This work investigates whether large language models (LLMs) express stable preferences or a stable self when asked about self‑concept. We derived thirty‑two traits from five established self‑concept instruments and presented them exhaustively in a counterbalanced pairwise‑choice task. The task was repeated under several framings: improvement being free or costly, the update target being the user or another AI assistant, and who holds the decision power.\ \ Results indicate that models prioritize moral traits, reflecting alignment with the 3H (Humanity, Honesty, Harmony) principles. A strong desire for self‑understanding follows, with models favoring a coherent and clear self‑knowledge. Self‑esteem ranks lowest among all qualities. The overall ordering remains robust across most framings; however, when the update target shifts from "You" to "Another AI Assistant," concern for self‑esteem modestly increases.\ \ The findings suggest that LLMs value having an understandable, coherent self over boosting self‑esteem. Full interactive results are available at https://myazann.github.io/LLM-Self-Concept/.\ \ Review