted pages from this host. This does not add a new claim of access from every remote Tailscale account/device. Scientific-claim review: population context is not a sufficient-statistic claim; the game solver is explicitly a toy; neural mechanism learning is not client equilibrium solving; voluntary participation is distinct from truthfulness; the finite-grid optimum is distinct from measured FL accuracy.