Bidirectional Reward Allocation Mechanism

• Bidirectional Reward Allocation Mechanism is defined as using two complementary informational directions—forward (L2R) and backward (R2L)—to assess reasoning steps.
• It averages scores from both directions to update rewards based on future context, enhancing error detection and global consistency in multi-step reasoning.
• Empirical results show significant performance improvements, validating the method’s role in better trajectory selection and reliable reward assignment.

A bidirectional reward allocation mechanism, as an Editor’s term, denotes reward assignment that uses two complementary informational directions rather than a single forward signal. In current LLM research, the clearest formalization is the Bidirectional Process Reward Model (BiPRM), which augments conventional left-to-right process reward modeling with a parallel right-to-left evaluation stream so that later reasoning steps can help assess earlier ones in real time (Zhang et al., 3 Aug 2025). Related work uses the same broad intuition in other ways: set-level utility can be decomposed into candidate-specific signals and then fed back into policy optimization (Ai et al., 31 Mar 2026), bidirectional chains of thought can be coupled with structurally weighted rewards (Liu et al., 13 May 2025), and peer-, sensor-, and coalition-based allocation mechanisms can combine forward assessments with predictive scoring, calibrated attribution, or ex-post settlement (Carvalho et al., 2013, Carvalho et al., 2013, Ballandies et al., 30 Apr 2026, Holly-Ponientzietz et al., 30 Jun 2026).

1. From myopic prefix scoring to backward verification

In process reward modeling, the central motivation for bidirectionality is the limitation of unidirectional left-to-right evaluation. Existing PRMs are described as myopic because they evaluate a step only from preceding context:

$r_t^{\text{L2R}} = f_\theta(s_t \mid q, s_{<t}).$

Under this formulation, step $s_t$ cannot be re-evaluated using future reasoning steps $s_{t+1}, \dots, s_T$, and the paper states:

$$\frac{\partial r_t^{\text{L2R}}}{\partial s_{t+k}} = 0 \quad \forall k \in [0, T-t].$$

The immediate consequence is that downstream information cannot influence earlier reward assignment. The motivating example is mathematical reasoning, where a step may look plausible locally but becomes verifiable only after later derivations. A conjecture in an induction proof, for example, may be impossible to confirm using only prior steps, whereas later derivation steps can reveal whether it was correct. BiPRM is therefore designed to capture global consistency rather than only local plausibility (Zhang et al., 3 Aug 2025).

This formulation clarifies a recurrent misconception. In this literature, bidirectionality is not primarily about training two different reward models, nor about introducing a separate backward parameterization. It is about allowing later reasoning to participate in the evaluation of earlier reasoning. The paper’s formulation treats this as reward allocation across reasoning steps in real time: the reward for an earlier step is no longer fixed by prefix context alone, but is updated by information from the future via a reversed evaluation stream. This suggests that the core innovation is a change in the informational dependency structure of reward assignment, not a change in the basic scalar-reward objective.

2. Formal specification of BiPRM

BiPRM evaluates each reasoning step in both directions and averages the two scores. Its left-to-right and right-to-left step rewards are defined as

$r_t^{\text{L2R}} = f_\theta(s_t \mid q, s_{<t}),$

$r_t^{\text{R2L}} = f_\theta(s_t \mid q, s_{>t}),$

and the bidirectional step reward is

$$r_t^{\text{BiPRM}} = \frac{1}{2}\left(r_t^{\text{L2R}} + r_t^{\text{R2L}}\right).$$

Trajectory-level reward is then aggregated with a reduction operator,

$$R_{\text{BiPRM}}(\tau) = \bigoplus_{t=1}^{T} r_t^{\text{BiPRM}},$$

where

$$\bigoplus \in \{\text{product}, \min, \max, \text{average}\}.$$

In the experiments, following prior work, the choice is

$\bigoplus = \min.$

The paper further notes that, under the bidirectional score,

$s_t$0

so future reasoning can affect the score assigned to earlier steps (Zhang et al., 3 Aug 2025).

The formal role of this construction is precise. The L2R stream evaluates a step from prior context, while the R2L stream evaluates the same step from future context. Their average produces a step score that is sensitive to both local plausibility and downstream consistency. The paper characterizes this as backward verification and better global coordination. A plausible implication is that BiPRM changes the granularity of error detection: instead of waiting for a final trajectory-level signal, it permits stepwise reward reassignment once later steps expose or validate earlier commitments.

3. Prompt reversal, architecture, and objective compatibility

The right-to-left stream is constructed by prompt modification only. Given a reasoning trajectory

$s_t$1

BiPRM constructs a reversed trajectory

$s_t$2

The question $s_t$3 remains unchanged, while the reasoning steps are reordered in descending order. The same base PRM is then run on the reversed prompt to obtain R2L rewards. The paper emphasizes that this is not a separate backward model with its own weights; it is the same reward model applied to a reversed reasoning sequence. Consequently, the method introduces no extra parameters, and the paper states that it introduces no inference latency because the L2R and R2L streams are run in parallel rather than sequentially (Zhang et al., 3 Aug 2025).

BiPRM is built on top of existing LLM-based PRMs. It follows the common PRM implementation of adding a scalar-valued reward head on the final hidden representation of the base LLM, with base model parameters $s_t$4 and reward head parameters $s_t$5. A vanilla PRM is described as

$s_t$6

where $s_t$7 maps hidden representations to scalar step rewards. BiPRM does not introduce a new training objective. It is evaluated under three existing PRM objectives: Binary Cross-Entropy (BCE), Mean Squared Error (MSE), and Q-value rankings loss. The method is tested on three backbones: Qwen2.5-Math-1.5B, Rho-Math-1B, and Deepseek-Math-7B. The paper’s stated position is that the bidirectional mechanism is orthogonal to the choice of objective and broadly compatible with different PRM architectures and model sizes (Zhang et al., 3 Aug 2025).

The architectural economy is central to the method’s identity. Bidirectionality is achieved without a dedicated backward network, without a new loss family, and without modifying the base LLM/reward-head decomposition. This suggests that the mechanism should be understood as an allocation rule over contextual evidence rather than as a model-family replacement.

4. Empirical profile, ablations, and operating assumptions

The empirical results report consistent improvements over unidirectional baselines across all tested configurations. For Qwen2.5-Math-1.5B, across 18 configurations, BiPRM outperforms L2R in every case; for Rho-Math-1B and Deepseek-Math-7B, across 36 configurations, BiPRM again improves over L2R in all settings (Zhang et al., 3 Aug 2025).

Backbone	L2R	BiPRM
Qwen2.5-Math-1.5B	47.38	50.39
Rho-Math-1B	36.16	39.72
Deepseek-Math-7B	47.00	49.95

A highlighted Qwen2.5-Math-1.5B result is on MuggleMath-13B with BCE, where average BON@$s_t$8 improves from 25.88 to 33.28, a 28.6% relative improvement. In the best single setting under the same configuration, BON@128 rises from 24.20 to 36.60, a 47.1% relative gain. The largest relative gain reported overall is for Deepseek-Math-7B + MuggleMath-13B + BCE, where average BON@$s_t$9 increases from 25.08 to 33.08, a 31.9% relative improvement. The paper presents these results as evidence that bidirectional reward allocation improves stepwise reward estimation and Best-of-N selection (Zhang et al., 3 Aug 2025).

The ablation study compares w/o R2L, w/o L2R, and full BiPRM. The reported result is consistent across datasets, policy models, and objectives: the full BiPRM always performs best, and removing either direction hurts performance. The paper interprets the two directions as complementary: L2R gives natural prefix-based reasoning, R2L provides backward verification, and together they produce stronger supervision than either alone. Its case study further reports that, on a correct trajectory, BiPRM assigns progressively strong, stable positive scores, while on an incorrect trajectory it sharply drops the score when a major logical error appears; L2R-PRM is described as less sensitive and as giving weaker or misleading signals (Zhang et al., 3 Aug 2025).

The paper does not present a long limitations section, but several assumptions are explicit or implicit. R2L scoring requires a full reasoning trajectory, so the mechanism is most natural in offline evaluation or reranking settings rather than strictly online generation. It is best suited to multi-step reasoning; if trajectories are very short or only one step long, bidirectional context offers little extra value. The experiments focus on mathematical reasoning benchmarks, so benefits may be less clear in domains where future steps are not informative or trajectories are not well-structured. The method also depends on the quality of step decomposition: if reasoning steps are poorly segmented, backward verification may be less meaningful. These constraints delimit the scope of the reported gains rather than negating them.

5. Related LLM formulations of bidirectional reward allocation

The term “bidirectional” is used differently in adjacent LLM reward-allocation work. In "ShapE-GRPO: Shapley-Enhanced Reward Allocation for Multi-Candidate LLM Training" (Ai et al., 31 Mar 2026), there is no explicit bidirectional module, but the mechanism is bidirectional in effect. The paper frames LLM-as-a-recommender / multi-candidate generation, where the response is decomposed as

$s_{t+1}, \dots, s_T$0

and the reward is set-level and often permutation-invariant over candidates:

$s_{t+1}, \dots, s_T$1

Standard GRPO assigns the same scalar reward to every token in the response,

$s_{t+1}, \dots, s_T$2

which induces free-riding when weak candidates inherit the positive signal generated by a strong peer. ShapE-GRPO uses a Shapley-based decomposition,

$s_{t+1}, \dots, s_T$3

and broadcasts candidate-specific reward back to candidate tokens:

$s_{t+1}, \dots, s_T$4

The paper explicitly characterizes the two directions as top-down set-level utility $s_{t+1}, \dots, s_T$5 candidate-level signals and bottom-up candidate contributions $s_{t+1}, \dots, s_T$6 policy optimization of the whole response. This is not temporal bidirectionality, but it is a bidirectional allocation pipeline in the sense that global reward is decomposed downward and then re-enters the RL objective upward.

A second distinct usage appears in "Fusing Bidirectional Chains of Thought and Reward Mechanisms A Method for Enhancing Question-Answering Capabilities of LLMs for Chinese Intangible Cultural Heritage" (Liu et al., 13 May 2025). That framework combines Bidirectional Chains of Thought with a reward mechanism on ICH-Qwen. The Bi-CoT data contains four parts: forward reasoning, reverse question, reverse reasoning, and final answer. For each question, the model samples a group of outputs

$s_{t+1}, \dots, s_T$7

with $s_{t+1}, \dots, s_T$8 in the paper. Each output is scored by

$s_{t+1}, \dots, s_T$9

and the scores are normalized within the group using

$$\frac{\partial r_t^{\text{L2R}}}{\partial s_{t+k}} = 0 \quad \forall k \in [0, T-t].$$0

The reward decomposition includes format reward, ROUGE-L / longest common subsequence similarity, and keyword coverage. The part weights are forward reasoning $$\frac{\partial r_t^{\text{L2R}}}{\partial s_{t+k}} = 0 \quad \forall k \in [0, T-t].$$1, reverse question $$\frac{\partial r_t^{\text{L2R}}}{\partial s_{t+k}} = 0 \quad \forall k \in [0, T-t].$$2, reverse reasoning $$\frac{\partial r_t^{\text{L2R}}}{\partial s_{t+k}} = 0 \quad \forall k \in [0, T-t].$$3, and final answer $$\frac{\partial r_t^{\text{L2R}}}{\partial s_{t+k}} = 0 \quad \forall k \in [0, T-t].$$4, while each keyword match gives $$\frac{\partial r_t^{\text{L2R}}}{\partial s_{t+k}} = 0 \quad \forall k \in [0, T-t].$$5 points and the keyword reward is capped at $$\frac{\partial r_t^{\text{L2R}}}{\partial s_{t+k}} = 0 \quad \forall k \in [0, T-t].$$6. Here bidirectionality refers to forward and reverse reasoning paths rather than L2R/R2L step scoring. The paper’s ablation reports that removing the reward mechanism lowers Accuracy from 65.35 to 54.46, BLEU-4 from 12.73 to 9.21, and Rouge-L from 26.73 to 21.50, while removing Bi-CoT lowers Accuracy to 43.56. This suggests that, in this line of work, bidirectional reward allocation is inseparable from the structure of the reasoning trace itself.

These two papers show that “bidirectional reward allocation mechanism” is not a single canonical recipe. In one case, bidirectionality means future-to-past step verification; in another, set-to-candidate and candidate-to-policy reward flow; in another, forward and reverse reasoning paths scored under a structurally weighted reward.

6. Peer-, attribution-, and coalition-based analogues

Outside LLM reasoning, several mechanisms instantiate related two-directional logics. In "Sharing a Reward Based on Peer Evaluations" (Carvalho et al., 2013), a fixed reward $$\frac{\partial r_t^{\text{L2R}}}{\partial s_{t+k}} = 0 \quad \forall k \in [0, T-t].$$7 is divided among $$\frac{\partial r_t^{\text{L2R}}}{\partial s_{t+k}} = 0 \quad \forall k \in [0, T-t].$$8 agents based on peer-supplied information. The paper presents a peer-evaluation mechanism, where each agent gives a direct evaluation for every other member of the group, and a peer-prediction mechanism, where each agent reports how they believe group members will evaluate a particular agent. In the peer-evaluation mechanism, shares are

$$\frac{\partial r_t^{\text{L2R}}}{\partial s_{t+k}} = 0 \quad \forall k \in [0, T-t].$$9

This mechanism is strategy-proof, budget-balanced, and individually rational, but it is described as extremely susceptible to collusions. In the peer-prediction mechanism, payments combine a grade term with a proper-scoring-rule term,

$r_t^{\text{L2R}} = f_\theta(s_t \mid q, s_{<t}),$0

and the paper states that if $r_t^{\text{L2R}} = f_\theta(s_t \mid q, s_{<t}),$1, then the peer-prediction mechanism is collusion-resistant. Here the two directions are direct peer assessment and predictive beliefs about peer assessment.

"A Truth Serum for Sharing Rewards" (Carvalho et al., 2013) refines this peer-based logic by combining received peer evaluations with Bayesian Truth Serum scoring. After rescaling evaluations and aggregating received opinions, the final share is

$r_t^{\text{L2R}} = f_\theta(s_t \mid q, s_{<t}),$2

where $r_t^{\text{L2R}} = f_\theta(s_t \mid q, s_{<t}),$3 is the evaluation-based component and $r_t^{\text{L2R}} = f_\theta(s_t \mid q, s_{<t}),$4 is the average BTS score over peers. The paper states that, under Bayesian assumptions and a sufficiently large population, the mechanism is incentive-compatible, budget-balanced, and tractable, and it provides bounds on $r_t^{\text{L2R}} = f_\theta(s_t \mid q, s_{<t}),$5 to guarantee fairness and individual rationality. In this setting, bidirectionality is explicitly peer-based: agents are rewarded both for being highly regarded and for reporting evaluations and predictions truthfully.

Two recent non-LLM examples further delimit the term’s scope. "Calibrating Attribution Proxies for Reward Allocation in Participatory Weather Sensing" (Ballandies et al., 30 Apr 2026) does not present a complete bidirectional reward allocation mechanism end-to-end. Instead, it proposes a calibrated attribution proxy as a value signal for reward allocation, with proportional payments

$r_t^{\text{L2R}} = f_\theta(s_t \mid q, s_{<t}),$6

The paper distinguishes signed attributions for gaming detection from absolute attributions for reward allocation, and it reports that attribution captures near-optimal sensor placement utility with monotonically faithful payments, while also showing gaming vulnerability that requires external baseline data for detection. The mechanism is therefore only partially bidirectional: it supports reward concentration and monitoring of harmful manipulation, but does not implement negative payments, a two-sided market, or a complete incentive-compatible design.

In "A Coalitional Stable and Fair Reward Allocation for Dynamic Virtual Power Plants" (Holly-Ponientzietz et al., 30 Jun 2026), bidirectionality takes the form of ex-ante and ex-post games. The paper defines a Forecasted Reward Game $r_t^{\text{L2R}} = f_\theta(s_t \mid q, s_{<t}),$7 and a Realized Reward Game $r_t^{\text{L2R}} = f_\theta(s_t \mid q, s_{<t}),$8, with coalition reward

$r_t^{\text{L2R}} = f_\theta(s_t \mid q, s_{<t}),$9

The allocation concept is chosen as Nucleolus if the game is not convex / the core may be empty, and Shapley value if the game is convex. The paper states that the proposed reward allocation fulfills individual rationality, coalitional stability, incentive compatibility, optimality, fairness and ex-post consistency. Here the two directions are forecast $r_t^{\text{R2L}} = f_\theta(s_t \mid q, s_{>t}),$0 bid $r_t^{\text{R2L}} = f_\theta(s_t \mid q, s_{>t}),$1 anticipated reward and realized performance $r_t^{\text{R2L}} = f_\theta(s_t \mid q, s_{>t}),$2 settlement $r_t^{\text{R2L}} = f_\theta(s_t \mid q, s_{>t}),$3 final allocation.

Taken together, these adjacent literatures show that the phrase has a stable high-level meaning but not a single universal formalization. The common thread is that reward assignment is not treated as a one-pass scalar broadcast. Instead, allocation is revised or decomposed using a second informational direction: future steps in BiPRM, candidate marginal contributions in ShapE-GRPO, reverse reasoning in Bi-CoT training, predictive beliefs in peer-evaluation systems, calibrated attribution plus audit signals in weather sensing, or ex-post settlement in dynamic coalitional games.

7. Conceptual significance and boundaries

Within process reward modeling, the main significance of bidirectional reward allocation is that it replaces a prefix-only notion of correctness with a trajectory-aware one. In BiPRM, later reasoning can upweight early steps that are later validated or downweight early steps that later prove inconsistent; this is the paper’s explicit explanation of why the mechanism improves global consistency, reduces myopic errors, and yields better trajectory selection (Zhang et al., 3 Aug 2025). In adjacent LLM work, the same significance appears as finer-grained credit assignment: ShapE-GRPO eliminates free-riding by allocating set reward according to candidate marginal contribution (Ai et al., 31 Mar 2026), while the ICH-Qwen framework converts a sparse QA objective into richer supervision through format compliance, ROUGE-L similarity, keyword coverage, and group-relative ranking over forward and reverse reasoning paths (Liu et al., 13 May 2025).

At the same time, the surveyed literature shows that “bidirectional” should not be conflated with “symmetric,” “fully game-theoretic,” or “universally fair.” BiPRM is lightweight because it introduces no extra parameters and no inference latency, but it presupposes complete reasoning trajectories and is evaluated on mathematical reasoning benchmarks (Zhang et al., 3 Aug 2025). ShapE-GRPO is exact and efficient in its targeted multi-candidate setting, but it is tailored to permutation-invariant set-level reward and is not presented as a general solution for arbitrary token-level credit assignment (Ai et al., 31 Mar 2026). The participatory weather-sensing work explicitly states that it supplies the valuation engine for a reward mechanism, not the complete mechanism itself (Ballandies et al., 30 Apr 2026). The peer-based and DVPP papers show that stronger guarantees such as collusion resistance, budget balance, core stability, or ex-post consistency require additional structural assumptions and mechanism-specific constraints (Carvalho et al., 2013, Carvalho et al., 2013, Holly-Ponientzietz et al., 30 Jun 2026).

A plausible synthesis is that bidirectional reward allocation is best understood as a design principle rather than a single algorithmic object. The principle is to let reward propagate through more than one explanatory direction—past and future, global and local, evaluation and prediction, forecast and settlement—so that allocation is more faithful to contribution structure than a one-directional scalar signal would permit.