{
 "artifacts": null,
 "category": "architecture",
 "description": "Give every token a valuation vector over experts and route it to the expert maximizing utility after subtracting a dynamically updated congestion price. The router is a neural analogue of the paper's measurable valuation-space partition: tokens with different valuations are assigned to different experts, while expert prices discourage overload without requiring a generic global load-balancing penalty.",
 "formulas_latex": [
  "$$V:=\\bigsqcup_{i\\in[k]}V_i,\\qquad \\mu\\!\\left(\\bigsqcup_{i\\in[k]}A_i\\right):=\\sum_{i\\in[k]}\\mu_i(A_i).$$",
  "$$V_{i,j}(\\tau_i):=\\left\\{v\\in V_i\\mid j\\in\\arg\\max_{\\hat j}U_i\\!\\left(v_{\\hat j},\\tau_{i,\\hat j}\\right)\\right\\}.$$",
  "$$r(x)=\\arg\\max_{j\\in[E]}U\\!\\left(v(x)_j,\\tau_j\\right),\\qquad \\tau_j^{t+1}=\\left[\\tau_j^t+\\eta_\\tau\\left(\\frac{\\ell_j^t}{C_j}-1\\right)\\right]_+.$$",
  "$$U\\!\\left(v_j,\\tau_j\\right)=v_j-\\tau_j\\quad\\Longrightarrow\\quad r(x)=\\arg\\max_j\\bigl(v(x)_j-\\tau_j\\bigr).$$"
 ],
 "id": 243,
 "implementation": "Integrate this at the router logits of a sparse MoE Transformer. For each token representation h_b, compute a valuation vector v_b = W_v h_b in R^E, where E is the number of experts. Maintain one nonnegative congestion price tau_j per expert, shared across the minibatch and updated with an exponential moving average of recent loads. Use this pseudocode: `v = Wv @ h`; `score[b,j] = v[b,j] - tau[j]`; `route[b] = argmax_j score[b,j]`; dispatch tokens to selected experts up to capacity C_j; compute `load[j] = number of assigned tokens`; update `load_ema = beta * load_ema + (1-beta) * load`; update `tau[j] = relu(tau[j] + eta_tau * (load_ema[j] / C_j - 1))`. In the forward pass use the one-hot argmax route. During training use a straight-through estimator whose backward pass substitutes the gradient of softmax(score divided by temperature T), or a softmax over the top two experts. Use U(v_j,tau_j) = v_j - lambda * tau_j, with lambda either fixed to one or tuned over a small grid. Estimate loads directly from each batch; the first implementation does not require a valuation-space density estimate or an optimal-transport solver. Compare against top-1 routing with Switch-style auxiliary load balancing on a 12-layer, 8-expert Transformer trained on a small language-model corpus. Measure validation perplexity, overflowed or dropped-token fraction, expert-load coefficient of variation, router assignment churn, and training throughput at equal FLOPs. The main falsifiable target is fewer overflowed tokens and lower load variance at the same auxiliary-loss weight, or equal load variance with less routing regularization and improved perplexity.",
 "math_summary": "The paper forms the joint valuation measure space as the disjoint union V := disjoint_union_i V_i, with measure mu(disjoint_union_i A_i) = sum_i mu_i(A_i). For an agent with valuation v in V_i, destination j in D_i, and congestion state tau_i = (tau_i,j), the choice region is V_i,j(tau_i) := {v in V_i : j belongs to argmax over j-hat of U_i(v_j-hat, tau_i,j-hat)}. Here v_j-hat is the token's valuation for destination or expert j-hat, U_i is utility as a function of valuation and congestion, and tau_i,j-hat is the congestion price or state of that destination. The paper argues that pairwise indifference sets are null when U_i is strictly monotone in valuation and the valuation measure mu_i is absolutely continuous, so the regions form an almost-everywhere partition. We use this partition as a hard router and update congestion prices from observed expert loads using a projected dual ascent rule.",
 "math_tags": [
  "optimal-transport",
  "measure-theory",
  "game-theory",
  "optimization"
 ],
 "ml_areas": [
  "moe-routing",
  "transformer",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2607.03625",
  "arxiv_url": "https://arxiv.org/abs/2607.03625",
  "summary_what_math_gives_to_ml": "The paper provides an explicit way to route heterogeneous agents through congestible alternatives: for a fixed congestion-price vector, the valuation space is partitioned into measurable destination regions by an argmax rule, with indifference boundaries negligible under absolute continuity. This suggests replacing uniform or purely heuristic load balancing in mixture-of-experts networks with a utility-aware congestion equilibrium, where each token has a vector of expert valuations and expert prices rise with batch load. The practical transfer is a hard or straight-through threshold router coupled to dual price updates, tested against top-k routing and standard auxiliary load-balancing losses at equal capacity and FLOPs.",
  "title": "Congestion Games with Heterogeneous Valuations: An Optimal Transport Approach",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 5,
  "usefulness": 6
 },
 "solves": [
  "scalability",
  "stability",
  "accuracy"
 ],
 "title": "Valuation-Threshold Congestion Router",
 "url": "https://synthcore.org/idea/243/valuation-threshold-congestion-router",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}