If the air is saturated, and you have water condensing (as you might with a tire pumped up with air from a hand pump) the compressibility will not follow the idea gas law very well because an increase in pressure results in water condensing. This is highly nonlinear.
For those that are really interested, there's an equation of state developed for most air. See here:
https://www.nist.gov/system/files/do.../CIPM-2007.pdf
An ideal gas is represented as P V = n R T. That is pressure times volume = amount of gas times temperature times the ideal gas constant "R". This "law" is a good approximation at low pressures (say 2 atmospherers or below) and high enough temperature (let's say above freezing). At higher pressures and/or lower temperatures a "fudge factor" called compressibility is added. Denoted "z", so we have
P V = z n R T
Obviously, if z = 1, you have the ideal gas loaw. Just to keep things reasonable, at the pressures and temperatures we see in road bike tires I'd be very surprised if z was more than 0.02 less than 1. So the non-idealities are really, really small. That is to say, it's kind of ridiculous to put all this effort into modulating z with expensive dry N2 or Ar. IMHO. If you want to figure out the compressibility of dry N2 or Ar vs moist air, the article above gives you the tools. Do the math!
In summary, if you are using air from a compressor (where some of the water condenses out and is drained) you are using reasonable dry air and things are the compressibility is pretty linear. Don't think you need dry nitrogen.
If you lived in the tropics were you have high absolute humidity and you currently can only inflate with a manual pump, then dry nitrogen (or getting a compressor) might make sense.
I see no advantage and a couple of disadvantages (high cost, higher weight by a couple of grams, and waste of a limited natural resource) in argon. Using argon in bike wheels is just plan silly.