Ambiguity and the variance of gambles

Ellsberg’s paradox shows that people prefer gambles with known probabilities to those where they are uncertain. Standard explanations rule out risk aversion by appealing to Savage’s (1954) subjective expected utility theory but this axiomatic approach leaves open other interpretations of the evidence. We provide a simpler argument: a routine application of the law of total variance shows that the variance of the payoff from a binary gamble is determined entirely by the mean probability belief, not by uncertainty about those beliefs. Ellsberg-type choices are not consistent with rational mean–variance evaluations of risk.