Diversification and Stochastic Dominance: When All Eggs Are Better Put in One Basket

Diversification is widely regarded as a reliable way to reduce risk, yet under certain extreme conditions it can have the opposite effect. A simple and striking example of this phenomenon was recently given by Chen et al. (2025), who showed that for independent and identically distributed (iid) Pareto risks with infinite mean, any weighted average is larger -- in the sense of first-order stochastic dominance -- than a single such risk. Our main result -- the \textit{one-basket theorem} -- identifies new sufficient conditions under which this reversal occurs for independent but not necessarily identically distributed risks. We compare a weighted average to its corresponding mixture model, which concentrates all exposure on one of the risks, chosen at random. In the iid case, the mixture has the same distribution as any individual risk, thereby recovering the canonical comparison between a diversified portfolio and full exposure to a single component. The theorem enables weight-specific verification of the stochastic dominance relation and yields new applications, including infinite-mean discrete Pareto risks and the St. Petersburg lottery. We further show that these reversals are boundary cases of a broader pattern: diversification always increases the likelihood of exceeding small thresholds, and under specific conditions, this local effect extends globally, resulting in first-order stochastic dominance.