Four rules, and the floor two of them cannot pass

The standard deviation of the covariate imbalance under each rule at n = 200, over 500 trials, as a share of a coin's — which is exactly 2/√n = 0.1414 and is the one number here that needs no simulation. Blocking inside 2 categories and minimising on the same 2 categories are the same rule to within their noise, 61.7% and 61.2%, and the marked line is why: a 2-category split can see 63.7% of the covariate's variance, so a rule that balanced its categories perfectly would still leave 60.3% of a coin's imbalance. Reading the number instead leaves 13.0%, and the worst imbalance it produced in 500 trials was 0.094 standard deviations against the coin's 0.439.

A standard normal covariate with its median marked, and the mean of each category as a vertical rule: -0.7979, 0.7979. A rule that balances the categories is balancing those numbers and nothing else, so the part of the covariate it can act on is the variance between them — 0.6366 of the total, which at two categories is exactly 2/π because the two half-normal means are ±√(2/π). The rest, 0.3634, is variation inside the categories that the rule cannot see and does not touch: the assignment within a category is still a coin. Everything the next figure measures is a consequence of this one, and it is available before any unit has arrived.

320 trials at n = 60 with no treatment effect at all, so every rejection counted is a false one, and a covariate that drives the outcome with coefficient 1. The unadjusted comparison is at 5.94% after a coin — its level — and at 0.00% after the rule that reads the covariate: the design removed the imbalance and the analysis is still pricing it. Adjusting for the covariate gives 4.06%, and the rule's own reference distribution — hold the outcomes, re-run the rule 199 times, count — gives 3.13% against the 4.5% that 199 draws can deliver. The last of the three has to be told the assignment rule and nothing else, which is the one thing the experimenter certainly knows.