Hardness Results for Signaling in Bayesian Zero-Sum and Network Routing Games
Bibliographic record
Abstract
We study the optimization problem faced by a perfectly informed principal in a Bayesian game, who reveals information to the players about the state of nature to obtain a desirable equilibrium. This signaling problem is the natural design question motivated by uncertainty in games and has attracted much recent attention. We present new hardness results for signaling problems in (a) Bayesian two-player zero-sum games, and (b) Bayesian network routing games. For Bayesian zero-sum games, when the principal seeks to maximize the equilibrium utility of a player, we show that it is NP-hard to obtain an additive FPTAS. Our hardness proof exploits duality and the equivalence of separation and optimization in a novel way. Further, we rule out an additive PTAS assuming planted clique hardness, which states that no polynomial time algorithm can recover a planted clique from an Erdős-Rényi random graph. Complementing these, we obtain a PTAS for a structured class of zero-sum games (where obtaining an FPTAS is still NP-hard) when the payoff matrices obey a Lipschitz condition. Previous results ruled out an FPTAS assuming planted-clique hardness, and a PTAS only for implicit games with quasi-polynomial-size strategy sets. For Bayesian network routing games, wherein the principal seeks to minimize the average latency of the Nash flow, we show that it is NP-hard to obtain a (multiplicative) $(4/3 - ε)$-approximation, even for linear latency functions. This is the optimal inapproximability result for linear latencies, since we show that full revelation achieves a $(4/3)$-approximation for linear latencies.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.017 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.008 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".