“Character skill” types would probably use some sort of system to resolve the game via dice rolls, probably modified by things like intelligence scores, chess skill levels, and the like. High roll wins. Match complete, the session moves on. This takes character ability scores and skills into account, adds a little randomness, and resolves it mechanically by what is essentially a “chess check.”
“Player skill” types might break out a real chess board and play the match out. This would be the way to do it for many reasons, particularly if it’s a “Chessmen of Mars”-type scenario or a puzzle test like the game of Wizard’s Chess in Harry Potter. Simply rolling a d20 and applying modifiers to get the result would be, to me at least, quite unsatisfying. Playing it out takes the player’s ability into account and makes a game of it.
The issue becomes, if you use the “player skill” method, how do you account for a PC with a high INT or WIS, particularly if the player isn’t a good chess player? What if a nine-year-old kid is playing a PC cleric with a 16 intelligence, a 15 wisdom, and a chess skill of +4? [UPDATE:] Or a player who is champion of his school’s chess club playing a half-orc with INT of 7? Is he obligated to ‘think dumb’ if the adventure is on the line? How does/should the DM enforce that?
I chose chess instead of something like poker because chess eliminates the element of chance; both sides are starting the match equal if you determine white randomly.
Obviously, the reasoning used for the chess example could be applicable to NPC encounters, searching for traps and secret doors, and many other things where thinking ability takes precedent over the physical element.
Note: While writing this up I changed an example but I changed it so it was a completely different point. So I changed it back.