Efron's conjecture on vulnerability to bias in a method for balancing sequential trials

Efron (1971) proposed a method for sequential assignment to treatments or control which is in many ways superior to traditional procedures. To analyse the method's susceptibility to accidental bias a criterion concerning the maximum eigenvalue of a fundamental covariance matrix was introduced. On the basis of numerical evidence, Efron conjectured an explicit formula for this eigenvalue. This note gives a proof of that conjecture.