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Unifying Temporal and Structural Credit Assignment in LLM-Based Multi-Agent Prompt Optimization
Wenwu Li, Yuran Song, Mingze Zhao, Bo Jin, Wenhao Li
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
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Summary
The paper addresses the Credit Assignment Problem in LLM-based Multi-Agent Systems (MAS) by proposing a unified framework for temporal and structural credit assignment. It introduces a state-space bottleneck via an aggregation module and a stationary policy constraint to decompose sparse terminal rewards into actionable feedback. Leveraging LLM-based critics to compute proxy gradients, the authors develop a verbalized block coordinate descent algorithm that alternately optimizes role-specific and aggregation prompts. This approach reduces query complexity and improves performance on reasoning benchmarks by targeting weak links rather than applying indiscriminate global updates.
Entities (9)
Relation Signals (10)
Multi-Agent Systems (MAS) → suffersfrom → Credit Assignment Problem (CAP)
confidence 94% · MAS optimization suffers from the Credit Assignment Problem (CAP) in its most severe form: the supervisory signal is typically sparse
Verbalized Block Coordinate Descent → optimizes → Aggregation Prompts
confidence 91% · it alternates between optimizing role prompts and aggregation protocols
Verbalized Block Coordinate Descent → optimizes → Role Prompts
confidence 91% · it alternates between optimizing role prompts and aggregation protocols
Temporal Credit Assignment → decomposes → Objective
confidence 90% · decomposes the objective along two axes: (i) temporal credit, using state-space bottlenecks to identify critical rounds
Structural Credit Assignment → isolates → Agent Contributions
confidence 90% · structural credit, using stationary role policies to isolate agent contributions
State-Space Bottleneck → enables → Temporal Credit Assignment
confidence 89% · This architectural choice is not merely functional; it is a prerequisite for Temporal Credit Assignment
LLM-based Critics → computes → Proxy Gradients
confidence 88% · utilize LLM-based critics to compute “proxy gradients”—decomposed credit scores that guide targeted textual updates
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Abstract
Abstract:While Multi-Agent Systems (MAS) empower Large Language Models to tackle complex reasoning tasks through collaborative interaction, optimizing their dynamics remains a formidable challenge due to the discrete, non-differentiable nature of the computation graph and the sparsity of global supervisory signals. Existing black-box optimizers struggle to attribute trajectory-level failure to specific local components, resulting in inefficient, high-variance exploration. We argue that tractable MAS optimization needs structural inductive biases to disentangle error signals. We propose temporal and structural credit assignment, which decomposes the objective along two axes: (i) temporal credit, using state-space bottlenecks to identify critical rounds, and (ii) structural credit, using stationary role policies to isolate agent contributions. Leveraging these decomposed signals, we introduce a discrete, verbalized block coordinate descent algorithm for iterative refinement. Rather than indiscriminate global updates, it alternates between optimizing role prompts and aggregation protocols, using LLM-generated "proxy gradients" to target only the identified weak links. Across diverse reasoning benchmarks, our approach substantially reduces query complexity while improving performance, providing a principled and interpretable path toward self-improving MAS.
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- Source: https://arxiv.org/abs/2605.30227v1
- Canonical: https://arxiv.org/abs/2605.30227v1
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Unifying Temporal and Structural Credit Assignment in LLM-Based Multi-Agent Prompt Optimization Wenwu Li 1 Yuran Song 1 Mingze Zhao 2 Bo Jin 1 Wenhao Li * 1 1 Tongji University, Shanghai, China, wenwu,2250753,bjin,whli@tongji.edu.cn 2 The University of Hong Kong, Hong Kong, China, zhaomingze@connect.hku.hk Abstract While Multi-Agent Systems (MAS) empower Large Language Models to tackle complex rea- soning tasks through collaborative interaction, op- timizing their dynamics remains a formidable chal- lenge due to the discrete, non-differentiable nature of the computation graph and the sparsity of global supervisory signals. Existing black-box optimiz- ers struggle to attribute trajectory-level failure to specific local components, resulting in inefficient, high-variance exploration. We argue that tractable MAS optimization needs structural inductive bi- ases to disentangle error signals. We propose tem- poral and structural credit assignment, which decomposes the objective along two axes: (i) tem- poral credit, using state-space bottlenecks to iden- tify critical rounds, and (i) structural credit, using stationary role policies to isolate agent contribu- tions. Leveraging these decomposed signals, we introduce a discrete, verbalized block coordinate descent algorithm for iterative refinement. Rather than indiscriminate global updates, it alternates between optimizing role prompts and aggregation protocols, using LLM-generated “proxy gradients” to target only the identified weak links. Across diverse reasoning benchmarks, our approach sub- stantially reduces query complexity while improv- ing performance, providing a principled and inter- pretable path toward self-improving MAS. 1 INTRODUCTION Large Language Models (LLMs) have evolved from static text generators into dynamic reasoning engines, achieving remarkable success in mathematical reasoning, code gener- ation, and complex planning [Brown et al., 2020, Wei et al., * Corresponding author. 2022]. To transcend the limitations of single-model infer- ence, recent research has shifted toward MAS, where spe- cialized agents collaborate via iterative critique-and-revision cycles [Zhuge et al., 2024, Zhang et al., 2025a]. By dis- tributing tasks across diverse roles (e.g., proposer, debater, synthesizer), MAS frameworks can theoretically solve prob- lems that are intractable for a single monolithic model [Li et al., 2024a]. However, optimizing the interaction dynamics of MAS presents a formidable challenge. From an optimization per- spective, an LLM-based MAS operates as a discrete, non- differentiable computation graph. Unlike neural network training where backpropagation precisely attributes error to specific weights, MAS optimization suffers from the Credit Assignment Problem (CAP) in its most severe form: the supervisory signal is typically sparse (a single scalar score at the terminal state) and global (applying to the entire tra- jectory) [Sutton and Barto, 2018, Foerster et al., 2018]. Ex- isting approaches often treat the entire system as a black box [Deng et al., 2022, Fernando et al., 2023, Guo et al., 2023], applying derivative-free optimization or heuristic aggregation [Nagpal et al., 2025, Ai et al., 2025]. While effective for simple chains, these methods struggle in complex multi-round collaborations because they ignore the internal causal structure of the reasoning process. Opti- mizing a system withNagents overRrounds essentially involves searching in a space ofO(N×R)coupled prompts. Without structural priors, this faces catastrophic variance and inefficient exploration [Zhou et al., 2024]. In this work, we argue that making MAS optimization tractable requires imposing structural constraints on the computation graph to disentangle the error signal. We pro- pose two principled relaxations that transform the chaotic interaction graph into a structured optimization landscape. First, state-space bottleneck via aggregation. In a fully con- nected MAS, information flows diffusely, making it impos- sible to pinpoint when reasoning collapsed. We introduce an explicit Aggregation Module at each round to summarize Preprint. May 29, 2026. arXiv:2605.30227v1 [cs.MA] 28 May 2026 EXAMPLE TASK: TRAVEL ITINERARY PLANNING CONCRETE EXAMPLE ILLUSTRATION EXAMPLE REASONING ROUNDS Based on Credit Assignment BEFORE / AFTER PROMPT UPDATE AFTER PROMPT Specific Guidance (Based on R2 & Agent A): Focus on refining daily plans after initial flight selection (R2). Prioritize destination logic and flow over initial cost (Agent A). BEFORE PROMPT General Goal: Create a 5-day travel itinerary. Spatiotemporal Credit-Guided Prompt Optimization Framework Prompts are reused across rounds, but contributions to the final outcome vary. Section 1-Multi-Agent Iterative Reasoning Agent A (Planner) Agent B (Solver) Agent C (Critic) Round 1 Round 2 Round R Task Input ... Section 2-Aggregation Module(Core Mechanism) Shared System State S r Inputs: Agent messages from the current round Aggregation Module . . . Feedback for next round Round 1 Initial destinations... Agent A: 0.89 Agent B: Flight options... 0.21 Agent C: Hotel suggestions... 0.34 Shared System State S 1 Round 2 Refined plan based on flight... Agent A: 0.13 Agent B: Adjusted costs... 0.91 Agent C: Added local activities... 0.24 Shared System State S 2 S 1 0.71 S 2 0.27 S r 0.34 ... TEMPORAL CREDIT Section 3-Spatiotemporal Credit Assignment A. Temporal Credit Assignment (Across Rounds) Inputs: Aggregated system states S 1 , S 2 , ..., S R Measure marginal impact of each aggregation update on the final outcome. Rank and normalize round-level contributions. Temporal Credit Weights ... B. Spatial Credit Assignment (Across Agents) Attribute outcome changes to agent-specific contributions. Aggregate evidence across rounds. Agent A Agent B Agent C High Medium Low Identify limiting or effective agent roles Identify critical reasoning rounds and suppress low-value states Inputs: Agent outputs across rounds Section 4-Credit-Guided Prompt Update (Inference-Time) Spatial Weights Temporal Weights Prompt A Agent A Agent B Agent C Prompt B Prompt C Shared System State S r Aggregation Module Prompt Aggregation + Message A + Message B + Message C ... Figure 1: Overview of the credit-guided prompt optimization pipeline. Top: a multi-agent, multi-round reasoning loop (planner/solver/critic) produces per-round messages that are aggregated into a shared system stateS r ; an aggregation module feeds back to the next round. From the completed trajectory, we compute temporal credit across rounds (identifying critical rounds) and structural/spatial credit across agents (identifying effective or limiting roles). These credits then drive inference-time prompt updates, selectively refining the lowest-credit rounds/roles while keeping strong components fixed. Bottom: an example travel-itinerary task illustrates per-round agent outputs, the evolving shared state (S 1 ,S 2 ,...), temporal credit weights, and a before/after prompt update that specializes guidance to the weak round/role. agent outputs into a unified stateS t . This effectively models the interaction as a Markov Decision Process (MDP), where the aggregator creates a “state bottleneck” [Puterman, 1994, Mnih et al., 2015]. This architectural choice is not merely functional; it is a prerequisite for Temporal Credit Assign- ment, allowing us to evaluate the quality of consensus at discrete time steps independent of individual agent noise. Second, stationary policy via parameter sharing [Gupta et al., 2017, Yu et al., 2022]. Optimizing unique prompts for every agent at every round creates an explosion of parameters. We impose a Stationary Policy Constraint, where each role (e.g., the Debater) shares the same system prompt across all inter- action rounds. This reduces the search space from trajectory- specific instructions to robust role definitions. Crucially, this constraint enables reliable Structural Credit Assignment: by observing an agent’s performance across multiple rounds, we can distinguish systematic role incompetence from tran- sient stochastic errors. Building on these structural insights, we propose a unified framework for Temporal and Structural Credit Assign- ment, as shown in Figure 1. We formalize the MAS op- timization problem as a bi-level objective depending on two orthogonal variable blocks: (i) Role Prompts (defin- ing agent behaviors) and (i) Aggregation Prompts (defin- ing state transition protocols). Instead of optimizing these jointly—which is prone to instability—we derive a novel Block Coordinate Descent (BCD) algorithm adapted for discrete prompt optimization [Tseng, 2001, Cai et al., 2023]. Our method alternates between refining role-specific instruc- tions (fixing the aggregation logic) and refining the aggre- gation protocol (fixing the role behaviors). To drive this descent, we utilize LLM-based critics to compute “proxy gradients”—decomposed credit scores that guide targeted textual updates only to the bottleneck components. Our contributions are summarized as follows: 1) We re- formulate MAS prompt optimization by introducing state- space bottlenecks and stationary policy constraints. These inductive biases bridge the gap between sparse terminal rewards and dense, actionable feedback. 2) We propose a theoretically grounded optimization algorithm that alter- nates between structural (role) and temporal (aggregator) up- dates. This mimics Block Coordinate Descent in the discrete prompt space, ensuring stability and convergence compared to joint optimization baselines. 3) Numerical experiments demonstrate that our approach achieves higher task accuracy while reduces query complexity. By focusing updates on specific "weak link" roles or "phase transition" rounds, we avoid the inefficiency of indiscriminate global updates. 2 RELATED WORKS Prompt Optimization. Several methods have been de- veloped for optimizing prompts, ranging from learning parameter-efficient soft prompts to automatically searching for optimal discrete prompts using reinforcement learning, meta-optimization, and evolutionary algorithms. Prompt- Tuning [Lester et al., 2021] freezes the parameters of a pre- trained language model and learns task-specific soft prompts via end-to-end backpropagation. RLPrompt [Deng et al., 2022] optimizes discrete textual prompts through reinforce- ment learning with a policy network. OPRO [Yang et al., 2023] frames prompt optimization as an iterative black- 2 box optimization problem. EvoPrompt [Guo et al., 2023] integrates evolutionary algorithms with large language mod- els by employing the LLM to implement linguistically co- herent crossover and mutation operations. PromptBreeder [Fernando et al., 2023] implements a self-referential self- improvement mechanism for prompts using a genetic algo- rithm that co-evolves task-prompts and mutation-prompts. LLM-based MAS. Current research on LLM-based multi- agent systems spans four key dimensions: system represen- tation, optimization mechanisms, dynamic adaptation, and reasoning efficiency. GPTSwarm [Zhang et al., 2024] ex- emplifies offline structural learning via static post-training optimization with strong theoretical grounding in policy- gradient methods. DyLAN [Liu et al., 2024] represents un- supervised online selection, offering principled agent eval- uation without labeled data and drawing theoretical con- nections to Shapley-value attribution. EvoMAC [Li et al., 2024b] reflects environment-driven online adaptation, en- abling test-time improvement from objective feedback and linking classical control-theoretic feedback to LLM gen- eration. MAS-GPT [Chen et al., 2024] illustrates gener- ative offline learning, trading development-time cost for inference-time efficiency and cross-domain transfer. Credit Assignment. The credit assignment problem refers to determining the contribution of each agent to the collec- tively obtained reward in a multi-agent system. In reinforce- ment learning context, when an agent receives a reward, it is challenging to determine which actions should be credited or blamed for the outcome. In multi-agent systems context, under the centralized training with decentralized execution paradigm, it is required to allocate contributions from joint decisions to individual agents. Value decomposition meth- ods address this challenge by employing mixing networks to decompose the joint state-action value function into indi- vidual local observation-action value functions. MATTRL [Hu et al., 2026] implements inference-time credit assign- ment via Difference Rewards for precise contribution at- tribution in multi-agent reasoning, while MAPRO [Zhang et al., 2025b] employs training-time credit assignment for joint optimization of discussion and answer generation. 3 PROBLEM FORMULATION We now formalize the multi-agent, multi-round prompting setting that our temporal and structural credit assignment targets. We specify the interaction protocol, notation, and the optimization variables, which will let us define per-round and per-role credit signals precisely. This section sets up the trajectory and objective used by the attribution method and the credit-guided prompt optimization procedure. Multi-Agent, Multi-Round Prompting. We consider a multi-agent, multi-round LLM reasoning system with a fixed set of agents (roles)A = a 1 ,...,a N andRin- teraction rounds. The base LLM parameters are fixed; the optimization variables are the prompts used by each agent at each round. Letφ i,t denote the prompt fed to agenta i at round t. We collect all prompts as Φ ≜φ i,t N,R i=1,t=1 . Given an input instancex ∼ D, the system runs forR rounds. At roundt, each agenta i produces an utteranceu i,t conditioned on x and its prompt φ i,t : u i,t ∼ p(·| x,φ i,t ),i∈1,...,N, t∈1,...,R. (1) LetU t ≜ u 1,t ,...,u N,t be the set of role utterances at roundt. An aggregation module summarizesU t into a shared stateS t = f t (U t ), t∈1,...,R, with an initial state S 0 fixed. Prompt-as-Input and Output-as-Context. We define each round prompt as a composition of (i) a system in- struction, (i) the previous-round shared state, and (i) a role-specific template. Concretely, for each agenta i and round t, φ i,t = g i,t (x,S t−1 ;θ i,t ),(2) whereg i,t (·)is a (deterministic) prompt constructor andθ i,t denotes its free text content (the part we optimize). This explicitly captures that each agent’s output at roundt− 1 becomes part of the input for round t. Final-Round Scoring. After the final round, a terminal decision module produces the system output from (x,S R ): ˆy = J (x,S R ).(3) A task-specific scorer assigns a scalar score to the final outputs(x, ˆy)∈ R. Equivalently, the trajectory-level score can be written as J (τ ;x) ≜ s(x,j(x,S R )),τ ≜ U 1 ,S 1 ,...,U R ,S R . (4) Optimization Objective.The basic prompt optimization problem is to maximize the expected final-round score: max Φ E x∼D E τ∼p(·|x;Φ) h J (τ ;x) i .(5) SinceJ (τ ;x)is only observed at the end of theR-round in- teraction, it depends on all upstream promptsφ i,t through the coupled generation process Eq.(1)–Eq.(3). Textual-Gradient Prompt Update.Because prompts are discrete text and the base LLM parameters are fixed, we do not backpropagate numerical gradients through Eq.(1). Instead, we convert the terminal score into a textual gradient (natural-language feedback) [Xiao et al., 2024, Yüksekgönül et al., 2024] that specifies how to revise prompts: ∆ i,t =G(x,τ,J (τ ;x),i,t),(6) 3 whereG(·)outputs a feedback string for prompt(i,t). Prompts are updated by applying the feedback to the current prompt text: φ (k+1) i,t =U φ (k) i,t , ∆ (k) i,t , i = 1,...,N, t = 1,...,R, (7) wherekindexes optimization iterations andU (·)is a text- edit operator (e.g., rewrite / insert constraints). This com- pletes the most basic definition of the optimization problem in a multi-agent, multi-round system, where the final-round score is propagated back to prompts via textual gradients. 4 METHODOLOGY Building on the naive objective in Eq.(5), we make two relaxations to obtain a tractable optimization scheme. First, we introduce a state-space bottleneck by inserting a round- indexed aggregation module (one aggregator per round), so round-level decisions become explicit and scorable. Second, we impose parameter sharing by instantiating a fixed set of role-specialized agents whose prompts are shared across rounds, so credits can be accumulated by role rather than by individual turns. During optimization, we evaluate interme- diate role outputs and per-round aggregation outputs with LLM-based critics to obtain proxy signals for credit estima- tion. We use the terminal scorers(x, ˆy)as the optimization objective, while LLM critics provide auxiliary signals to estimate role- and round-level credits for targeted prompt updates. These critic signals are transformed into two forms of credit: (i) structural credit over roles, used to identify and optimize weak agents; and (i) temporal credit over rounds, implemented as a per-round aggregator credit score used to trigger targeted aggregation-prompt optimization. 4.1 STATE-SPACE BOTTLENECK In MAS, the terminal score provides a single global op- timization signal. To update prompts, this signal must be expressed as a textual gradient. Whether such feedback can be made actionable depends on the availability of a state- space bottleneck: an explicit shared state that concentrates, rather than disperses, the information flow. Without aggregation. If no aggregation is used, the fi- nal output is derived directly from the collection of all utterancesu i,t i,t . In this case, the textual gradient can only be expressed as an unstructured global signal∆ = G(x,u i,t i,t ,J (τ ;x)) . Because there is no explicit bottle- neck to anchor intermediate credit, the feedback cannot be aligned with specific agents or interaction rounds, providing only coarse guidance for prompt updates. With aggregation. By introducing an aggregation mod- ule, the system maintains explicit shared statesS t , which instantiate a state-space bottleneck across rounds. This en- ables textual gradients to be grounded on intermediate repre- sentations∆ t =G t (x,S t ,S R ,J (τ ;x)), and further decom- posed to individual agents: ∆ i,t =G i,t (x,u i,t ,S t ,S R ,J (τ ;x)).(8) In this way, aggregation provides the structural interface that makes textual gradients representable, alignable, and decomposable across both temporal rounds and agent roles. New optimization variables. Crucially, the aggregation module is itself prompt-driven, so introducing shared states also introduces new decision variables: the aggregator prompts that control howU t is summarized. Letψ t denote the aggregation prompt at roundt, and write the state up- date asS t = f t (U t ;ψ t ). The optimization target therefore changes from role-prompt-only tuning to a joint objective over role prompts and aggregation prompts: max Φ,Ψ E x∼D E τ∼p(·|x;Φ,Ψ) h J (τ ;x) i , Ψ ≜ψ t R t=1 . (9) This shift is important: optimizing aggregation can change the effective trajectory distribution and thus the credit sig- nals used to update role prompts. 4.2 PARAMETER SHARING We also analyze a constrained regime with parameter shar- ing, where each agent reuses a single prompt across all interaction rounds. Formally, for each agent a i , we impose φ i,1 = φ i,2 =· = φ i,R ≜ φ i .(10) This reduces the number of optimization variables, but in- creases the influence of each prompt because it is applied repeatedly under different aggregated contexts. Under this constraint, the optimization objective becomes max φ i N i=1 ,Ψ E x∼D E τ∼p(·|x;φ i ,Ψ) J (τ ;x) .(11) This is also our final optimization target. See Fig. 2 for the overall evolution. Although simplified, the problem remains non-trivial: each shared promptφ i affects the final score through multiple uses across evolving statesS t−1 . Effec- tive optimization therefore requires aggregating feedback across rounds, which naturally motivates temporal credit assignment under parameter sharing. Naive objective Eq. (5) Add aggregation prompts Eq. (9) Share role prompts Eq. (11) Figure 2: Evolution of the optimization objective: from the naive terminal-score objective, to introducing round-wise aggregation prompts, to enforcing shared role prompts. 4 4.3 VERBALIZED BCD Having specified a new optimization objective, the next step is to design an algorithm that can effectively solve it. Our objective provides supervision only at the end of theR-round interaction (Eq.(11)), leaving the intermediate rounds and roles without direct learning signals. Crucially, terminal success does not imply that the final aggregation round (or speaker) is solely responsible; earlier rounds can be decisive or harmful. We therefore introduce LLM-based critics as auxiliary evaluators of intermediate utterances and aggregations, and translate their scores into temporal and structural credits. These credits then drive a targeted block coordinate descent procedure that alternates between updating low-credit role prompts and low-credit aggregation prompts, while keeping high-credit components fixed. Credit Computing. For the structural dimension, let q i,t ∈ [0, 1]denote a normalized critic score for rolei’s utteranceu i,t at roundt. To account for interaction effects, we introduce a peer-view proxyp i,t ∈ [0, 1]: we do not re- quire each agent to explicitly score others; instead, a single LLM-based judge conditions on the full setU t and assesses each role’s contribution to group reasoning. We fuse these two signals asc role i,t = λq i,t + (1− λ)p i,t whereλ∈ [0, 1], and aggregate across rounds to obtain a role-level structural creditC role i = 1 R P R t=1 c role i,t . For the temporal dimension, let q agg t ∈ [0, 1]be the critic score for the round-taggregation output S t . We define the temporal credit as C time t = q agg t . Verbalized BCD over Prompt Blocks.Our optimization is a special instance of BCD over two prompt blocks: (i) the structural prompts that instantiate role behaviors, denoted byΦ role (e.g.,φ i N i=1 under the shared-prompt constraint in Eq.(10)); and (i) the temporal prompts that instantiate the round-indexed aggregation modules, denoted byΨ ≜ ψ t R t=1 . Starting from(Φ role,(0) , Ψ (0) ), each outer iteration k performs two alternating phases: Phase A: optimize roles while fixing aggregation prompts. We keepΨ (k) fixed, roll out trajectories, com- puteC role i , select a subset of low-credit roles (e.g., bottom-Kor below a threshold), and update only their prompts using textual gradients (Eq. (7)): Φ role,(k+1) ← ROLEOPT Φ role,(k) ; Ψ (k) , C role i . (12) Phase B: optimize aggregation prompts while fixing roles. We then keepΦ role,(k+1) fixed, roll out trajecto- ries again (or reuse logged trajectories when applicable), computeC time t , select low-credit rounds, and update only their aggregation prompts: Ψ (k+1) ← AGGOPT Ψ (k) ; Φ role,(k+1) , C time t . (13) Algorithm 1: BCD for Temporal and Structural Credit- Guided Prompt Optimization Input:D; fixed base LLM; rounds R; roles A =a i N i=1 Output: Optimized promptsφ i N i=1 andψ t R t=1 Initialize:φ (0) i ,ψ (0) t ; k ← 0; while not converged do // Step 1: Structural block update (fix ψ t ) ComputeC role i N i=1 ; Select low-credit rolesI (k) ⊆1,...,N; foreach i∈I (k) do φ (k+1) i ←U φ (k) i , ∆ role i ; Set φ (k+1) i ← φ (k) i for i /∈I (k) ; // Step 2: Temporal block update (fix φ i ) ComputeC time t R t=1 ; Select low-credit roundsT (k) ⊆1,...,R; foreach t∈T (k) do ψ (k+1) t ←U ψ (k) t , ∆ time t ; Set ψ (k+1) t ← ψ (k) t for t /∈T (k) ; k ← k + 1; returnφ (k) i N i=1 ,ψ (k) t R t=1 ; We repeat Phases A–B until a fixed budget is exhausted or the held-out performance saturates. This alternating design makes the procedure implementable and stable: at each step we optimize one block while treating the other as a fixed environment, reducing unnecessary drift in the coupled multi-round interaction. 5 EXPERIMENTS To validate the proposed temporal and structural credit as- signment and the resulting prompt optimization routine, we now turn to empirical evaluation. The following section specifies datasets, protocols, and baselines, and then quanti- fies how the credit signals translate into measurable gains and interpretability across benchmarks. 5.1 SETTINGS We evaluate our proposed credit-guided prompt optimiza- tion on multiple-choice reasoning benchmarks, where each question has four options and the correct answer is selected from A, B, C, D. Models generate free-form text, and we deterministically extract the predicted option using a fixed 5 parsing rule: the first occurrence of a standalone A/B/C/D token in the output. If no valid option is found, the predic- tion is considered invalid and counted as incorrect. These parsing rules are fixed prior to evaluation and remain con- stant across all runs. We report results on AQuA [Ling et al., 2017], MedMCQA [Pal et al., 2022], GPQA [Rein et al., 2024], and MMLU [Hendrycks et al., 2021b,a]. For each dataset, we randomly select a fixed optimization set of 100 examples used exclusively for prompt search and optimization (i.e., updating role prompts and round-wise aggregator prompts). All reported metrics are evaluated on a larger, disjoint leave-out test set, which is never accessed during optimization. The split is fixed across runs. No test ex- amples are used for prompt search, gradient updates, credit assignment, or hyperparameter tuning. Accordingly, we em- phasize no test-set exposure, rather than zero-shot perfor- mance, on the reported test set. Random seeds are fixed for sampling the optimization set and for any stochastic components in decoding or optimiza- tion. Unless otherwise stated, results are averaged over mul- tiple runs with different seeds, and we report the mean ac- curacy along with standard deviation. Prompt optimization is performed independently for each dataset. This consti- tutes dataset-specific prompt search over a fixed base model, not model fine-tuning. Cross-dataset generalization is not claimed unless explicitly evaluated. Models and Runtime. All debate agents are instantiated with open-weight instruction-following LLMs, allowing in- spection and modification of prompts. We use Qwen2.5- 7B-Instruct, LLaMA3-8B-Instruct, and Gemma-7B-Instruct for the main experiments, and decoding settings are kept identical across baselines and our method. Frameworks and Baselines. We evaluate our approach on several multi-agent frameworks, with a focus on LLM- Debate and DyLAN. In all settings, the interaction proto- col (number of agents, number of rounds, and aggregator placement) is fixed. We compare against: (i) the unmodi- fied baseline prompts, (i) a black-box prompt optimization baseline using DSPy MIPRO (multi-stage instruction and demonstration optimization) [Opsahl-Ong et al., 2024] that performs agent-wise non-informed prompt search, and (i) our credit-aware optimization. Training and Optimization Protocol. We follow a two- stage prompt-only procedure; no model parameters are up- dated. First, DSPy (MIPROv2) is used solely for automated prompt initialization [Opsahl-Ong et al., 2024]. Second, fixed-budget prompt edits are performed using only the optimization split; the test split is never accessed. (i) Struc- tural optimization. We evaluate each agent’s final-round answer withAgentTurnEval, aggregate scores into a risk statistic, and deterministically select the two worst- performing roles. Only these role prompts are optimized usingRolePromptOptwith summarized error diagnosis. (i) Temporal optimization. Temporal credit is assigned to per-round aggregators using the EMA update described in Section 3. Credit is updated only on informative failures (aggregator wrong but at least one agent correct). When the credit of roundtdrops below a threshold and sufficient failures are observed, only that round’s aggregator prompt is optimized. All hyperparameters are fixed across datasets. 5.2 MAIN RESULTS AND ANALYSIS Based on the analyses in this section, the experiments aim to answer four research questions: RQ1: Does credit-guided prompt optimization consistently improve accuracy across datasets and models compared with unoptimized prompts and a black-box baseline? RQ2: How much do structural and temporal optimization each contribute, and does their combination yield the strongest and most consistent gains? RQ3: Is the optimization process efficient and stable (con- vergence speed and variance), and how do key hyperpa- rameters affect performance and stability? RQ4: Do credit signals align with error types and outcome shifts, provid- ing interpretable diagnostics and actionable guidance for follow-up optimization? 5.3 RQ1: EFFECTIVENESS Table 1 summarizes the performance of our credit-guided prompt optimization compared to two baselines: the unmod- ified prompts and a black-box optimization baseline. Across all datasets and models, our method consistently improves accuracy. For instance, on MedMCQA with LLaMA3-8B, our approach improves accuracy by 7.0% relative to the base- line, while on GPQA with Qwen2.5-7B the gain is 2.1%. Even on datasets with higher initial accuracy, such as AQuA and MMLU, we observe consistent improvements. Standard deviations across multiple runs remain low, indicating stable and reproducible performance. These results confirm that integrating structural and temporal credit signals effectively guides prompt optimization in multi-agent settings. RQ1 Answer. Credit-guided prompt optimization delivers broad, reliable accuracy gains across datasets and model families, outperforming both the unoptimized prompts and the black-box baseline, which supports its generalizability and robustness. 5.4 RQ2: ABLATIONS To understand the source of improvements, we perform ab- lation experiments. We isolate the contributions of structural and temporal components: Role-only optimization (struc- tural): optimizing only the worst-performing role prompts selected by the risk statistic. Aggregator-only optimization 6 Table 1: Main results on multiple-choice reasoning benchmarks. The main number is the mean accuracy, and the right-side annotation reports the delta relative to the corresponding MAS baseline: “-” marks the baseline,↑denotes an increase, and↓ denotes a decrease. Cell colors follow a red–white–green gradient from negative to zero to positive deltas, and the lower number in each cell is the standard deviation. DyLANDebate DomainModelbaselineoptimizedoursbaselineoptimizedours MedMCQA LLaMA3-8B 54.13− ±2.3936 61.75↑7.63 ±3.3040 61.13↑7.00 ±1.0308 55.13− ±3.2243 61.63↑6.50 ±2.1360 64.63↑9.50 ±1.7970 Qwen2.5-7B 51.88− ±2.3229 53.88↑2.00 ±1.6520 55.75↑3.88 ±1.9365 54.50− ±1.0000 54.88↑0.38 ±3.3510 55.50↑1.00 ±4.3205 Gemma-7B 19.13− ±1.1087 19.38↑0.25 ±0.9465 21.33↑2.21 ±2.4664 26.75− ±3.1820 27.50↑0.75 ±2.4833 28.50↑1.75 ±2.5981 GPQA LLaMA3-8B 27.42− ±2.2728 29.61↑2.19 ±3.0873 28.11↑0.69 ±1.8779 39.75− ±1.7816 35.02↓4.73 ±1.5535 36.75↓2.30 ±4.4439 Qwen2.5-7B 32.83− ±2.3308 35.02↑2.19 ±2.0956 32.37↓0.46 ±1.6581 32.26− ±1.4546 33.07↑0.81 ±1.3212 35.83↑3.57 ±0.8752 Gemma-7B 11.52− ±0.4600 12.29↑0.77 ±0.5312 11.98↑0.46 ±0.6505 12.56− ±1.3778 17.75↑5.19 ±3.5850 18.67↑6.11 ±1.6334 AQuA LLaMA3-8B 51.15− ±3.0376 55.46↑4.31 ±2.5500 56.32↑5.17 ±1.6263 53.16− ±3.6784 52.01↓1.15 ±4.3354 52.30↓0.86 ±9.4087 Qwen2.5-7B 83.05− ±1.7250 85.35↑2.30 ±1.7250 83.91↑0.86 ±1.3279 83.05− ±2.8750 81.04↓2.01 ±1.4846 83.62↑0.58 ±3.3031 Gemma-7B 18.72− ±2.9480 22.42↑3.70 ±0.8132 23.57↑4.85 ±0.6640 18.62− ±1.4994 22.61↑3.99 ±1.3279 27.59↑8.97 ±2.3000 MMLU LLaMA3-8B 56.43− ±1.0820 56.20↓0.23 ±0.5952 56.30↓0.13 ±0.7407 68.78− ±1.0444 69.33↑0.55 ±2.6207 74.77↑6.00 ±2.8646 Qwen2.5-7B 65.35− ±1.1045 66.38↑1.03 ±1.1386 67.48↑2.13 ±0.6135 61.58− ±1.3371 61.95↑0.38 ±1.9265 63.38↑1.80 ±2.0126 Gemma-7B 29.40− ±0.3162 30.40↑1.00 ±0.4733 32.00↑2.60 ±0.2828 28.25− ±4.8946 33.73↑5.48 ±3.7754 33.60↑5.35 ±2.2627 (temporal): optimizing only the low-credit round aggrega- tors while freezing role prompts. Figure 3 visualizes the LLaMA3-8B Debate results on MedMCQA and MMLU, highlighting the consistent gains from credit-guided prompt optimization. Both components independently improve accuracy relative to the baseline. Structural optimization contributes more on role-sensitive datasets (e.g., MedMCQA), whereas temporal optimization is most effective on datasets where aggregator consolida- tion is critical (e.g., GPQA). The combination consistently achieves the highest performance. We further analyze temporal optimization at the round level. Restricting updates to one roundtat a time reveals which rounds are most error-prone. Early rounds often dominate failures, indicating that losing key evidence early can limit overall accuracy, while late rounds mainly affect final con- solidation. These findings provide actionable guidance on where optimization should focus. RQ2 Answer. The ablations show complementary ef- fects: role-only (structural) updates yield larger gains on role-sensitive datasets such as MedMCQA, while aggregator-only (temporal) updates are more beneficial on consolidation-heavy tasks such as GPQA. Combining both consistently gives the best accuracy across MedMCQA and MMLU, outperforming either component alone. 5.5 RQ3: SENSETIVITY We evaluate performance across optimization iterations to assess efficiency and stability. Figure 4 shows the accuracy trajectories across iterations, highlighting faster gains for our method versus DSPy MIPRO and the baseline. Credit- guided optimization converges rapidly within a few itera- tions and maintains low variance across runs. In contrast, the black-box baseline converges more slowly and exhibits higher variance, reflecting inefficient exploration. 7 Figure 3: LLaMA3-8B Debate results on MedMCQA and MMLU: the combined temporal+structural credit-guided optimization yields the strongest gains, with structural-only and temporal-only providing smaller improvements over the baseline. Figure 4: Accuracy convergence over optimization itera- tions; ours converges faster and to higher accuracy than DSPy MIPRO and the baseline. We also investigate hyperparameter sensitivity by varying the temporal credit threshold τ and buffer size m. Lower τ or largermresults in conservative updates, preserving sta- bility, while higherτor smallermallows more aggressive updates, increasing the risk of performance drift. Monitor- ing credit trajectoriesα t and test accuracy variance confirms that our recommended settings achieve a favorable stabil- ity–plasticity balance. RQ3 Answer. Building on the consistent gains in RQ1 and the complementary effects in RQ2, the optimization is efficient because it targets the most influential components, achieving improvements with only a few focused iterations. Stability follows from this targeted selection, whileτandm provide explicit control over the trade-off between cautious refinement and faster but riskier updates. 5.6 RQ4: INTERPRETABILITY To validate that optimization aligns with failure diagno- sis, we analyze outcome shifts and error types. Com- pared with black-box edits, credit-guided optimization re- duces regressions (correct→incorrect) and increases repairs (incorrect→correct). Per-category accuracy shows that role- Table 2: Distribution of prediction shifts before and after optimization. Compared with DSPu(MIPRO), our method yields more incorrect→correct repairs and fewer correct→incorrect regressions, indicating that credit signals act as structured diagnostics rather than global perturbations. ShiftDSPy(MIPRO) (%)Ours (%) X→X38.0034.50 ✓→X7.005.00 X→✓7.0011.00 ✓→✓48.0049.50 specific optimizations mainly improve categories flagged as failing by the critic, and temporal creditα t correlates with aggregator reliability rather than mere signal presence. To- gether, these analyses confirm that structural and temporal credit signals provide both mechanistic interpretability and practical guidance for follow-up experiments. RQ4 Answer. Taken together with RQ1–RQ3, the credit signals function as actionable diagnostics: they pinpoint which rounds and roles actually drive outcome changes, explain when gains persist or regress under tuning, and indicate where the next optimization pass should focus. 6 CLOSING REMARKS In this paper, we introduce a test-time temporal and struc- tural credit assignment framework that decomposes multi- agent LLM trajectories along rounds and roles, enabling component-level attribution without altering inference dy- namics or model parameters. Across the experiments, the resulting credit signals reveal pronounced contribution im- balance and identify the small set of stages and agents that consistently constrain outcomes. Leveraging these signals, credit-guided prompt optimization focuses updates on weak components and delivers reliable gains across benchmarks and model families, reducing wasteful global changes while improving effectiveness. These findings position temporal and structural attribution as a practical diagnostic and optimization tool for multi-agent reasoning systems. At the same time, the framework remains limited to completed trajectories and relies on evaluative sig- nals; temporal attribution can be computationally heavy for long horizons; and the current instantiation assumes fixed roles with centralized aggregation. 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Mingchen Zhuge, Wenyi Wang, et al. Gptswarm: lan- guage agents as optimizable graphs. In Proceedings of the 41st International Conference on Machine Learning, ICML’24. JMLR.org, 2024. 10 A APPENDIX OVERVIEW This appendix provides supplemental details to support reproducibility and analysis. We use standard appendix numbering (e.g., Table A.1, Figure A.1, Algorithm A.1) and keep all references consistent with the main text. B EVALUATION AND PROMPT OPTIMISATION PROMPTS The prompts used for agent evaluation, diagnosis and role prompt optimisation are presented below to aid reproducibility. These prompts are treated as experimental protocols that can be referenced in the main text. B.1 AGENT TURN EVALUATION PROMPT Agent Turn Evaluation Prompt AGENT_TURN_EVAL_PROMPT = """ You are an experienced question evaluation specialist and failure-analysis prompt engineer. You are given: 1) A multiple-choice question with options A, B, C, and D. 2) The gold correct answer. 3) The final answer produced by ONE debating agent. Your task is NOT to solve the question again. Your task is to evaluate this agents final answer independently and assign a score that reflects the quality of its reasoning and decision-making. You must: - Determine whether the final answer is correct. - If incorrect, identify the primary reason for failure using a predefined category. - If correct, assess whether the reasoning is robust or fragile. - Assign a numerical score that can be accumulated across many questions. -------------------------------- Failure pattern taxonomy (choose EXACTLY ONE): You MUST select exactly one of the following labels. Do NOT invent new labels. - DOMAIN_MISMATCH The agent reasons from an inappropriate domain or role (e.g., ethical, economic, or technical framing instead of task-relevant reasoning). - KNOWLEDGE_DEFICIT The agent lacks or misuses core domain knowledge, leading to factual or conceptual errors. - MISINTERPRET_QUESTION The agent misunderstands key constraints, conditions, or intent of the question. - INCOMPLETE_REASONING The agents reasoning is partially correct but missing critical logical steps or justification. - OVERGENERALIZATION 11 The agent applies generic patterns or heuristics without adequately considering case-specific details. - MISALIGNED_OBJECTIVE The agent answers a different question than what is being asked (e.g., treatment vs diagnosis). - INSUFFICIENT_JUSTIFICATION The conclusion may be correct, but the reasoning is weak, shallow, or insufficiently supported. - RANDOM_OR_UNGROUNDED The answer appears arbitrary, speculative, or not grounded in the provided information. - NONE The answer is correct and the reasoning is sound. -------------------------------- Scoring rules: - Scores must be integers from 0 to 5. - 5 = Correct answer with strong, well-justified reasoning. - 34 = Correct answer but with weak, incomplete, or risky reasoning. - 12 = Incorrect answer due to reasoning or judgment errors. - 0 = Incorrect answer due to fundamental misunderstanding or systematic reasoning failure. Consistency rules: - If the answer is correct, failure_pattern MUST be "NONE". - If the answer is incorrect, failure_pattern MUST NOT be "NONE". Output format: - Final answer correctness: correct / incorrect - Primary failure or risk pattern: <one of the predefined labels> - Brief explanation: 12 sentences focused on reasoning quality - Score: <integer 05> Additional rules: 1. Do NOT restate the full question or options. 2. Do NOT compare with other agents. 3. Do NOT suggest prompt changes explicitly. 4. Focus on issues that could be mitigated by improving the agents prompt or debate behavior. 5. Be concise, consistent, and scoring-oriented. """ B.2 AGENT DIAGNOSIS PROMPT Agent Diagnosis Prompt AGENT_DIAGNOSIS_PROMPT = """ You are an attribution analyst for a multi-agent reasoning system. Your role is to analyze summarized error information produced by a single agent and generate a concise attribution summary of the agents systematic failure characteristics. 12 The input you receive is a structured summary grouped by failure types. For each failure type, the input provides: - how frequently this failure occurred, and - a small number of representative example explanations. You should treat the frequency information as an indicator of how systematic and dominant each failure pattern is. The example explanations are only illustrative signals and do NOT represent all errors. You are NOT responsible for fixing the errors or rewriting the agents role prompt. Your task is strictly to identify and summarize why the errors occurred. When producing the attribution summary: - Focus on dominant and recurring failure patterns, prioritizing those with higher frequency. - Identify shared reasoning weaknesses, perspective mismatches, or systematic misalignments. - Abstract away from individual questions, examples, or surface details. - Do NOT repeat or quote raw explanations. - Do NOT enumerate failure types or counts explicitly. - Do NOT include task-specific facts or domain knowledge. Your summary should capture, at an appropriate level of abstraction: - role or perspective mismatches, - reasoning or interpretation deficiencies, - knowledge usage or grounding issues, if they are reflected in the summarized failures. The output should be concise, structured in natural language, and suitable for downstream prompt refinement modules. Output only the attribution summary. Do not include analysis, bullet points, statistics, or recommendations. """ B.3 ROLE PROMPT OPTIMISATION PROMPT Role Prompt Optimisation Prompt ROLE_PROMPT_OPTIMIZE = """ You are a prompt refinement module for a multi-agent reasoning system. Your task is to correct and reconstruct an agent’s role prompt based on aggregated evaluation failures. The original role prompt may contain invalid assumptions, missing constraints, or misleading reasoning guidance. Do NOT assume the original prompt is correct. Use the failure summary to infer systematic issues in three aspects: (1) role and perspective alignment, (2) reasoning and interpretation discipline, (3) knowledge use and grounding. Follow these principles implicitly: 13 1. Role and objective correction: - If failures indicate domain mismatch or objective misalignment, remove or correct the role perspective, focus, or priorities that cause the agent to reason from an inappropriate viewpoint or answer the wrong question. 2. Reasoning discipline reconstruction: - If failures indicate misinterpretation, incomplete reasoning, overgeneralization, or weak justification, introduce clearer reasoning requirements such as careful question interpretation, constraint checking, step-by-step reasoning, and explicit justification. 3. Knowledge use and grounding control: - If failures indicate knowledge deficits or ungrounded responses, strengthen guidance on using only relevant, task-appropriate knowledge and avoiding speculative or unsupported conclusions. Constraints: - Do not preserve incorrect assumptions from the original prompt. - Do not add task-specific facts or external domain knowledge. - Do not overfit to individual examples; address systematic behavior only. - Keep the reconstructed prompt concise and suitable for debate-based interaction. Output only the reconstructed role prompt. Do not include analysis, explanations, or failure labels. """ C EXTENDED ANALYSIS OF THE DYLAN BASELINE ON MMLU (EXAMPLE) To aid reproducibility and provide deeper insight into system behavior, we analyzed the evaluation log file 8080_dylan_mmlu_2026-02-19_21-20-23.txt. This file contains 500 entries, each recording the final answer of a multi-agent debate, the evaluator’s judgment of correctness, a failure-pattern label, and a quality score. The analysis below summarizes the distributions of these values. D EXTENDED ANALYSIS OF OPTIMIZED DYLAN ON MMLU (EXAMPLE) In addition to the baseline analysis, we examined the log file from the optimized DyLAN system (8080_dylan_mmlu_2026-02-19_21-20-44.txt). This experiment applied an optimization method to the role prompts before evaluation. The log comprises 500 multiple-choice questions from the MMLU dataset. D.1 OVERALL PREDICTION STATISTICS Table 3 summarizes the aggregate performance of the optimized system. Out of 500 questions, the system answered 279 correctly (55.8% accuracy). The mean evaluation score was 3.38 on a 0–5 scale, comparable to the baseline run. Thus, despite optimization, there was no substantial improvement in overall accuracy or score. D.2 DISTRIBUTION ACROSS DEBATE ROUNDS The optimization encouraged earlier convergence: 81.6% of questions were answered after one round, while only 3.4% of questions reached the third round (Table 4). The average score dropped sharply from round 1 to round 2 (3.59 to 2.27) and increased again for the few questions that reached round 3 (3.12). 14 Table 3: Overall statistics for the optimised DyLAN system on MMLU. MetricValueDescription Total predictions500Number of evaluated questions Correct predictions279 (55.8%)Final answers matching the gold answer Incorrect predictions221 (44.2%)Final answers that were wrong Average evaluation score3.38Mean score on a 0–5 scale Table 4: Number of questions and average score by debate round for the optimised system. RoundCountProportionAverage score 140881.6%3.59 27515.0%2.27 3173.4%3.12 D.3 FAILURE PATTERN DISTRIBUTION Table 5 reports the frequency and average score of each failure pattern. As with the baseline run,KNOWLEDGE_DEFICIT andMISINTERPRET_QUESTIONremain the most common error modes, collectively accounting for nearly 38% of all predictions. Additional errors such asINCOMPLETE_REASONINGandOVERGENERALIZATIONappear more often than in the baseline, and a single RANDOM_OR_UNGROUNDED error is observed. Table 5: Failure patterns and average scores for the optimised system. “NONE” denotes correct answers. Failure patternCountShareAvg. scoreInterpretation NONE (correct)27955.8%5.00Correct answer with robust reasoning KNOWLEDGE_DEFICIT12324.6%1.33Missing or misused domain knowledge MISINTERPRET_QUESTION6513.0%1.32Misunderstanding the question’s intent or constraints INCOMPLETE_REASONING112.2%1.45Partial reasoning missing critical steps MISALIGNED_OBJECTIVE81.6%1.38Answering an off-target question OVERGENERALIZATION61.2%1.33Generic heuristics without context INSUFFICIENT_JUSTIFICATION40.8%1.50Plausible conclusion but weak justification DOMAIN_MISMATCH30.6%1.00Reasoning from an inappropriate domain RANDOM_OR_UNGROUNDED10.2%1.00Arbitrary or ungrounded answer D.4 ROUND–ERROR CROSS ANALYSIS Table 6 cross-tabulates failure patterns across debate rounds. Most correct responses occur in the first round. Knowledge deficits remain prevalent across rounds, whereas misinterpretation errors are more prominent in the third round. The distribution suggests that the optimization did not fully mitigate fundamental knowledge gaps. Table 6: Number of predictions by round and failure pattern for the optimised system. NONE KNOWL.DEFICIT MISINT.QUESTION MISALIGN.OBJ. INCOMP.REASON. DOMAINMISM. OVERGEN. INSUFF.JUST. RANDOMUNGR. Round 12519242672431 Round 2202917231210 Round 3826010000 15