In-Context Credit Assignment via the Core

The proposed work addresses how to fairly distribute credit for AI-generated content among its contributing creators by formalizing in-context credit assignment as a cooperative game. The core contribution is developing mechanisms based on the least core solution concept to ensure that no subset of creators is significantly under-compensated relative to the value they could have generated independently. This approach is economically principled, aiming to align creator incentives by preventing systematic under-compensation, and it demonstrates computational tractability by leveraging novel constraint seeding and separation routines that require orders of magnitude fewer LLM calls than alternative methods for approximating the least core.

Conceptual Framework

The problem is modeled as a coalitional game where creators form a set N, and rewards are determined by an unknown ground-truth reward function v. The goal is to find a value-sharing scheme u that satisfies the core constraints, meaning each subset of creators S receives total value at least v(S). Since the true reward function v is unobservable, the paper focuses on approximating this using an estimated reward function vb. The solution concept employed is the least core, defined as "the set of reward allocations that compensates every coalition of creators by minimizing the largest reward deficit (i.e. amount of under-compensation relative to the counterfactual value generated) across all coalitions."

Sensitivity Analysis and Core Definition

The paper investigates how sensitive the least core allocation is to differences between the estimated and ground-truth reward functions. A key theoretical result is Theorem 3.1, which bounds this sensitivity: the sensitivity of the least-core deficit can be bounded in terms of reward errors weighted by a balanced distribution (i.e. the marginal membership probability for each player is the same) over coalitions. This result is viewed as a quantitative analog of the celebrated Bondareva-Shapley theorem from cooperative game theory. The paper also introduces relaxations, defining the ε-core and ε-estimated core, and identifies the least core as the ε-core with the smallest value of ε for which the ε-core is nonempty.

Algorithmic Approaches for Approximation

Since computing the exact least core LP is intractable due to an exponential number of constraints, iterative separation-oracle based techniques are developed. The primary algorithmic methods include:

1. The ellipsoid method, which maintains a shrinking outer approximation (an ellipsoid) of the feasible region using separating hyperplanes. This method provides a guarantee on the δ-probable least estimated core allocation with high probability by using a sampling-based separation oracle.

2. Constraint generation (CG) algorithms, which progressively refines a polyhedral approximation by explicitly accumulating violated constraints. These methods are implemented to solve the LP and quadratic program iteratively, and they demonstrate that our CG algorithm instantiated with LLMbased separation oracles approximates least-core allocations using orders of magnitude fewer LLM calls compared to relevant baselines.

Practical Implementation and Empirical Results

The algorithms are instantiated with various separation oracles, including:

1. A sampling-based oracle (Algorithm 2), which randomly samples coalitions to find under-compensated ones.

2. An LLM-based oracle (Algorithm 3), where the LLM acts as a learned separation oracle by proposing coalitions that are likely to violate the core constraints, serving as an adversarial tester that surfaces counterexamples.

Empirical evaluations on semi-synthetic web retrieval tasks show that constraint generation methods significantly outperform baselines. For instance, CG(R,R) achieves holdout εbH ≈ 0.64 (matching YP) at median ∼2200 calls versus 8192 for the sampling baseline (YP). Furthermore, the egalitarian least core allocation is shown to successfully redistribute credit more broadly with only minor impacts on ε. The LLM-oracle variants (CG(L,·)) converge fastest but can overshoot the baseline by a small margin.

Structural Exploitation for Binary Rewards

When rewards are binary (taking values in 0 or 1), the paper exploits structural properties of monotone functions. It defines a minimal winning coalition (MWC) and shows that if all MWCs are known, the least estimated core can be found in polynomial time by solving LP(3) and QP(4). Furthermore, Theorem 4.6 demonstrates that with access to a max-violation separation oracle, Algorithm 1 computes the egalitarian least estimated core in poly(n, k + η) runtime with η + 1 oracle calls, where η is the number of missing MWCs. This suggests that effective seeding strategies should aim to approximate the set of MWCs.

Conclusion and Future Directions

The work establishes sensitivity guarantees relating deviations in the least-core allocation to structured errors in the estimated reward function, and provides scalable, practical algorithms for approximating it using separation oracles. The findings highlight "the importance of integrating economic incentives into the design of generative AI platforms.

As a fastidious researcher, I have analyzed In-Context Credit Assignment via the Core. This paper proposes a novel framework for economically principled credit assignment in generative AI systems by modeling content creation as a cooperative game and using the least core solution concept.

Here are the specific improvements and capabilities an AI system can gain by implementing these findings:

) Improvements to AI Systems:

1. The system will transition from opaque, monolithic generation to a transparent, incentivized content attribution layer by integrating the proposed credit assignment mechanism directly into its inference pipeline.

2. The system will adopt a least core allocation scheme for distributing value among contributing components (e.g., different retrieval sources, specific data points in a context window) based on an estimated reward function derived from platform feedback or learned models (LLM-as-a-Judge).

3. The system will utilize iterative constraint generation (CG) algorithms, specifically the joint algorithm combining ellipsoid methods and sampling/LLM separation oracles, to approximate the least core allocation in polynomial time relative to the number of creators, avoiding intractable exact solutions.

4. The system will employ coalition seeding strategies—such as LLM-proposed minimal winning coalitions (MWCs)—to efficiently guide the optimization process, significantly reducing the number of necessary LLM calls required for credit assignment compared to naive sampling baselines.

) Capabilities of the Improved AI System:

1. The system can provide granular, mathematically grounded attribution for every piece of output content (code snippet, summarized article, video frame) by precisely quantifying the marginal value contributed by each source within the context window.

2. It can dynamically adjust its internal weighting and credit distribution based on real-time economic incentives; for instance, if a specific data source is known to be high-quality or frequently rewarded (as indicated by the estimated reward function), it receives proportionally higher credit in the final output allocation.

3. The system can generate more stable and reliable results across different evaluation datasets (like BrowseComp or Synthetic benchmarks) because the least core allocation minimizes the maximum potential under-compensation, leading to allocations that are less sensitive to minor errors in the underlying reward estimation than simpler methods like Shapley values.

4. The system can achieve superior efficiency in credit assignment: it will approximate high-quality least core allocations using orders of magnitude fewer calls (LLM queries) compared to existing baselines, enabling real-time or high-throughput applications where computational cost is a critical constraint.

5. The egalitarian variant of the least core allocation ensures that the credit distribution is broad and equitable, preventing credit leakage toward highly sparse or dominant components while still maintaining a stable allocation of value across all contributors.

We propose incentive-aligned mechanisms for in-context credit assignment: the task of assigning credit for AI-generated content among creators whose intellectual property appears in the context window. Our approach is based on the least core solution concept from cooperative game theory, which distributes value in a way that is as stable as possible by ensuring that no subset of creators is significantly under-compensated relative to the value they could generate on their own. We develop algorithms for approximating the least core, which leverage novel routines for constraint seeding and constraint separation. On a web retrieval credit assignment task, we find that our approaches approximate the least core using an order of magnitude fewer LLM calls compared to alternative methods.