Trajectory Shapley Attribution (TSA)

• TSA is a family of Shapley-based methods that assigns contributions along trajectories rather than only at a final endpoint.
• It adapts the player set, value function, and admissible coalitions to respect temporal order, process structure, and optimization dynamics.
• TSA has practical applications in federated learning, traffic analysis, and manufacturing, offering granular insights into sequential contributions.

Trajectory Shapley Attribution (TSA) denotes a class of Shapley-based attribution methods for ordered, path-dependent, or process-structured settings in which contribution is assigned along a trajectory rather than only at a final endpoint. In recent arXiv literature, the term is used for federated optimization trajectories, traffic routes, and sequential manufacturing processes, while closely related work applies the same logic to frames in videos, time steps in sequences, retrieved documents in generation, and agent sub-trajectories in multi-agent systems (Kuznetsov et al., 28 May 2026, Feng et al., 2024, Miyaguchi et al., 27 Jul 2025). This suggests that TSA functions less as a single standardized algorithm than as a family of constructions that preserve the Shapley perspective while modifying the player set, the value function, and the admissible coalitions to respect temporal order, process structure, or optimization dynamics.

1. Conceptual scope

Across the literature, TSA is motivated by a recurrent limitation of ordinary Shapley analysis in sequential domains: standard Shapley values treat inputs as an unordered set of players, whereas many real systems are intrinsically ordered. In semiconductor manufacturing, this is described as a failure to respect process order and physically admissible trajectories; in federated learning, endpoint-only utilities ignore whether a client helped or hurt early or intermediate optimization steps; in traffic interpretation, region-level explanations do not answer the route-level question of where flows come from (Miyaguchi et al., 27 Jul 2025, Kuznetsov et al., 28 May 2026, Feng et al., 2024).

A second recurrent motivation concerns the semantics of feature removal. The general Shapley literature distinguishes baseline, marginal, and conditional value functions, each corresponding to a different treatment of missing information. When transferred to trajectories, these become choices about how to construct partial trajectories, hybrid paths, or conditionally completed sequences, and therefore about what “contribution” means in a sequential setting (Chen et al., 2022).

The consequence is that TSA is best understood as a methodological pattern. It attributes a scalar quantity of interest to units that lie on, generate, or influence a trajectory. Depending on the application, those units may be clients, measurement items, trajectories, frames, time steps, documents, or agents. The value being decomposed may be a validation-guided descent utility, a model prediction, a prediction loss, a log-likelihood, or a terminal task reward (Kuznetsov et al., 28 May 2026, Miyaguchi et al., 27 Jul 2025, Nematov et al., 6 Jul 2025).

2. Common game-theoretic structure

Most TSA formulations retain the standard cooperative-game definition of the Shapley value. With player set $D=\{1,\dots,d\}$ and coalitional game $v:\mathcal{P}(D)\to\mathbb{R}$, the attribution of player $i$ is

$$\phi_i(v)=\sum_{S\subseteq D\setminus\{i\}} \frac{|S|!(|D|-|S|-1)!}{|D|!}\bigl(v(S\cup\{i\})-v(S)\bigr).$$

Equivalent permutation, weighted least-squares, and multilinear-extension characterizations are also used in the broader Shapley literature, and these carry over directly once “players” are reinterpreted as trajectory elements (Chen et al., 2022).

What changes in TSA is the definition of the game. In some settings, the players are the trajectory elements themselves: frames in a video sequence, time steps in a generic sequence, or ordered measurement items in a manufacturing route. In others, the players are not temporal indices but entities whose influence is evaluated against a trajectory: clients in federated learning are scored by how their updates align with a validation-guided optimization path, and retrieved documents in retrieval-augmented generation are scored by how they support a generated answer trajectory through token log-likelihood (Price et al., 2020, Kuznetsov et al., 28 May 2026, Nematov et al., 6 Jul 2025).

The design of the value function is therefore the central modeling choice. Baseline Shapley uses a fixed reference input; marginal Shapley draws missing components from a marginal distribution; conditional Shapley draws them from $p(x_{\bar S}\mid x_S^e)$. In sequential settings, these become, respectively, hybrid trajectories against a baseline path, off-manifold randomizations of missing trajectory elements, or plausible completions conditioned on the observed subsequence (Chen et al., 2022). Several TSA formulations also impose additional structure on admissible coalitions. The wafer-diagnosis variant restricts attention to prefix-like sequential subsets, and the video setting evaluates ordered subsequences that preserve the original sequence order (Miyaguchi et al., 27 Jul 2025, Price et al., 2020).

3. Major formulations

The term TSA is attached to several distinct constructions rather than a single canonical estimator.

Setting	Players	Trajectory/value construction
Federated learning	Clients	Per-round Shapley over clients with trajectory utility $v^t(\mathcal S)$, accumulated as $\varphi_i^T=\sum_t \phi_i(v^t)$
Traffic interpretation	Trajectories	Region SHAP redistributed to routes via $$\phi(Y\rightarrow T_r)=\sum_{i,j}\frac{1}{|C^{c_2,i,j}|}\phi(Y\rightarrow X_t^{c_2,i,j})$$
Wafer diagnosis	Ordered measurement items	Closed-form sequence-aware score $$s_i(x^t)=p(x_{1:i}^t,x_{(i+1):D}^0)-p(x_{1:(i-1)}^t,x_{i:D}^0)$$
Video and generic sequences	Frames or time steps	Shapley over ordered subsequences with $v_c(X')=f_c(X')$
RAG and multi-agent systems	Documents or agents	Log-likelihood utility over an answer trajectory, or leave-one-agent-out marginal credit over a planner-plus-workers trajectory

In federated learning, "Trajectory Shapley Value" is a concrete instantiation of TSA specialized to client contribution. At each round $v:\mathcal{P}(D)\to\mathbb{R}$0, a coalition update $v:\mathcal{P}(D)\to\mathbb{R}$1 is compared with a server-side validation reference update $v:\mathcal{P}(D)\to\mathbb{R}$2, yielding a bounded utility $v:\mathcal{P}(D)\to\mathbb{R}$3. The standard Shapley value is computed over participating clients in that round and then accumulated over rounds as $v:\mathcal{P}(D)\to\mathbb{R}$4. Time enters through the utility definition and the cumulative aggregation, not by treating rounds themselves as players (Kuznetsov et al., 28 May 2026).

In TrafPS, TSA is a region-to-trajectory explainer for traffic prediction. Region SHAP first attributes a target prediction $v:\mathcal{P}(D)\to\mathbb{R}$5 to historical grid-flow features $v:\mathcal{P}(D)\to\mathbb{R}$6. A theorem then states that every trajectory contributing to a grid and channel receives an equal share of that grid’s explanation, so trajectory SHAP becomes a path-wise aggregation of region SHAP. The paper explicitly characterizes this as “not a new Shapley estimator; rather a reallocation layer” on top of SHAP (Feng et al., 2024).

In good-bad wafer diagnosis, TSA is a sequence-aware extension of Shapley values for manufacturing routes. Standard SV is criticized for averaging over non-physical coalitions and depending on arbitrary baselines. TSA instead defines physically admissible trajectories as prefix-like subsets that respect process order, treats non-participating variables as missing, and imputes them through Lot-Aware Kernel Imputation (LAKI). Under the imposed sum rule, the attribution reduces to a closed-form path difference that compares the classifier’s output before and after revealing the $v:\mathcal{P}(D)\to\mathbb{R}$7-th ordered measurement item (Miyaguchi et al., 27 Jul 2025).

Related sequence formulations broaden the concept. "Play Fair" defines the Element Shapley Value (ESV) for frames in a variable-length video, with class score $v:\mathcal{P}(D)\to\mathbb{R}$8 decomposed over ordered subsequences. The RAG attribution paper states that document-level Shapley in generation can be viewed as a special case of TSA, because the utility is already a sum of token-level log-likelihood contributions along the generated answer trajectory. SHARP in multi-agent systems applies a trajectory-level Shapley-style leave-one-agent-out credit over planner and worker sub-trajectories, using counterfactual masking rather than full coalition averaging (Price et al., 2020, Nematov et al., 6 Jul 2025, Li et al., 9 Feb 2026).

4. Utility design and computation

TSA methods differ most sharply in how they define utility. In federated learning, the utility is validation-based and geometric:

$v:\mathcal{P}(D)\to\mathbb{R}$9

This makes the score bounded, monotone in distance to the validation-guided descent direction, and temporally normalized across early and late rounds. The authors argue that such a bounded, validation-based, temporally normalized utility mitigates instability near convergence and supports meaningful accumulation over time (Kuznetsov et al., 28 May 2026).

In traffic interpretation, the value function remains the model’s prediction or its region-SHAP proxy, but the attribution is redistributed to trajectories by linearity and symmetry. In manufacturing, the value function is a defect-probability model $i$0 evaluated on prefix-revealed trajectories with LAKI-imputed suffixes. In video, the value function is the class score of an ordered subsequence. In generic trajectory-oriented Shapley theory, these choices map naturally onto baseline, marginal, and conditional semantics, which respectively correspond to fixed reference trajectories, randomizations that break temporal dependence, and on-manifold completions that preserve it (Feng et al., 2024, Miyaguchi et al., 27 Jul 2025, Price et al., 2020, Chen et al., 2022).

Exact computation is typically infeasible. Several approximation families recur. Federated TSV uses per-round decomposition and Monte Carlo sampling over permutations, exploiting the fact that coalition updates are averages of precomputed client updates. TrafPS computes SHAP once at the grid-flow level, then performs linear-time redistribution to trajectories. ESV for video uses subsequence sampling together with a dynamic-programming-style recurrence for a multi-scale model, reducing the exponential burden of exact enumeration. In RAG, the paper compares exact Shapley with TMC-Shapley, Beta Shapley, Kernel SHAP, ContextCite, and leave-one-out under a costly log-likelihood utility. SHARP avoids exact coalition averaging entirely and uses a leave-one-agent-out counterfactual reward over the realized multi-agent trajectory (Kuznetsov et al., 28 May 2026, Price et al., 2020, Nematov et al., 6 Jul 2025, Li et al., 9 Feb 2026).

These computational choices alter the semantics only to the extent that they alter the underlying game. TrafPS retains SHAP semantics but changes the explained object through redistribution; SHARP is explicitly an approximation to a Shapley-style marginal credit rather than an exact Shapley computation; FedTSV remains an exact Shapley construction at the per-round level but approximates it by Monte Carlo over client permutations (Feng et al., 2024, Li et al., 9 Feb 2026, Kuznetsov et al., 28 May 2026).

5. Applications and empirical behavior

In federated learning, FedTSV replaces fixed aggregation weights with weights derived from cumulative trajectory Shapley values. On MNIST and CIFAR-10 with 100 clients comprising 70 benign IID, 10 benign non-IID, and 20 malicious clients, FedTSV is reported to achieve higher final accuracy and lower loss than FedAvg, CGSV, and LOO, while producing a weight distribution in which IID benign clients concentrate in a high positive-contribution region, non-IID benign clients are intermediate, and malicious clients are strongly suppressed near zero (Kuznetsov et al., 28 May 2026).

In traffic analysis, TSA is used to identify top-$i$1 trajectories contributing positively or negatively to predicted congestion in a selected grid. The fine-grained grid view combines a grid matrix, parallel coordinates, map overlays of top-$i$2 trajectories, a details table, and a bi-directional bar chart. In the reported case studies, analysts use radar glyphs and TSA-ranked routes to identify congestion-causing routes on National Day morning and to support reversible-lane planning during weekday morning peaks (Feng et al., 2024).

In wafer diagnosis, TSA is applied to experimental front-end-of-line processes at the Albany NY CREATES fab. The dataset contains $i$3 wafers, $i$4 measurement items, an average missing rate of 94%, and $i$5. With LAKI, an $i$6-regularized logistic regression model achieves a true positive rate of 0.80 for classifying bad wafers, compared with 0.75 without LAKI. TSA cumulative attribution curves are then used to identify upstream measurements that increased predicted defect risk, including examples where the wafer appears to have “become bad” significantly earlier than the process-limited yield checkpoint (Miyaguchi et al., 27 Jul 2025).

In sequence and trajectory-prediction settings, TSA-style analysis often serves a diagnostic rather than an optimization role. ESV in video models decomposes a class score into fair per-frame contributions and reports that, in 8-frame sequences, about half the frames are distracting and that removing frames with the lowest ESV can improve accuracy more than Grad-CAM or Integrated Gradients. In multi-agent trajectory prediction, a Shapley-based attribution over target past trajectory and neighboring agents shows that on ETH–UCY, SDD, and nuScenes the social interaction score is insignificant and close to zero, whereas on SportVU the same models assign non-negligible contribution to neighbors and the ball (Price et al., 2020, Makansi et al., 2021).

In LLM-related systems, TSA-like constructions are used for credit assignment under expensive black-box utilities. In RAG, the utility is $i$7, and the paper finds that Kernel SHAP and ContextCite approximate exact Shapley well under tight sample budgets, whereas synergy remains difficult because the utility is dominated by explicit answer tokens. In multi-agent tool use, SHARP combines a broadcast-accuracy reward, a Shapley-based marginal-credit reward, and a tool-process reward, and reports average match improvements of 23.66% and 14.05% over single-agent and multi-agent approaches, respectively (Nematov et al., 6 Jul 2025, Li et al., 9 Feb 2026).

6. Limitations, assumptions, and open problems

The principal limitation of TSA is that the attribution depends on the chosen game. The player set may be clients, routes, measurements, frames, documents, or agents; the value function may be a loss, a probability, a log-likelihood, or a binary terminal reward; and removal semantics may be baseline, marginal, or conditional. The general Shapley literature emphasizes that these choices are inequivalent, especially under strong dependence, and that conditional expectations in high dimension are difficult to estimate (Chen et al., 2022).

The domain-specific papers make these dependencies explicit. FedTSV provides no convergence proof and acknowledges that the usual FedAvg analysis does not directly apply because the effective objective is time-varying; robustness remains empirical, and the method requires a reliable server-side validation set. TrafPS assumes equal sharing among trajectories within a grid and channel, which may blur heterogeneous vehicle effects; it also inherits the computational burden and model dependence of region-level SHAP. The wafer-diagnosis TSA simplifies physical admissibility to prefix-like sequential subsets and depends on accurate lot metadata for LAKI. The trajectory-prediction study stresses that positive Shapley values indicate what a trained model uses, not necessarily what is truly causal in the data-generating system (Kuznetsov et al., 28 May 2026, Feng et al., 2024, Miyaguchi et al., 27 Jul 2025, Makansi et al., 2021).

Related TSA-like work exposes further issues. In RAG, attribution explains the model’s own generated answer rather than ground-truth correctness, and synergy is systematically under-credited. In SHARP, leave-one-agent-out counterfactual masking is only a trajectory-specific approximation to full Shapley averaging and assumes that one can preserve the environment and the rest of the interaction while removing an agent. More broadly, long trajectories create a large-player regime in which exact coalitional reasoning is impossible without grouping, adaptive sampling, or amortized explainers (Nematov et al., 6 Jul 2025, Li et al., 9 Feb 2026, Chen et al., 2022).

The open problems are correspondingly heterogeneous but structurally related. The cited papers repeatedly call for more rigorous convergence and game-theoretic analysis, better handling of dependence and causal structure, scalable approximations for large trajectories, and richer hierarchical schemes that move from whole-trajectory attribution to step-wise or token-wise credit. This suggests that TSA is evolving toward a general language for Shapley analysis in sequential systems, but that its semantics remain tightly coupled to application-specific assumptions about trajectory construction, counterfactual completion, and the social or predictive objective being decomposed (Kuznetsov et al., 28 May 2026, Nematov et al., 6 Jul 2025, Chen et al., 2022).