These results are all well within each other's margin of error, i.e. with another programmer their order might be reversed.
Since these games were all meant for humans to play them (initially), we could also define some kind of "human-centered complexity", i.e. "how complex are the rules, given their players". There could be rules that are complex in a vacuum, but that mimick some behavioral rule that's already implemented in the (average) human player, and thus may less add complexity than a supposedly simpler (but not human-natural) rule.
(The ordering of two strings A and B by their complexity may yield a different result compared to the ordering of two strings (by complexity) AH and BH. Let A and B be game rules and H be some human component, then the games were devised as AH and BH, not as A and B.)
"with another programmer their order might be reversed"
This seems unlikely to me, having done it, but I would love to see someone else's attempt.
I've seen various contenders for the title of simplest abstract game that's interesting enough that a professional community could reasonably play it full time. While Go probably has the best ratio of interest to complexity, Checkers and Dots and Boxes might be simpler while remaining sufficiently interesting. [1] But is Checkers actually simpler than Go? If so, how much? How would we decide this?
Initially you might approach this by writing out rules. There's an elegant set for Go and I wrote some for Checkers, but English is a very flexible language. Perhaps my rules are underspecified? Perhaps they're overly verbose? It's hard to say.
A more objective test is to write a computer program that implements the rules. It needs to determine whether moves are valid, and identify a winner. The shorter the computer program, the simpler the rules of the game. This only gives you an upper bound on the complexity, because someone could come along and write a shorter one, but in general we expect that shorter programs imply shorter possible programs.
To investigate this, I wrote ones for each of the three games. I wrote them quickly, and they're kind of terse, but they represent the rules as efficiently as I could figure out. The one for Go is based off Tromp's definition of the rules while the other two implement the rules as they are in my head. This probably gives an advantage to Go because those rules had a lot of care go into them, but I'm not sure how much of one.
The programs as written have some excess information, such as comments, vaguely friendly error messages, whitespace, and meaningful variable names. I took a jscompiler-like pass over them to remove as much of this as possible, and making them nearly unreadable in the process. Then I ran them through a lossless compressor, gzip, and computed their sizes:
(The programs are on github. If you have suggestions for simplifying them further, send me a pull request.)
[1] Go is the most interesting of the three, and has stood up to centuries of analysis and play, but Dots and Boxes is surprisingly complex (pdf) and there used to be professional Checkers players. (I'm having a remarkably hard time determining if there are still Checkers professionals.)
I also posted this on my blog.