Now we may be in a semantics discussion. Is "the choice is it matches the constant A or does not." one choices or two? I clearly see two choices: (1) X is exactly A, and (2) X is anything but A. One problem with these choices is that at least one of them demands an exact answer: "X is exactly equal to A".
In real world listening tests the listener is often unprepared, even after considerable listening, to answer questions that demand an exact answer. Instead ABX asks the question: "Is X more like A or B?". This is a question with an inexact or approximate answer. In 30 years we have found that it is a far easier question to answer than "Is X exactly equal to A?"
Really?
I'm thinking that your lack of real world experience with ABX is tripping you up big time.
First, lets look at how order bias can exist in an ABX test.
If I present two known references to the listener, it would be logical that one might be A and the other might be B. I would then present X and ask the question: Which does X sound more like, A or B? If I always present A first and B second, then there is a bias since X is always presented right after B.
But that is not how an ABX test as they are commonly done. or how they have ever been done. Very early on we told people to try selecting ABX, and then BAX, and mix that up all the time. We also told them to try AB and BA to refresh their minds about what the audible difference actually was. Add to that the fact that X is either A or B, but is itself randomly selected. So in the case that someone disregards our instructions and always does ABX, he in in fact doing a random selection of ABB and ABA.
So given all that and in view of the fact that we give people an ABX hand held controller and tell them to "have it it", where is the experimental bias?
BTW, here is a picture of some ABX hardware:
http://home.provide.net/~djcarlst/abx_hdwr.htm
And some inline pictures of the key components, the control module and the hand held controller:
While these are artist's renderings, about 60 of each were built and distributed. I believe that the Meyer Moran JAES article was based on Meyer's set of this hardware as well as some other hardware from the web page I linked. He added some other hardware that he developed - namely a large display that can be easily seen from a goodly distance. Meyer has a pretty big listening room, I was in it years back.