1. The players and gm put their characters on a grid in whatever position the gm dictates is appropriate for the circumstances.
2. In a battle, each side gets 1 turn at a time. So if there are several players, they must decide amongst themselves what character should go. Players control npcs that are on their side. The gm will determine what player gets to control the npc(s).
3. A character can only move 1 square at a time, or they can use one of their special abilities. If a character wants to overtake a piece on a square (or take pieces using a ranged attack), a rock-paper-scissor exchange occurs. The more powerful a character is in relation to his opponent, the more do-overs he gets should he fail an exchange. Attacking from the flank gets 1 do-over bonus. Attacking from behind gets 2.
4. If you get your piece taken, then you're out of the fight until your side wins or a player restores you. If all players are taken, the gm wins. Or you could just run away I suppose.
5. Therefore, there is very little to keep track of (no initiative, no hp, no math, etc.)
6. Some weapon types are superior to other types. If you have a superior weapon type, then you win on even exchanges (e.g. rock-rock; scissor-scissor).
A new diceless mechanic
Running this...
I would very much like to run a game that employs this system or a very similar one. I like how combat would become an even more communal effort for the players, tactics would be highly important and players would have to work together well.
Have you played in a White Wolf LARP? They use a similair system (RPS) as well as an attribute bid system, where a character has (say) 3 mental attributes that might aid him in noticing the out of the ordinary, he can overbid on someone else making (say) a stealth check against him to try and stab him in the back, forcing the person to do-over their RPS test.
One thing I would change is that no character PC or NPC could go on two consecutive turns, this would benefit in that it would force more tactical choices, would stop one character standing there with a gun/bow/cannon and just dropping opponents from afar and would stop on Player getting all the action.
Otherwise I really like this idea. Good work.
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Fleshing out the mechanics
I've been working on the mechanics for combat, and I noticed something that you pointed out: a single player acting on consecutive turns really unbalances things. Thanks for your insight.
Another thing I've noticed: overtaking a square may work for chess, but not for skirmish-type combat; it may force you into a pincer. If a character wanted to defend a narrow area against multiple opponents, he wouldn't be able to take advantage of the bottleneck effect.

This could easily be converted into dice.
When you try to take a square, equal ability gets (percentile dice) 50% chance of success, +10% for every increment greater, -10% for every increment lower. +10% for flank, +20% for behind. +10% for every weapon rank greater, -10% for every armor rank. The maximum success rate is 95%.
Personally, it sounds much easier and more fun with hand gestures!