Interesting Chess Math/Simulation Problem

[CRoberson]

In the movie "Queen of Katwe", there was a kid that insisted that her queen can not be captured and wouldn't let the other kids capture it.
Of course, this gives a great advantage. But, how much of an advantage?
Is it the same as a queen odds handicap or more or less?
You start with both having queens, but one side having a queen that can't be captured. It could sit on h6 without concern for the g7 bishop or pawn.

[lkaufman]

In the movie "Queen of Katwe", there was a kid that insisted that her queen can not be captured and wouldn't let the other kids capture it.
Of course, this gives a great advantage. But, how much of an advantage?
Is it the same as a queen odds handicap or more or less?
You start with both having queens, but one side having a queen that can't be captured. It could sit on h6 without concern for the g7 bishop or pawn.

You haven't thought this through. The queen would just capture every enemy piece, being immune to capture. It would be ridiculous, far more than a queen handicap. Even an invulnerable knight would be overwhelming. Now if you said the queen cannot be captured except immediately after it moved, that wouldn't be so ridiculous.

[tmokonen]

How about a queen (or other piece) that can check and give mate, but cannot capture, or be captured? This sort of queen would be deadly in an endgame, but not as powerful in the opening, where every piece would be a barrier. Maybe still too powerful.

[hgm]

The usual terminology for this is 'iron pieces'. An Iron Queen would indeed (trivially) decide the game in just a few moves, (e.g. e4 - Qh5 - Qxf7) by driving the enemy King to the edge with contact checks until it is mated, with nothing being able to stop it. Even if both had an Iron Queen it would simply mean that white wins, as Iron Queens can still not capture each other, and it is impossible for a single obstacle to prevent a Queen from reaching a given target.

Note that some Shogi variants have (in comparison very limited) rules to protect the strongest piece from capture: In Chu Shogi a Lion cannot be captured by another Lion if it is protected (i.e. if recapture would be possible). But other pieces can still capture it in this situation, so it isn't much like an Iron Lion; it is more to prevent Lion trading. (Chu Shogi would be a pretty boring game after the Lions are gone...)

In Chess you could adopt a similar rule w.r.t. Queens: a Queen is not allowed to capture a Queen when (pseudo-legal) recapture is possible, and other pieces cannot capture a Queen in a move following the capture of their own Queen by a non-Queen. (The latter to prevent indirect trading.)

Another type of anti-trading mechanism is 'contageon': any piece capturing a Queen becomes a Queen itself. This doesn't forbid you to trade Queens, but strongly discourages it (as after the recapture the opponent would have his Queen back, but you would have lost yours).

[Raphexon]

It would likely be very/extremely trivial to prove the side with the iron queen can force a win.