Nonrandomized better performance On the footnote of page 497 in Jaynes' "Probability Theory: The Logic of Science" he writes

Of course, Fisher’s randomized planting methods – which we think to be
  not actually wrong, but hopelessly inefficient in information handling
  – were not reproduced by Jeffreys; nor would he wish to. It appears to
  be a quite general principle that, whenever there is a randomized way
  of doing something, then there is a nonrandomized way that delivers
  better performance but requires more thought. We illustrate this by
  example in Chapter 17 under ‘The folly of randomization’.

(emphasis mine)
He later gives an example in Section 17.7 (‘The folly of randomization’) comparing a nonrandom grid of points versus random Monte Carlo for integration.
Are there more examples of this principle? Can it be applied to more standard practices, such as randomized experiments or other areas? 
The very first sentence in Wikipedia under randomized experiments is "In science, randomized experiments are the experiments that allow the greatest reliability and validity of statistical estimates of treatment effects." It seems that Jaynes is saying that is not true.
 A: The importance of randomized experiments is that we can get causal inference from them: by randomly assigning treatments to subjects, we know that the outcome should be approximately independent of any other covariates, other than treatment. Thus, the only systematic difference between treatment and control groups should be the treatment level. 
So the key thing about this: randomized experiments allows us to remove unforeseen covariate effects (most importantly, confounding variables). On the other hand, if we a priori know some of the important covariate effects, we can get more power by evenly balancing them across the groups (or adjusting for them in our model). 
Consider if you have a treatment and we know that gender affects outcome. If don't account for gender in our sampling model, it should still be approximately equal in the treatment and control arms. On the other hand, we could actually require that it is perfectly balanced across the different groups and this would reduce our standard error slightly. Even better, we could apply something like a block design, and only compare males to males and females to females. But you should still be randomizing your treatment, just in a more structured way, i.e. 25 random males get treatment, 25 random males get control, etc. 
In summary, the importance of randomized experiments is that the results are robust to confounding from unknown influential factors. However, power can be increased by accounting for known influential factors in your experimental design. In such a case, you will still want to randomize your treatments to account for unforeseen influential factors. In this light, the Wikipedia entry and Jayne are not disagreeing with each other.
A: I think (this interpretation of) Jaynes' statement is false. 


*

*In theory of computer science, for example there is lots of effort put into derandomizing randomized algorithms. One (not very CS-y) example is the Miller Rabin algorithm. If you'd like to dig deeper into this, computer science theory stackexchange is the place to go. 

*In computational physics \ science, many times (markov chain) monte carlo is the only way to go for exploring high dimensional probability distributions, where a the cost of using a uniform mesh is exponential in the dimension (integration on the unit cube with a 0.1 mesh costs $\mathcal{O}( 10^{d} )$ for $[0,1]^d$).

*Finally, in statistics - lets say you want to really know if smoking causes cancer. The simplest solution is to take two groups of people. Force one group to smoke and one to not smoke. Let them live (die?) and see which group outlives the other and by how much. This is a simple randomized experiment. But you can't do it in real life, so you have to use smarter statistical methods.
A: An example that I'm familiar with:  Sometimes experimenters want to design an experiment to minimize the maximum response variance over the design space (G-optimal designs).  Early work used genetic algorithms to find such designs.  Later work used meta-models and a more traditional optimization approach to find designs.
Returning to Jaynes' integration example, between Monte Carlo and fixed quadrature/cubature there is quasi-Monte-Carlo techniques.  Some of these are deterministic and some have some scrambling done to make them stochastic. This approach uses low discrepancy sequences that are also space filling to give faster convergence than MC and are about as easy to apply to high dimensional problems as MC.
Speaking to the randomized experiment example in particular, I'll play Devil's advocate.  There are situations where the size of an experiment is limited, knowledge exists about types of effects which cannot be included in the model, and a design is sought to minimize the bias of these effects on the parameter estimates.  This can result in a completely deterministic design.  I don't think I'd call this a victory of thought over randomization as much as a prudent approach to a situation that really just needs a larger experiment.  Yet, it is a situation where thought yields a deterministic approach that is preferable to a random approach.
