Randomize or not the order of items? (It is a question about an optimal design, rather than a statistical question. So if it's not allowed by the rules, I am sorry in advance: please don't downvote me, but just let me know and I'll delete the question)
So simplifying things as much as possible:
I want to measure whether information about crime rate affects the preferences towards a region. To do this in between-subject design I let participants choose between Region 1, and Region 2, providing them several pieces of information. In a control treatment these pieces do not include crime rate. In a T_A, these pieces do include crime rate. Other pieces of information are the same in both treatments.
So schematically it looks like that:

So, the hypothesis is that in T_A the share of those who prefer R1 to R2 will be higher than in the baseline.
The question is:

*

*should I randomize the order of Regions between subjects? (so sometimes Region 1 goes first, sometimes Region 2 goes first).

*should I randomize the order of pieces of information between subjects? (so sometimes the information is shown in order Population->Crime rate->Average wage, and sometimes Crime rate->Average wage->Population etc.)

*If I do both or one of randomizations, does it substantially increase the sample size?


Anchoring may make your participants decide differently on the second item they see because of what they saw in the first item. The simplest way to reduce this effect (more precisely, to balance it across participants) is indeed to randomize the order.
Also: it's much easier to randomize before you start the experiment... than to rebut a reviewer who complains about the lack of randomization because of a possible anchoring effect (or other bias).
So: yes, do randomize. Especially the order of regions. Possibly also the order of information shown, but that may confuse your participants, so the benefit is less clear-cut here.
As user2974951 points out, there are no sample size implications. Or if there are, they would go the other way: if you do not randomize but account for possible sequence or anchoring effects in your model, you will likely need a larger sample size than if you simply randomize.
