The various definitions of "arbitrary" in the non-RPG context revolve around who or what has an influence on how a choice is made. Many times, it means something is determined by chance or whim, or without any plan, in contrast to choices made based on rational decisions or in accordance with some system or need. A lot depends on what people are expecting as a selection criteria, which is why personal preferences can sometimes be arbitrary and sometimes not at all.
It's all about the perceived plan. Are you following the plan? Then you aren't making arbitrary decisions. Are you ignoring the plan? Then you are being arbitrary.
In RPGs, some people object to GMs who ignore challenge ratings or who let their personal feelings about players affect the decisions they make about characters. The infamous "Rocks fall, everyone dies", for example. If the players feel like the event happened because the GM was having a bad day, or just to exert some petty dominance, or because the players deviated from "the story", then they will say "That's so arbitrary!"
And so, people have kind of redefined "arbitrary" to mean "evil, spiteful GMing". But that wasn't what I was talking about in reference to sandbox GMing. The defining criteria of a sandbox is that it does not follow any GM plans or preference: no predefined story, no momentary anger, no punishment of players for heading into uncharted territory, no spur-of-the-moment encounters to strip characters of benefits or to show off something fancy. The GM is supposed to set up the world and any random tables in advance, but once play begins, the players or the dice or some other arbitrary method of selection have control over what happens next, NOT the GM.
This is why I made such a big deal about meaningful geographical freedom. If the players can go north, south, east, or west, but will encounter the same thing no matter what (because the GM just moved the "next" encounter to the appropriate location,) then the players had no real choice at all: their freedom was an illusion.
If, on the other hand, the GM has a list of encounters arranged in a random table, or if there's an unordered list and the GM uses a table like this:
- Players go North: Use encounter on top of list, cross it off.
- Players go South: Use encounter on bottom of list, cross it off.
- Players go East or West: Use encounter in middle of list, cross it off.
- Players go Northeast or Northwest: Use encounter above the one in the middle of the list, cross it off.
- Players go Southeast or Southwest: Use encounter below the one in the middle of the list, cross it off.
... Then that removes petty GM preferences from the equation and makes what happens next arbitrary.
Which is good.