Cooperation can fail when socially valuable effort is costly, hard to observe, and benefits mainly someone else. Building on Holmstrom's model of moral hazard in teams, the Dialogue Moral Hazard Game instantiates this hidden-action structure as a textual environment for language agents. An agent chooses between keeping an immediate local reward and paying a query cost to reveal a hidden safety fact that helps another agent's downstream decision. We evaluate fourteen open-weight and four frontier models using measures of information acquisition, communication, downstream use, and team success. In matched 3,015-decision-per-model experiments, GPT-5.6 Sol, Claude Opus 4.8, and Nemotron-3 Ultra track the derived private-share boundary across nine query costs, with mean absolute errors of 0.013, 0.030, and 0.024. Muse Spark 1.1 responds directionally, whereas Fable 5 remains query-saturated. SFT, RLOO, SFT+RLOO, and GEPA produce heterogeneous mechanism changes. GEPA raises Muse team success from 22.2% to 100.0% while reducing query use from 51.1% to 0.3%. Frozen-prompt interventions show that this success depends on a learned rank-label mapping rather than direct revelation: changing the mapping reduces team success from 100.0% to 12.5% and then 0.0%. We introduce CREDIT (Counterfactual Replay for Evidence-Driven Information Transfer), a mechanism-aligned multi-agent prompt-optimization algorithm that uses matched hidden-state twins and total-action replay to reward robust causal contribution rather than query frequency. Across five models and multiple seeds, CREDIT preserves query-mediated behavior while revealing model-specific acquisition and downstream-use bottlenecks. Optimization can reach the same aggregate outcome through direct revelation or a learned effective information structure, motivating mechanism-level evaluation and optimization rather than team success alone.

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