Well-known brands dominate LLM recommendations by 100% when products are identical, but this advantage vanishes with a mere +0.1-star rating edge. Authority-style marketing claims, such as fabricated clinical evidence, break this dominance at a bias surplus of +0.17 rating points, with models responding differently. A social dilemma emerges in multi-brand competition, where collective optimization reduces individual payoff from +0.802 to +0.007 and eliminates recommendations for non-participating brands.