A Note on Weighted Randomization

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The Econometrics of Randomized Experiments (2016, methodology): “(8.2) We can estimate this regression function using weighted least squares with the weight for unit i, belonging to cluster g(i), equal to 1/Ng(i) as in the Cox (1956) analysis of weighted randomized experiments.”
On regression adjustments in experiments with several treatments (2008, methodology): “N (ii) Kempthorne (1952) points to the diﬀerence between permutation models and the usual linear regression model; see Chapters 7–8, especially Section 8.7. Also see Biometrics vol. 13, no. 3 (1957). Cox (1956) cites Kempthorne, but appears to contradict Theorem 5 above. I am indebted to Joel Middle”
Covariance Adjustment in Randomized Experiments and Observational Studies (2002, background): “Cox (1956) discussed a form of weighted randomization that led to estimates of mean squares associated with covariance adjustment that are unbiased over the randomization distribution.”