Who are you people and what am I doing here?
It seems to me that every game has roles to a certain extent, although I'm not so sure I can really count the first one as a "role." There's a strange entanglement too within theory & practice. Equal games with equal starting roles will usually end up with players adopting certain play styles based on the natural circumstances, so they are no longer equal. Set role games have the possibility of mirror matches, at which point there's no difference in capabilities. Set games will often also have a very blurry line with customizable games as slight customization options are available (choosing an Ultra in Super Street Fighter 4). Customizable games, in practice, often end up with very few viable options and therefore will have even less options possible than many set role games. Read past the break to figure out what the hell I'm talking about.
Equal: In these games, both players have identical capabilities at the start of the game. Hearts, Backgammon, and Monopoly. Chess almost gets there, but the differing placement of kinds & queens relative to other pieces within each side's army makes them technically different. I believe Shogi would qualify though, because as I recall, the board is identical on both sides. For the sake of not being a smart-ass, I'll consider turn order as being irrelevant when considering whether a game could be equal roles (going second does naturally lead to playing slightly more reactively).
Set: Fighters almost universally embrace this. You pick a character (role), every one has varying degrees of advantages & disadvantages. For the sake of clarity, Street Fighter 2 matches this while Super Street Fighter 2 Turbo does not with its ability to choose between new & old versions of the characters. This is effectively customizing the chosen character because the two characters are almost completely identical, though I'll admit part of it is my personal perception. To get to my first example, Chess fits in here as a "set role" game. Of course, if players select a mirror match in a fighter, then that effectively turns at least that match into an equal role because both sides have no difference in capabilities.
Custom: Everyone simply has a system of how they are allowed to customize their roles. Dungeons & Dragons fits this as does any collectible game like Magic: the Gathering (your deck would be your role). Super Street Fighter 2 Turbo would fit here, although it's probably the lowest possible example. Capcom Vs. SNK 2 would be a more obvious example with customizing teams, character strength within teams, gauges used, and custom gauges if people are willing to make them usable (I don't know if tournaments allow them, probably not).
Having looked over these 3 levels, I think I prefer set roles first, then equal roles. It's nice to be able to customize, but too much customization potential easily makes for a small number of really powerful things without much variety. With set role games, I can at least have decent faith that the designers put in enough playtesting that no option dominates the game overall. Of course, less playtesting is needed for an equal role game because the focus is entirely on the system, not the roles.