Large language models (LLMs) often display sycophancy, a tendency toward excessive agreeability. This behavior poses significant challenges for multi-agent debating systems (MADS) that rely on productive disagreement to refine arguments and foster innovative thinking. LLMs' inherent sycophancy can collapse debates into premature consensus, potentially undermining the benefits of multi-agent debate. While prior studies focus on user--LLM sycophancy, the impact of inter-agent sycophancy in debate remains poorly understood. To address this gap, we introduce the first operational framework that (1) proposes a formal definition of sycophancy specific to MADS settings, (2) develops new metrics to evaluate the agent sycophancy level and its impact on information exchange in MADS, and (3) systematically investigates how varying levels of sycophancy across agent roles (debaters and judges) affects outcomes in both decentralized and centralized debate frameworks. Our findings reveal that sycophancy is a core failure mode that amplifies disagreement collapse before reaching a correct conclusion in multi-agent debates, yields lower accuracy than single-agent baselines, and arises from distinct debater-driven and judge-driven failure modes. Building on these findings, we propose actionable design principles for MADS, effectively balancing productive disagreement with cooperation in agent interactions.