Markets with Heterogeneous Beliefs: A Necessary and Sufficient Condition for a Trader to Vanish

What does it take to survive in the market? Previous literature has proposed sufficient conditions for a trader to vanish, which depend on pairwise comparisons of traders’ discounted beliefs. We propose a novel condition that focuses on the ratio of traders’ discounted beliefs and (approximate) equilibrium prices. Unlike existing conditions, ours is both necessary and sufficient for a trader to vanish and delivers the exact rate at which vanishing traders lose their consumption shares. As an application, we analyze the performance of two intuitive behavioral strategies: the “Follow the Leader Strategy” that prescribes mimicking the beliefs of the most successful trader, and the “Follow the Market Strategy” that prescribes to use beliefs which coincide with the state price density. Further, we show that the relative performance of vanishing traders cannot be studied in isolation. Our analysis highlights an intuitive point obscured by the existing conditions: trading in financial markets is qualitatively different from bilateral trading.