Multi-Agent Routing as Set-Valued Prediction: A WildChat Benchmark and Cost-Aware Evaluation

# Multi-Agent Routing as Set-Valued Prediction: A WildChat Benchmark and Cost-Aware Evaluation
Source: [https://arxiv.org/html/2606.28925](https://arxiv.org/html/2606.28925)
Ananto Nayan BalaandFaisal Muhammad ShahAhsanullah University of Science and TechnologyDhakaBangladesh[faisal\.cse@aust\.edu](https://arxiv.org/html/2606.28925v1/mailto:[email protected])

###### Abstract\.

Tool and agent routing from natural\-language prompts is naturally a set\-valued prediction problem: a single query may require multiple agents, while over\-selection increases execution cost\. The benchmark introduced here is derived from WildChat and contains 3,000 prompts over a fixed 12\-agent catalog, with AI\-assisted heuristic labels under a fixed schema and controlled rebalancing for multi\-label evaluation\. The evaluation protocol combines set\-level metrics \(Precision, Recall, F1, Jaccard, and Exact Match\), latency, an execution\-oriented capability\-coverage simulation, and a constrained weighted\-routing setting based on ordinal agent\-cost tiers\. Compared methods include nearest\-neighbor matching, linear multilabel classification, dependency\-aware baselines, a fine\-tuned encoder, deterministic weighted post\-scoring via Weighted Agent Routing \(WAR\), and a zero\-shot LLM baseline\. Results show that supervised routers substantially outperform nearest\-neighbor and zero\-shot LLM routing\. The fine\-tuned encoder achieves the strongest unconstrained set accuracy, while the linear multilabel model provides the strongest practical baseline\. In the constrained setting, the weighted routing layer improves utility when applied on top of strong supervised scorers, with the largest gain observed for Encoder\+WAR\. Overall, the benchmark and evaluation protocol support reproducible study of accuracy–cost trade\-offs in fixed\-catalog multi\-agent routing\.

††copyright:acmlicensed††journalyear:2026## 1\.Introduction

Modern AI systems increasingly rely on catalogs of tools or agents, where the system must select one or more agents to fulfill a user request \(e\.g\., query a database, fetch an API, run statistical analysis, or generate a plot\)\. This setting maps naturally to set\-valued routing: given a query, the system predicts a small set of relevant agents that jointly satisfy the request\. Unlike top\-1 routing, this formulation captures real multi\-step workflows and enables explicit trade\-offs between coverage and execution cost\.

Despite growing interest in tool\-augmented assistants, there is limited work that treats routing as a multi\-label set prediction problem with set\-level evaluation\. Prior routing pipelines often select a single agent or rank tools without a principled decision rule for multi\-agent execution\. We address this gap by treating agent routing explicitly as set\-valued prediction over a fixed inventory and evaluating it with standard set metrics and cost\-aware utility\.

We build a WildChat\-derived benchmark with controlled agent coverage and set\-size distribution\. Starting from real user prompts, we assign AI\-assisted heuristic labels under a fixed 12\-agent catalog and rebalance the pool for stable multi\-label evaluation, then split it into train/dev/test partitions\. Because prompt\-to\-agent routing can admit more than one defensible routed set depending on redundancy tolerance, cost sensitivity, and user preference, these labels are best interpreted as protocol\-defined reference sets for comparative evaluation\. We evaluate three families of methods: \(i\) content\-based nearest neighbor retrieval, \(ii\) supervised multi\-label classification, and \(iii\) a fine\-tuned encoder that provides stronger semantic matching\. We also study a cost\-aware selection policy that trades off prediction quality and execution cost\. Our contributions are:

- •A set\-valued prediction formulation of agent routing that makes multi\-agent selection and cost\-aware evaluation explicit\.
- •A WildChat\-derived benchmark with real prompts, heuristic labels under a fixed 12\-agent catalog, and controlled set\-size/agent\-balance targets\.
- •A systematic empirical comparison of KNN, linear, dependency\-aware, encoder, and LLM baselines under a shared set\-evaluation protocol, together with a constrained weighted\-routing study based on deterministic WAR post\-scoring, identifying clear accuracy and cost\-aware operating regimes\.

An anonymous repository is provided for review:[https://anonymous\.4open\.science/r/multi\-agent\-routing\-D655/](https://anonymous.4open.science/r/multi-agent-routing-D655/)

## 2\.Related Work

### 2\.1\.Tool/Agent Routing in LLM Systems

Routing user queries to specialized tools or agents has been explored in dialogue systems, intent classification, and LLM tool\-use\. Classical intent detection work studies how utterances are mapped to downstream actions or APIs\(Gooet al\.,[2018](https://arxiv.org/html/2606.28925#bib.bib3); Casanuevaet al\.,[2020](https://arxiv.org/html/2606.28925#bib.bib4)\)\. More recent work extends this line to multi\-turn intent classification and intent\-conditioned dialogue generation\(Liuet al\.,[2024](https://arxiv.org/html/2606.28925#bib.bib1),[2025](https://arxiv.org/html/2606.28925#bib.bib2)\)\. In parallel, tool\-using LLMs and API\-centric systems make the routing problem explicit by requiring the model to choose an external tool, API, or expert model before execution\(Schicket al\.,[2023](https://arxiv.org/html/2606.28925#bib.bib5); Qin and others,[2024](https://arxiv.org/html/2606.28925#bib.bib6); Patilet al\.,[2024](https://arxiv.org/html/2606.28925#bib.bib7); Haoet al\.,[2023](https://arxiv.org/html/2606.28925#bib.bib20)\)\. Frameworks such as TaskWeaver\(Qiao and others,[2023](https://arxiv.org/html/2606.28925#bib.bib8)\)and related multi\-agent orchestration systems\(Wu and others,[2024](https://arxiv.org/html/2606.28925#bib.bib15); Hong and others,[2024](https://arxiv.org/html/2606.28925#bib.bib16); Qian and others,[2024](https://arxiv.org/html/2606.28925#bib.bib25)\)further motivate treating the agent catalog as a fixed item set that must be selected from given a prompt\. Our work differs from these systems papers by isolating the routing stage and evaluating it directly with set\-based metrics\.

### 2\.2\.Bundle/Slate and Set Selection

Set\-valued outputs are common in RS through bundle, basket, and slate construction\. Beladev et al\.\(Beladevet al\.,[2016](https://arxiv.org/html/2606.28925#bib.bib26)\)study bundle construction that jointly optimizes relevance and revenue using CF and pricing signals\. This perspective is directly relevant because our router also returns a small set rather than a single item\. The main difference is that our objective is semantic adequacy under execution\-cost constraints rather than revenue or basket composition, but the underlying decision structure is still closely related to selecting a compact set from a fixed catalog\.

### 2\.3\.Multi\-Label Prediction and Session\-Based Models

Multi\-label prediction generalizes top\-1 classification to multiple relevant outputs\(Tsoumakas and Katakis,[2007](https://arxiv.org/html/2606.28925#bib.bib32); Zhang and Zhou,[2014](https://arxiv.org/html/2606.28925#bib.bib33)\)\. Session\-based RS, such as GRU4Rec\(Hidasi and others,[2016](https://arxiv.org/html/2606.28925#bib.bib27)\), models short\-term context to predict multiple plausible next items under limited interaction history\. Dependency\-aware multilabel methods such as classifier chains\(Read and others,[2011](https://arxiv.org/html/2606.28925#bib.bib30)\)and ML\-kNN\(Zhang and Zhou,[2007](https://arxiv.org/html/2606.28925#bib.bib31)\)explicitly model label correlations, which makes them natural baselines when prompts may legitimately activate more than one agent\. Our routing task can therefore be seen as a controlled multi\-label routing problem where each prompt implies a compact set of relevant agents rather than a single intent label\.

### 2\.4\.Learning\-to\-Rank, Retrieval, and Evaluation Methodology

Learning\-to\-rank methods are standard in RS and retrieval for optimizing top\-N quality\. Pairwise and listwise ranking work\(Rendle and others,[2009](https://arxiv.org/html/2606.28925#bib.bib17); Caoet al\.,[2007](https://arxiv.org/html/2606.28925#bib.bib36)\)motivates viewing agent routing as score\-based selection over a fixed catalog rather than pure classification\. Dense and neural retrieval work\(Karpukhin and others,[2020](https://arxiv.org/html/2606.28925#bib.bib9); Khattab and Zaharia,[2020](https://arxiv.org/html/2606.28925#bib.bib12); Nogueira and Cho,[2019](https://arxiv.org/html/2606.28925#bib.bib18); Lin and others,[2021](https://arxiv.org/html/2606.28925#bib.bib19); Thakur and others,[2021](https://arxiv.org/html/2606.28925#bib.bib14)\)is also relevant because our KNN and encoder baselines rely on semantic text representations and prompt–agent matching rather than collaborative signals\. Steck\(Steck,[2013](https://arxiv.org/html/2606.28925#bib.bib28)\)shows that rating\-prediction accuracy does not necessarily correlate with ranking performance, while Jannach et al\.\(Jannach and others,[2010](https://arxiv.org/html/2606.28925#bib.bib29)\)provide broader guidance on offline evaluation protocols and metrics\. Our evaluation follows this practice by using set\-based precision, recall, F1, Jaccard, and exact match instead of only pointwise accuracy\.

### 2\.5\.Cost\-Aware Routing and Utility\-Aware Selection

Cost\-aware decision layers are important when routing choices trigger retrieval, analysis, external calls, or model invocations with different latency and monetary costs\. Related dialogue\-policy and decision\-making surveys emphasize the importance of balancing usefulness against downstream execution cost\. In LLM applications, cost\-aware routing and model\-selection policies have been explored for expert selection and cost reduction\(Lu and others,[2024](https://arxiv.org/html/2606.28925#bib.bib22); Ding and others,[2024](https://arxiv.org/html/2606.28925#bib.bib23); Chen and others,[2023](https://arxiv.org/html/2606.28925#bib.bib24)\)\. Our WAR variant adapts this utility\-aware perspective to agent routing by applying a deterministic weighted post\-scoring rule over ordinal agent tiers, enabling controllable trade\-offs between routing quality and execution cost\.

### 2\.6\.Multi\-Agent Systems and Task Decomposition

Broader literature on multi\-agent LLM systems addresses task decomposition, coordination, and execution\(Wu and others,[2024](https://arxiv.org/html/2606.28925#bib.bib15); Hong and others,[2024](https://arxiv.org/html/2606.28925#bib.bib16); Qian and others,[2024](https://arxiv.org/html/2606.28925#bib.bib25)\)\. These systems typically couple routing, planning, and execution inside a larger orchestration loop\. We instead isolate the initial routing decision and evaluate it with controlled baselines and a shared protocol\. This decomposition helps separate “who should handle this prompt” from later orchestration concerns, enabling clearer diagnosis of routing errors before downstream planning and execution are introduced\.

## 3\.Method

### 3\.1\.Problem setup and notation

Let𝒜=\{a1,…,aM\}\\mathcal\{A\}=\\\{a\_\{1\},\\dots,a\_\{M\}\\\}be a fixed catalog of agents and letxxbe an input prompt\. Each prompt has a gold setG​\(x\)⊆𝒜G\(x\)\\subseteq\\mathcal\{A\}with one or more valid agents\. The training data is𝒟=\{\(xn,Gn\)\}n=1N\\mathcal\{D\}=\\\{\(x\_\{n\},G\_\{n\}\)\\\}\_\{n=1\}^\{N\}\. The router outputs per\-agent scores and then a predicted setS^​\(x\)⊆𝒜\\hat\{S\}\(x\)\\subseteq\\mathcal\{A\}\. Each agent also has an ordinal cost tierc​\(a\)∈\{1,2,3\}c\(a\)\\in\\\{1,2,3\\\}representing relative execution cost\. The objective is to maximize overlap betweenS^​\(x\)\\hat\{S\}\(x\)andG​\(x\)G\(x\)while controlling cost\.

### 3\.2\.System overview

![Refer to caption](https://arxiv.org/html/2606.28925v1/hldx2.png)Figure 1\.Deployment view of the routing flow\. The benchmark implemented in this paper evaluates the routing stage directly and supplements it with a downstream capability\-coverage simulation; route\-map resolution \(agent ID to endpoint address\) and full downstream execution are shown as separate deployment components\.At inference, the system embeds the prompt, scores each agent, and produces a dynamic\-size setS^​\(x\)\\hat\{S\}\(x\)via a threshold rule with top\-1 fallback when the set is empty\. This set can be interpreted as the routed agent set for execution\.

### 3\.3\.Router internals

![Refer to caption](https://arxiv.org/html/2606.28925v1/ragent.png)Figure 2\.Router internals for set evaluation\. An input prompt is converted into shared prompt features and routed through the evaluated baselines\. Score\-producing routers \(KNN, linear ML, classifier chains, ML\-kNN, and the fine\-tuned encoder\) output a per\-agent score vector, which is converted into a predicted set by thresholding\. WAR is a deterministic weighted post\-scoring layer applied to these score vectors before constrained set selection\. The zero\-shot LLM and Majority baselines predict sets directly\. All predicted sets are then evaluated with set\-based accuracy metrics and deployment\-oriented utility measures\.The routing pipeline shares a common prompt\-feature view across methods, but the decision policies differ: Majority emits a fixed default set, KNN performs non\-parametric semantic matching, ML applies independent supervised per\-agent scoring, CC and ML\-kNN add explicit dependency\-aware multilabel structure, the fine\-tuned encoder supplies a stronger semantic scorer, the zero\-shot LLM predicts a set directly from the catalog, and WAR acts as a deterministic cost\-aware post\-scoring layer on top of score\-producing routers under ordinal tier costs\.

### 3\.4\.Embedding and agent profile representation

A sentence\-transformer encoderf​\(⋅\)f\(\\cdot\)maps text to vectors inℝd\\mathbb\{R\}^\{d\}; sentence\-transformer style encoders are standard for semantic matching and dense retrieval\(Reimers and Gurevych,[2019](https://arxiv.org/html/2606.28925#bib.bib10)\), and the implementation here uses the MPNet family backbone\(Song and others,[2020](https://arxiv.org/html/2606.28925#bib.bib11)\)\. For each promptxx, we computep=f​\(x\)p=f\(x\)\. For each agentaia\_\{i\}, we build a profile text \(role description, capability hints, and intent cues\) and computevi=f​\(profilei\)v\_\{i\}=f\(\\text\{profile\}\_\{i\}\)\. Embeddings areℓ2\\ell\_\{2\}\-normalized so cosine similarity is well\-defined and comparable across methods\.

### 3\.5\.KNN content\-based baseline

KNN computes per\-agent similarity as

\(1\)scoreknn​\(x,ai\)=p⊤​vi‖p‖​‖vi‖\.\\text\{score\}\_\{\\text\{knn\}\}\(x,a\_\{i\}\)=\\frac\{p^\{\\top\}v\_\{i\}\}\{\\\|p\\\|\\,\\\|v\_\{i\}\\\|\}\.Agents are ranked by this score and converted to a predicted set under the same threshold\-based set\-construction rule used throughout this paper\. KNN is interpretable and low\-latency but cannot exploit dataset\-specific decision boundaries between frequently co\-occurring labels\.

### 3\.6\.One\-vs\-Rest Linear SVM

Our primary baseline is a one\-vs\-rest linear SVM trained on prompt embeddings for multilabel agent prediction\. For each prompt, the target is a multi\-hot vectory∈\{0,1\}My\\in\\\{0,1\\\}^\{M\}, whereyi=1y\_\{i\}=1iffai∈G​\(x\)a\_\{i\}\\in G\(x\)\. We train one binary linear SVM per agent in a one\-vs\-rest scheme, using the prompt embedding as input and the corresponding agent indicator as the target\. At inference, each agent gets a scoresi​\(x\)s\_\{i\}\(x\)and predicted sets are produced by

\(2\)S^​\(x\)=\{ai\|si​\(x\)≥t\},\\hat\{S\}\(x\)=\\\{a\_\{i\}\\,\|\\,s\_\{i\}\(x\)\\geq t\\\},with top\-1 fallback when the set is empty\.

### 3\.7\.Dependency\-aware multilabel baselines

We include two standard multilabel baselines that explicitly model label structure\. Classifier Chains train a sequence of binary classifiers in which earlier label predictions are exposed as features for later labels, allowing the model to capture agent co\-occurrence dependencies\(Read and others,[2011](https://arxiv.org/html/2606.28925#bib.bib30)\)\. ML\-kNN estimates per\-agent posterior probabilities from the label distribution of nearest neighbors in embedding space\(Zhang and Zhou,[2007](https://arxiv.org/html/2606.28925#bib.bib31)\)\. Both methods use the same thresholded set\-construction rule as ML and KNN, keeping the comparison consistent across score\-producing routers\.

### 3\.8\.Fine\-tuned encoder baseline

We also evaluate a stronger content\-based model by fine\-tuning a sentence\-transformer encoder with a linear classification head for multilabel prediction\. The encoder directly maps each prompt to a dense embedding optimized for the task, and a sigmoid layer produces per\-agent probabilities for thresholded set prediction\. This baseline provides a higher\-capacity comparison against the linear SVM while retaining the same set\-evaluation protocol\.

### 3\.9\.Weighted Agent Routing \(WAR\)

Weighted Agent Routing \(WAR\) is a deterministic weighted post\-scoring layer applied on top of a score\-producing router such as ML or the encoder\. For a promptxx, the backbone produces per\-agent scoressi​\(x\)s\_\{i\}\(x\), and each agentaia\_\{i\}has an ordinal tier costc​\(ai\)∈\{1,2,3\}c\(a\_\{i\}\)\\in\\\{1,2,3\\\}\. WAR converts these scores into cost\-aware adjusted scores

\(3\)s~i​\(x\)=si​\(x\)−λ​c​\(ai\)\.\\tilde\{s\}\_\{i\}\(x\)=s\_\{i\}\(x\)\-\\lambda\\,c\(a\_\{i\}\)\.The routed set is then produced by

\(4\)S^WAR​\(x\)=\{ai\|s~i​\(x\)≥t\},\\hat\{S\}\_\{\\mathrm\{WAR\}\}\(x\)=\\\{a\_\{i\}\\,\|\\,\\tilde\{s\}\_\{i\}\(x\)\\geq t\\\},with top\-1 fallback when the set is empty\. Thus, WAR does not replace the base relevance model; instead, it changes the decision rule so that expensive agents must clear a higher effective score threshold\.

In the constrained setting reported later, we apply WAR to both ML and Encoder, tune\(t,λ\)\(t,\\lambda\)on the dev split by utility, and report test performance under the same ordinal tier environment\. This isolates whether a simple weighted selection layer can improve utility without retraining the underlying scorer\.

## 4\.Dataset

We construct a benchmark from public WildChat prompts\(Zhao and others,[2024](https://arxiv.org/html/2606.28925#bib.bib34)\)over a fixed catalog of 12 agents\. Prompts are filtered for agent\-relevant content and assigned AI\-assisted heuristic reference labels under the fixed inventory, after which we rebalance the pool to obtain controlled set\-size and agent\-coverage targets\. Because prompt\-to\-agent routing is not always uniquely determined, these labels are best read as protocol\-defined reference sets for controlled comparative evaluation\. The benchmark therefore emphasizes stable catalog coverage, set\-size diversity, and repeatable evaluation while still retaining the linguistic variability of real user prompts\. The benchmark containsN=3000N=3000prompts with a controlled set\-size distribution \(1/2/3 agents = 1800/900/300; average\|G\|=1\.50\|G\|=1\.50\) and balanced agent coverage \(375 occurrences per agent\)\.

Each sample is stored as a CSV row with prompt text, source metadata, and one or more gold agents\. Multi\-agent labels are retained to reflect workflows where a prompt may require retrieval plus analysis, or analysis plus reporting\. We keep this multi\-label structure throughout evaluation rather than collapsing labels to a single class\.

We use a standard stratified split by gold set size into 2,400 train, 300 dev, and 300 test prompts\. Dataset statistics \(N, M, split sizes, average\|G\|\|G\|, and per\-agent label counts\) are reported in the experimental setup section\.

To assess whether the labeling protocol is systematic rather than arbitrary, we also run a inter\-prompt consistency analysis over all 3,000 prompts\. Using a TF\-IDF cosine\-similarity graph over prompt text, we compare each prompt to its nearest neighbors and ask whether text\-similar prompts receive similar routed sets\. The result is strongly positive: the rank\-1 nearest neighbor has mean label\-set Jaccard 0\.601 versus 0\.091 for random pairs, share\-any\-label rate 0\.732 versus 0\.179, and exact set\-match rate 0\.490 versus 0\.034\. Across top\-5 neighbors, mean Jaccard remains 0\.494, and the Pearson correlation between prompt similarity and label\-set Jaccard is 0\.517\. This does not imply that every prompt has a single universally correct routed set, but it does indicate that the benchmark labels follow a coherent and learnable routing protocol rather than arbitrary assignment noise\.

![Refer to caption](https://arxiv.org/html/2606.28925v1/fig_gold_setsize_donut.png)Figure 3\.Gold set\-size distribution in the benchmark dataset\. The balanced split follows a 60/30/10 mix of 1\-, 2\-, and 3\-agent labels\.![Refer to caption](https://arxiv.org/html/2606.28925v1/fig_agent_pct_lollipop.png)Figure 4\.Per\-agent appearance rate \(percentage of prompts\)\. Controlled rebalancing yields nearly uniform agent coverage\.Table 1\.Example prompts with gold agent sets\.
## 5\.Experimental Setup

#### Data and splits\.

We use the WildChat\-derived 12\-agent benchmark defined in the Dataset section\. The evaluation uses the fixed 2,400/300/300 train/dev/test split overN=3000N=3000prompts andM=12M=12agents\. All methods are trained and evaluated against the same AI\-assisted heuristic reference labels under the fixed inventory\.

#### Evaluation protocol \(set prediction\)\.

For each method, per\-agent scores are converted to a predicted set viaS^​\(x\)=\{ai:si​\(x\)≥t\}\\hat\{S\}\(x\)=\\\{a\_\{i\}:s\_\{i\}\(x\)\\geq t\\\}with top\-1 fallback if empty\. We sweep thresholdst∈\{0\.4,0\.5,0\.6,0\.7,0\.8,0\.9\}t\\in\\\{0\.4,0\.5,0\.6,0\.7,0\.8,0\.9\\\}on the dev split and report the best global threshold on the test set while also reporting the sweep curve\. Per\-method optima are reported in results; we use a singlettfor fair comparison\.

#### Metrics\.

We report sample\-averaged Precision, Recall, F1, Jaccard, Exact Match, and average predicted set size\|S^\|\|\\hat\{S\}\|\. These metrics directly evaluate overlap between predicted and gold agent sets\.

#### Baselines\.

We include Majority \(most frequent agent\), KNN semantic matching, a linear multilabel classifier \(ML\), dependency\-aware multilabel baselines \(Classifier Chains\(Read and others,[2011](https://arxiv.org/html/2606.28925#bib.bib30)\)and ML\-kNN\(Zhang and Zhou,[2007](https://arxiv.org/html/2606.28925#bib.bib31)\)\), a fine\-tuned encoder baseline, a zero\-shot GPT\-4o router constrained to the fixed catalog, and a cost\-aware WAR policy layer\.

#### Models\.

The embedding\-based baselines use the same sentence\-transformer backbone \(all\-mpnet\-base\-v2\)\. ML uses a one\-vs\-rest LinearSVM with class\-weight balancing and sigmoid\-mapped margins for thresholding\. The encoder baseline fine\-tunes the same backbone with a linear multilabel head \(3 epochs, batch size 8, learning rate2×10−52\\times 10^\{\-5\}, weight decay1×10−21\\times 10^\{\-2\}\)\.

#### Weighted Agent Routing \(WAR\)

In the constrained study, WAR is a deterministic weighted post\-scoring layer applied to ML and Encoder\. For a backbone score vectorsi​\(x\)s\_\{i\}\(x\)and ordinal tier costc​\(ai\)∈\{1,2,3\}c\(a\_\{i\}\)\\in\\\{1,2,3\\\}, WAR usess~i​\(x\)=si​\(x\)−λ​c​\(ai\)\\tilde\{s\}\_\{i\}\(x\)=s\_\{i\}\(x\)\-\\lambda\\,c\(a\_\{i\}\)and predicts a set by thresholdings~i​\(x\)\\tilde\{s\}\_\{i\}\(x\)with top\-1 fallback\. We assign each agent to a coarse ordinal tier \(low/medium/high = 1/2/3\), sweep thresholdst∈\{0\.4,0\.5,0\.6,0\.7,0\.8,0\.9\}t\\in\\\{0\.4,0\.5,0\.6,0\.7,0\.8,0\.9\\\}and penaltiesλ∈\{0,0\.02,0\.05,0\.10,0\.15\}\\lambda\\in\\\{0,0\.02,0\.05,0\.10,0\.15\\\}on the dev split, and select the best setting by utility\. Because WAR only rescales existing scores, its additional inference cost is negligible relative to the base scorer\.

#### Seeds and reproducibility\.

We report the WildChat\-12 threshold analysis from the completed dev sweep and summarize the main test table with three\-seed aggregates\. Deterministic methods \(Majority, KNN, ML, MLkNN, and ML\+WAR\) are invariant under the fixed split and therefore show zero variance\. CC, Encoder, and Encoder\+WAR show modest seed\-to\-seed variation\. Embeddings areℓ2\\ell\_\{2\}\-normalized and cosine similarity is used for KNN\. All methods share the same splits and evaluation protocol\.

#### Execution\-oriented simulation\.

To complement set\-level metrics, we run a capability\-coverage simulation on the test split\. For a prompt with reference setG​\(x\)G\(x\)and predicted setS^​\(x\)\\hat\{S\}\(x\), task success is𝟏​\[G​\(x\)⊆S^​\(x\)\]\\mathbf\{1\}\[G\(x\)\\subseteq\\hat\{S\}\(x\)\], coverage is\|G​\(x\)∩S^​\(x\)\|/\|G​\(x\)\|\|G\(x\)\\cap\\hat\{S\}\(x\)\|/\|G\(x\)\|, cost is the sum of ordinal agent tiers overS^​\(x\)\\hat\{S\}\(x\), and utility isCoverage−0\.10⋅Cost−0\.05⋅\|S^​\(x\)∖G​\(x\)\|\\mathrm\{Coverage\}\-0\.10\\cdot\\mathrm\{Cost\}\-0\.05\\cdot\|\\hat\{S\}\(x\)\\setminus G\(x\)\|\. This tests whether a predicted routing set covers the required capabilities under a simple cost\-sensitive proxy\.

## 6\.Results

Main set\-evaluation results\.We sweep thresholds from 0\.4 to 0\.9 in increments of 0\.1 on the dev split and select a globalt=0\.60t=0\.60\(optimal for ML and Encoder\)\. KNN is nearly flat, improving slightly from F1 39\.41 att=0\.40t=0\.40to 39\.72 fromt=0\.50t=0\.50onward, while Majority is effectively constant over this sweep range; we keept=0\.60t=0\.60for comparability\. Table[2](https://arxiv.org/html/2606.28925#S6.T2)reports the current test\-set results at this shared operating point\. The ranking is clear at the shared operating point: the encoder is the strongest overall method \(F189\.59±0\.3389\.59\\pm 0\.33, Jaccard85\.52±0\.3085\.52\\pm 0\.30, Exact Match73\.11±0\.1973\.11\\pm 0\.19\), while the linear ML model substantially outperforms KNN and Majority \(F1 71\.79 versus 42\.56 and 13\.89\)\. CC reaches F165\.13±1\.0465\.13\\pm 1\.04with higher recall \(71\.96±0\.6171\.96\\pm 0\.61\) and larger set size \(1\.65±0\.011\.65\\pm 0\.01\), while MLkNN yields F1 63\.10 with stronger precision \(71\.44\) but lower recall\. This places the dependency\-aware baselines in a useful middle regime: modeling label interactions improves over simple retrieval, but stronger semantic representations still matter more than label\-dependency modeling alone\. Cost\-aware WAR is evaluated separately in the constrained\-routing study of Table[4](https://arxiv.org/html/2606.28925#S6.T4)rather than in this unconstrained set\-accuracy table\. In the reported three\-seed summary, the deterministic baselines remain at 0\.00 standard deviation at the displayed precision, while CC and Encoder show small but nonzero variation\.

Zero\-shot LLM baseline\.We evaluate a zero\-shot GPT\-4o router using a constrained JSON schema over the fixed agent catalog and instructions to choose a minimal sufficient set of 1–3 agents\. Table[2](https://arxiv.org/html/2606.28925#S6.T2)shows that the zero\-shot LLM trails the supervised baselines by a wide margin on this benchmark \(F1 41\.51\)\. Its precision \(46\.22\) and recall \(42\.33\) are both far below ML and the encoder, while Avg\|S\|\|S\|\(1\.40\) remains close to the gold average \(1\.50\)\. This result makes a strong case that zero\-shot prompting alone is not sufficient for reliable routing under the fixed 12\-agent inventory used here\.

Threshold behavior\.Figure[5](https://arxiv.org/html/2606.28925#S6.F5)visualizes the sweep used to choose the operating threshold\. The dev sweep shows the expected precision–recall trade\-off: ML improves from F1 58\.31 att=0\.40t=0\.40to 67\.64 att=0\.50t=0\.50and peaks at 70\.09 att=0\.60t=0\.60, after which recall falls and F1 drops to 67\.00 att=0\.70t=0\.70\. The encoder shows the same pattern more sharply, rising from F1 73\.41 att=0\.50t=0\.50to 90\.42 att=0\.60t=0\.60, then declining to 83\.89 att≥0\.70t\\geq 0\.70\. KNN is effectively flat fromt=0\.50t=0\.50onward \(F1 39\.72\), while Majority is constant; we therefore keept=0\.60t=0\.60as the common operating point for comparison\.

![Refer to caption](https://arxiv.org/html/2606.28925v1/fig_threshold_sweep.png)Figure 5\.Threshold sweep on the dev split \(mean±\\pmstd over three seeds\)\. ML and Encoder both peak neart=0\.60t=0\.60, while lower thresholds over\-select and higher thresholds suppress recall\.![Refer to caption](https://arxiv.org/html/2606.28925v1/fig_pr_scatter.png)Figure 6\.Precision–recall scatter under the selected unconstrained set\-evaluation protocol \(t=0\.60t=0\.60\)\. Each point denotes one unconstrained routing method at the shared operating point\.![Refer to caption](https://arxiv.org/html/2606.28925v1/fig_avg_set_size.png)Figure 7\.Average predicted set size\|S^\|\|\\hat\{S\}\|att=0\.60t=0\.60for the aggregated three\-seed unconstrained results\. Larger values indicate more multi\-agent dispatch decisions\. The cost\-aware WAR variants are analyzed separately in the constrained study\.Table 2\.Main set\-eval results at thresholdt=0\.60t=0\.60\. Deterministic baselines report identical values across the three completed test seeds\. CC and Encoder are reported as mean±\\pmstd over those same three seeds and show modest variation\. The LLM row is single\-run\. Cost\-aware WAR is evaluated separately in Table[4](https://arxiv.org/html/2606.28925#S6.T4)\.Table 3\.Per\-query latency and execution\-oriented capability\-coverage simulation on the test split \(300 queries\)\. Success denotes full capability coverage, Coverage reports\|G∩S^\|/\|G\|\|G\\cap\\hat\{S\}\|/\|G\|, Cost sums ordinal agent tiers, Extra counts\|S^∖G\|\|\\hat\{S\}\\setminus G\|, and Utility isCoverage−0\.10⋅Cost−0\.05⋅Extra\\mathrm\{Coverage\}\-0\.10\\cdot\\mathrm\{Cost\}\-0\.05\\cdot\\mathrm\{Extra\}\. Latency is reported in milliseconds from a representative run\. KNN and ML are deterministic under the fixed split, Encoder reports mean±\\pmstd over three seeds, and the LLM row is single\-run\.Execution\-oriented simulation\.Table[3](https://arxiv.org/html/2606.28925#S6.T3)complements set accuracy with a downstream capability\-coverage simulation\. The encoder again dominates, reaching task success 74\.33±\\pm1\.20%, coverage 86\.07±\\pm0\.68%, and the highest utility \(0\.654±\\pm0\.006\) while adding almost no extra agents \(0\.03±\\pm0\.00\)\. ML is the strongest practical baseline by utility \(0\.465\), while the zero\-shot LLM remains lower\-performing both in utility \(0\.183\) and latency\. The gap between ML and the encoder is especially informative here: the encoder’s higher set accuracy translates into more complete capability coverage without materially increasing unnecessary dispatch, which is exactly the behavior desired in a deployment\-oriented router\. Table[3](https://arxiv.org/html/2606.28925#S6.T3)evaluates the base routed sets only; explicit cost\-aware decision\-time adaptation is studied separately in Table[4](https://arxiv.org/html/2606.28925#S6.T4)\.

Table 4\.Constrained weighted\-routing study under ordinal tier costs\. Base thresholds and WAR hyperparameters are selected on the dev split by utility\. WAR is a deterministic weighted post\-scoring layer applied to the base scorer\.Constrained weighted routing\.Table[4](https://arxiv.org/html/2606.28925#S6.T4)distinguishes cost\-aware evaluation from cost\-aware decision making by applying WAR directly to the base score vectors\. For ML, WAR improves utility from 0\.465 to 0\.480 by lowering average cost \(2\.67 to 2\.48\) while preserving success and coverage, although it gives up some F1 and Exact Match\. This is the expected behavior of a constrained post\-scoring layer: once cost enters the decision rule, the selected set becomes slightly more selective and therefore trades some unconstrained overlap for better utility\. For the encoder, WAR yields the strongest constrained result overall, raising success from 74\.33% to 92\.11%, coverage from 86\.07% to 96\.39%, and utility from 0\.654 to 0\.670, at the cost of higher dispatch cost and more extra agents\. The stronger gain on top of the encoder suggests that WAR benefits most when the underlying score distribution is already sharp and semantically well calibrated: in that regime, a simple tier\-aware adjustment can shift the operating point toward more complete capability coverage without destabilizing the ranking\. This makes WAR a useful constrained\-selection layer rather than a replacement for the base scorer\. WAR should therefore be interpreted as utility\-oriented operating\-point adjustment rather than cost minimization alone: for ML the gain comes mainly from lowering dispatch cost at nearly unchanged coverage, whereas for Encoder the gain comes from a much larger increase in capability coverage that outweighs the additional cost and extra\-agent penalties\. Figure[8](https://arxiv.org/html/2606.28925#S6.F8)makes this trade\-off explicit: utility peaks at a moderate penalty for ML \(λ=0\.10\\lambda=0\.10att=0\.40t=0\.40\), while the encoder reaches its best setting under a lighter penalty \(λ=0\.02\\lambda=0\.02att=0\.50t=0\.50\), showing that stronger base scorers require less aggressive cost adjustment\.

![Refer to caption](https://arxiv.org/html/2606.28925v1/fig_war_tradeoff.png)Figure 8\.WAR utility trade\-off on the dev split\. Each curve fixes the method\-specific selected threshold and varies the cost penaltyλ\\lambda; markers indicate the selected operating point for ML and Encoder\.
## 7\.Discussion

The results establish three primary conclusions about set\-valued routing under the reported benchmark protocol\.

First, the fine\-tuned encoder is the clear best model at the selected operating point \(t=0\.60t=0\.60\)\. In Table[2](https://arxiv.org/html/2606.28925#S6.T2), the encoder reaches F189\.59±0\.3389\.59\\pm 0\.33and Jaccard85\.52±0\.3085\.52\\pm 0\.30, showing that task\-specific fine\-tuning of the encoder yields markedly more reliable routing sets than linear scoring or nearest\-neighbor matching\. This ordering is also consistent with prior dense retrieval and semantic\-matching results showing that higher\-capacity encoder models, especially when fine\-tuned, typically outperform simpler linear or non\-parametric baselines on meaning\-sensitive matching tasks\(Karpukhin and others,[2020](https://arxiv.org/html/2606.28925#bib.bib9); Khattab and Zaharia,[2020](https://arxiv.org/html/2606.28925#bib.bib12); Lin and others,[2021](https://arxiv.org/html/2606.28925#bib.bib19); Thakur and others,[2021](https://arxiv.org/html/2606.28925#bib.bib14)\)\. The same pattern holds under the execution\-oriented simulation in Table[3](https://arxiv.org/html/2606.28925#S6.T3), where the encoder achieves the highest task success \(74\.33±\\pm1\.20%\) and the highest utility \(0\.654±\\pm0\.006\) with almost no extra\-agent dispatch\.

Second, the linear ML model is a strong practical baseline with a favorable efficiency–accuracy trade\-off\. ML substantially outperforms KNN and Majority while remaining easy to retrain as the agent catalog evolves\. In Table[3](https://arxiv.org/html/2606.28925#S6.T3), ML is markedly faster than the LLM baseline and retains a solid downstream utility score \(0\.465\), making it the strongest practical alternative when a fine\-tuned encoder is not available\. CC and MLkNN occupy a middle ground between retrieval and the stronger supervised scorers: they confirm that modeling label dependencies is helpful, but the main bottleneck in this benchmark is still semantic representation quality rather than dependency structure alone\. That ML remains ahead of both CC and MLkNN suggests that, on this benchmark, most of the usable routing signal is already captured by the prompt representation and per\-agent scoring; explicit dependency modeling helps, but not enough to overcome its added rigidity and, for CC, possible chain error propagation\. The zero\-shot LLM result adds a useful contrast: broad generative capability by itself is not enough for dependable fixed\-catalog routing, where calibrated set selection matters more than free\-form tool choice\.

Third, WAR is most useful as a deterministic constrained\-selection layer rather than as a standalone router\. In Table[4](https://arxiv.org/html/2606.28925#S6.T4), ML\+WAR improves utility from 0\.465 to 0\.480 by lowering average cost while preserving success, though it gives up some unconstrained set accuracy\. The effect is stronger for Encoder\+WAR, which raises success from 74\.33% to 92\.11%, coverage from 86\.07% to 96\.39%, and utility from 0\.654 to 0\.670\. This suggests that when routing must respect tier\-weighted costs, a simple score\-adjustment layer can improve decision quality without retraining the backbone\.

The threshold sweep reinforces these findings: low thresholds over\-select and reduce precision, while higher thresholds suppress recall\. In the reported dev sweep, ML peaks at F1 70\.09 and the encoder at F1 90\.42 whent=0\.60t=0\.60; both decline when the threshold is raised further\. This interior optimum provides a practical default and suggests that score calibration is reasonable but not perfect\.

Although the benchmark labels are protocol\-defined reference labels, the empirical behavior is internally coherent: stronger supervised routers consistently outperform simpler and zero\-shot alternatives, the relative ordering is stable across seeds, and the threshold curves show smooth precision–recall trade\-offs rather than erratic swings\. The inter\-prompt consistency analysis points in the same direction: text\-similar prompts have far higher label overlap than random pairs \(rank\-1 Jaccard 0\.601 versus 0\.091\), which is what one would expect from a benchmark with meaningful routing structure rather than arbitrary noise\.

This interpretation is important because prompt\-to\-agent routing does not always admit a single universally preferred target set\. Depending on deployment assumptions, different but still defensible routed sets may be chosen because of differences in task decomposition, desired output form, redundancy tolerance, or cost sensitivity\. In that setting, the most important property of the benchmark is not uniqueness of every individual label set, but whether the labeling protocol is systematic, learnable, and stable enough to support comparative evaluation\. The results here suggest that it is\.

Taken together, these outcomes suggest a pragmatic deployment recipe: use the encoder when maximum unconstrained accuracy is required, use Encoder\+WAR when tier\-weighted constraints matter, use ML when efficiency and retrainability are primary constraints, use ML\+WAR when low\-overhead cost\-aware selection is desired, and retain KNN as a transparent baseline\.

## 8\.Limitations

Our benchmark uses real WildChat prompts together with AI\-assisted heuristic reference labels under a fixed routing protocol\. Because prompt\-to\-agent routing can admit multiple defensible routed sets, some assignment uncertainty remains, especially for semantically overlapping or borderline multi\-agent cases\. Different reasonable deployments may also prefer different sets depending on how they trade off concise versus formal outputs, descriptive analysis versus forecasting, or minimal versus redundant multi\-agent dispatch\. The benchmark is therefore best interpreted as a controlled comparative testbed for routing methods rather than as a claim that every prompt has a single universally preferred routed set\.

This ambiguity is partly mitigated by two observations\. First, all methods are evaluated under the same protocol, so the benchmark remains suitable for controlled comparison\. Second, the main empirical patterns recur across unconstrained set metrics, execution\-oriented simulation, the constrained weighted\-routing study, and the inter\-prompt consistency analysis, all of which point to a coherent and learnable labeling scheme\. This trade\-off between ecological realism and tighter annotation control is common in open\-ended evaluation collections, where broader prompt diversity improves coverage but also increases annotation ambiguity and benchmark heterogeneity\(Zhao and others,[2024](https://arxiv.org/html/2606.28925#bib.bib34); Thakur and others,[2021](https://arxiv.org/html/2606.28925#bib.bib14)\)\. In addition, the benchmark is rebalanced for evaluation stability, so performance under the natural WildChat distribution may differ from the numbers reported here\.

The benchmark focuses on a fixed 12\-agent catalog over 3,000 prompts, providing a controlled setting for evaluation; broader catalog scaling remains future work\. Set prediction depends on score calibration and threshold choice; different operating environments may require re\-tuning\.

WAR uses ordinal cost tiers rather than measured, context\-dependent costs \(latency spikes, rate limits, provider outages\), so its trade\-off curves are best interpreted as deployment\-oriented guidance rather than universal optima\. Our downstream analysis uses a capability\-coverage simulation rather than real agent execution, which keeps the study controlled but leaves execution ordering, API failures, online feedback, and user studies to future work\.

## 9\.Broader Impact

This work targets practical gains in multi\-agent systems by improving routing quality with efficient, retrainable models\. Better routing can reduce unnecessary tool calls, lower latency, and improve reliability in downstream task execution, especially when prompts require multiple coordinated agents\. In settings where agent invocation carries monetary cost, latency overhead, or external\-call risk, routing quality directly affects both user experience and system efficiency\.

The main risk is incorrect set selection: under\-selection can omit required capabilities, while over\-selection can trigger unnecessary actions, increased cost, or broader attack surface through avoidable external calls\. In high\-stakes domains, these errors may propagate to user\-facing decisions\. WAR\-style cost\-aware selection is relevant here because it makes the accuracy–cost trade\-off explicit rather than leaving it implicit in a fixed threshold\.

Operational safeguards are therefore essential: confidence\-aware fallbacks, threshold calibration on held\-out traffic, rate limits on fan\-out dispatch, endpoint authorization, and audit logs for post\-hoc error analysis\. For sensitive applications, human review should remain in the loop for low\-confidence or high\-impact prompts\. It is also important to monitor routing behavior across prompt types, domains, and languages so that a router does not systematically under\-serve less frequent tasks simply because they appear less often in the benchmark\.

The benchmark used in this study is derived from public WildChat prompts and fixed agent labels\. Real deployments must still enforce secure handling of prompt logs, access control for agent endpoints, and monitoring for misuse or drift that could bias routing decisions over time\. More broadly, framing routing as set\-valued prediction may help future systems move from ad hoc tool selection toward more transparent, auditable, and resource\-aware orchestration policies\.

## 10\.Conclusion

This paper studies agent routing as a set\-valued prediction problem over a fixed catalog and evaluates that formulation on a WildChat\-derived benchmark built from real prompts and AI\-assisted heuristic reference labels under a fixed 12\-agent inventory\. Across KNN, linear multilabel, dependency\-aware, encoder, WAR\-augmented, and zero\-shot LLM baselines, the results show a clear and stable pattern: stronger supervised semantic models produce better routed sets, while fixed\-catalog routing quality is best understood through set overlap, downstream coverage, and cost\-aware utility rather than top\-1 accuracy alone\.

In the unconstrained setting, the fine\-tuned encoder is the strongest router by a substantial margin, while the linear ML model remains a strong alternative with favorable efficiency and retrainability\. In the constrained tier\-weighted setting, WAR is most useful as a deterministic post\-scoring decision layer: it improves utility for both ML and Encoder, with the largest gain on top of the encoder, showing that simple cost\-aware selection rules can define practical operating points once the base scorer is strong\. Taken together, the results support three claims: first, fixed\-catalog agent routing is naturally and usefully set\-valued; second, separating relevance scoring from constrained selection provides a clean way to study accuracy–cost trade\-offs in multi\-agent systems; and third, even when prompt\-to\-agent assignments are not uniquely determined, a protocol\-defined benchmark can still be systematic and learnable enough to support meaningful comparative evaluation\.

Future work includes evaluation under natural prompt distributions, broader catalog scaling, measured execution costs, and end\-to\-end task success studies beyond the capability\-coverage simulation reported here so that routing quality can be tied more directly to downstream user outcomes\.

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