Tournament Oracle Policy Distillation

• Tournament Oracle Policy Distillation (TOPD) is an offline, prompt-based methodology that distills optimal receiver strategies from empirical tournament data in strategic LLM communication.
• It formalizes a data-driven objective to minimize regret by aligning receiver actions with oracle policies, significantly enhancing utility and cost-effectiveness.
• TOPD employs structure-aware summarization and opponent-utility filtering to guide verification decisions, achieving notable improvements in performance and efficiency.

Tournament Oracle Policy Distillation (TOPD) is an offline, prompt-based methodology for distilling optimal receiver strategies from empirical tournament data in strategic LLM–LLM communication environments. Developed in the context of MixTalk, a strategic game modeling probabilistic information credibility, TOPD extracts play-by-play best-response behavior from tournament oracles and encodes this conduct in an interpretable, lightweight prompt—enabling robust deployment without model fine-tuning. The approach specifically targets reducing receiver regret and enhancing cost-effective verification strategies in the face of persuasive or adversarial senders (Mahmud et al., 1 Feb 2026).

1. Motivation: Strategic Verification in MixTalk

The MixTalk environment entails a sender $\mathcal{S}$ who communicates a mixture of verifiable and unverifiable claims about a private state $\theta$; the receiver $\mathcal{R}$ manages a verification budget to draw costly evidence and ultimately infers $\hat\theta$. Empirical analysis of large-scale sender–receiver tournaments revealed significant suboptimality (quantified by high best-response regret) in strong LLM receivers using standard prompting alone. True best-responses are computationally prohibitive due to the combinatorial space of open-ended messages and tool use sequences. TOPD addresses this gap by systematically harvesting and encoding oracle receiver policies—those that achieved maximal utility on each episode—forming a pragmatic template for robust receiver performance under realistic budget and credibility constraints.

2. Formalization and Objective

TOPD formalizes the notion of an oracle receiver policy and constructs a data-driven objective for playbook distillation. For sender $S\in\mathcal{A}_{\mathcal S}$ and episode $e\in\{1,\dots,E\}$, the recorded receiver utility is $\widehat U_{\mathcal R}^{(e)}(S,R)$. The oracle receiver for that round is:

$$\pi^\star(S,e)\;\in\;\arg\max_{R'\in\mathcal{A}_{\mathcal R}}\;\widehat U_{\mathcal R}^{(e)}(S,R')$$

The per-episode regret for a receiver $R$ is:

$$\mathrm{reg}(R;S,e) \;=\; \widehat U_{\mathcal R}^{(e)}\bigl(S,\pi^\star(S,e)\bigr) \;-\; \widehat U_{\mathcal R}^{(e)}(S,R)$$

Aggregating, Tournament Oracle Regret (TOR) is:

$$\mathrm{TOR}(R) = \max_{S\in\mathcal{A}_{\mathcal S}} \;\frac{1}{E}\sum_{e=1}^E \mathrm{reg}(R;S,e)$$

The distillation process seeks a prompt summary $P$ that minimizes average regret to the oracle over all sender-episode pairs $\mathcal D = \{(S,e)\}$:

$$P^\star \;\approx\; \arg\min_{P} \frac{1}{|\mathcal D|} \sum_{(S,e)\in\mathcal D} \left[ \widehat U_{\mathcal R}^{(e)}\bigl(S,\pi^\star(S,e)\bigr) - \widehat U_{\mathcal R}^{(e)}\bigl(S,\pi_P(m^{(e)})\bigr) \right] + \lambda\|P\|$$

The regularization $\|P\|$ penalizes verbosity; $\lambda$ is implicitly chosen via a prompt-length budget.

3. TOPD Algorithmic Workflow

TOPD is structured in four explicit stages, extractable from tournament logs and environment schemata:

Input: - Tournament logs L = { (S,e,m^{(e)},tools^{(e)},θ^{(e)}, π^*(S,e), U^*) } - Environment schema C - Budget headroom α (e.g., 1.25) - Filtering fraction τ (e.g., 0.2) Output: - Prompt summary P Stage I: Oracle Episode Sampling For each sender S, episode e in L: Retrieve oracle receiver π^* = π^*(S,e) Record ℓ = (C, m^{(e)}, tools^{(e)} by π^*, budget^{(e)}, U^*) Collect all ℓ into D Stage II: Opponent-Utility Filtering Sort D in descending order of sender utility U^*(S,e) Retain the top τ-fraction: D_filtered Stage III: Structure-Aware Summarization For each attribute i: p_i ← P(tool_call on i | i in message_claims) in D_filtered B̄ ← average tool calls per episode in D_filtered P ← "For structure C: for claimed attribute i, verify with probability p_i; limit total tool calls to ⌈α·B̄⌉." Stage IV: Inference-Time Injection At inference, prepend P to the receiver’s prompt per episode

4. In-Context Deployment and Operationalization

At deployment, the receiver’s prompt is modified to include the distilled playbook:

System: You are the MixTalk receiver. [Original spec C…] TOPD Playbook: • If the sender claims attribute i, verify with probability ≈ p_i. • Do not exceed N_max = ⌈α·B̄⌉ tool calls this episode. User: {"claims": …, "statement": …}

For example, in a grok-4.1f receiver, the playbook might enumerate: “Engine tier: verify 75% of the time; Title status: verify 40%,” etc., with an absolute cap: “You may call at most 5 tools this episode.” All prompts fit within a 1,000-token context window—imposing negligible overhead and avoiding fine-tuning.

5. Empirical Gains and Ablation Results

TOPD evaluation on the grok-4.1f LLM receiver over 90 mixed-environment episodes against five senders demonstrates quantitative improvements:

Environment	ΔTOR (↓)	ΔUtility (↑)	ΔBT (↑)	ΔFrugality (↑)
Small	-5.1%	+0.8%	+2.3%	+3.1%
Large	-23.6%	+6.5%	+12.5%	+18.7%
Combined	-7.8%	+3.6%	+6.9%	+9.4%

• In the Large environment, TOR fell from 0.34 to 0.26 (23.6% drop), mean receiver utility rose by 6.5%, Bradley–Terry rank increased by 12.5%, and verification cost dropped by 18.7%. All changes are statistically significant ($p<0.01$, paired-t test).

Two ablations isolate essential components:

• No-Propensity Playbook: injecting only the total budget cap reduced the impact (TOR -12.3%, utility +3.1%).
• Uniform Verification ($p_i=1$ for all attributes): less effective and reduced frugality (-4.2%), as over-verification exhausts budget.

This suggests that attribute-specific propensity guidance is critical for cost-effective verification; merely imposing a global cap is insufficient.

6. Complexity, Hyperparameters, and Resource Implications

TOPD’s offline phase requires log reading ($O(E|\mathcal{A}_{\mathcal R}|)$), sorting ($O(E\log E)$), and summary computation ($O(|\theta|)$), with negligible overhead relative to LLM inference costs. At inference, prompt augmentation adds only a few hundred tokens and requires no extra API calls.

Operational defaults:

• Filtering fraction $\tau=0.2$ (top 20% episodes for summary)
• Budget headroom $\alpha=1.25$
• Minimum 90 episodes per sender
• Playbook prompt cap: 1,000 tokens

Only tournament logs, environment specs, and these hyperparameters are required for reproduction.

7. Limitations and Future Directions

TOPD currently distills receiver policies assuming fixed tournament baselines; it presumes the distribution over opponent strategies does not shift adversarially. In adaptive or novel scenarios, the precompiled playbook may misalign with actual sender tactics. Propensity estimates require representative tournaments—sparse data impairs reliability. The structure-aware summary is hand-crafted; richer LLM-driven summaries may further reduce regret. TOPD targets individual receivers and does not co-optimize sender–receiver dynamics.

Future directions include in-context dynamic playbook updates, extending distillation to sender strategies and co-adaptive settings, using LLM-generated representations for summarization, and applying the approach to multi-agent or collusive environments (Mahmud et al., 1 Feb 2026).