Crazyhouse is a chess variant where pieces you capture can be dropped back on the board - as your pieces! Many would-be checkmates are well defended in Crazyhouse, and sacrifices that do not succeed usually lead to a huge disadvantage.
In Crazyhouse, the element of material is more important than anything else. Why is this? Because you can literally change the position, fighting against positional advantages. Also note that the element of space is less important, because captured pieces come back to occupy more squares.
This means that in Crazyhouse we only need to focus on one thing - gaining material, and using it to setup a quick mate. This is one of the best winning strategies.
First of all, we have to reassign all the pieces new values - why? Because you'll see these values are important, not just for trading, but reflect how well you are doing and tell you what extra force you have - value is based on mobility.
In a normal chess game, the pawn is worth about one point. This means it is worth 100 centipawn. In Crazyhouse, you can still consider it 1 point or 100 centipawn, as it is what other piece's values are based on, but I see that sometimes you can trade a bishop for a pawn. This is because pawns can lie deep in your territory and give checks, even build super powerful pawn chains that are worth a bishop to destroy. When placed where it can safely promote or strengthen the position greatly, the pawn is generally worth a bit more than 1 point, sometimes increasing to two points (100-200 centipawn).
What about the bishop? Because the position is often quite cramped (due to pieces always being dropped back), the bishop is far less mobile. This reduces its value to a bit less than its original value - 2 or 3 points. In very locked up positions a bad bishop can be traded for a pawn. The bad bishop is worth approximately the same as the pawn because the closed and locked up position will likely remain that way.
The knights shine out in Crazyhouse. They can jump over pieces and can be dropped near the center, making them some of the strongest pieces. They are always very mobile and you should NEVER trade them for bad bishops - the knight's value rises to an estimation of 4 or 5 points, being almost as powerful as a rook in standard chess.
Rooks, on the other hand, are weak as an open file can be closed. They are slightly less valuable then knights in certain positions, but in the endgame when dropped are highly valued. Thus I estimate them to be of equal value to the knight, 3-4 points when not developed or in closed positions and 4-5 points when dropped on a clear rank or open file.
Now we come to the queens. A queen should NEVER be developed too early because of the threat of knights. If your opponent has 3 or 4 knights, chances are you'll have to defend against the threat of strong forks in a position where there are little safe or unoccupied squares.
The queen is slightly less mobile but can still maneuver. If it starts acting like a bad bishop or gets locked in on a wing, expect its value to greatly decrease. It may be worth only 7 or 8 points then, because it still has potential.
The values we just looked over are based on the MOBILITY and STRENGTH of a piece in most Crazyhouse positions. However, we actually have different values for trading, because A CAPTURED PIECE CAN BE DROPPED BACK ON ANY UNOCCUPIED SQUARE, with the exceptions of pawns (no dropping on 1st or 8th ranks).
For example, a queen that is dominated by 3 knights has a mobility value of a bad bishop or locked pawn, thus being pretty much worthless. So can we trade the queen for the knights? Certainly not! The queen can be USED by your opponent on a better square, so you MUST watch out.
With trading, the values of pawns is matched with the mobility of the strongest position for the dropped pawn - usually around 2 points, 1 point in some positions.
Bishops can be dropped in certain positions that are good but are usually blocked in anyway, so its value is still dropped to 2 or 3 points (more than 3 points in some positions!).
Knights are far more mobile because they don't have to move to squares - you can drop them there if you've captured them. This means that its value is usually 4-5 points. Where there is a tactical opportunity its mobility remains the same.
With rooks, they are often dropped near the enemy king if there are safe squares, so they usually remain at their 5 points. Threatening mate makes them more valuable and they should not be traded if they support a mate.
Queens are worth about the same amount as them in chess, because even though they are less mobile they can be dropped on certain squares that create huge threats, perhaps winning material (the most important element in Crazyhouse!).
By playing many games of Crazyhouse, I have decided that the exact value of a piece really depends on the position. But in general, I think their value is near my estimates.
Have you played a game where you think their value is different? And what do you think of these values for mobility and trading? Do you think the knight is more valuable, or the rook less valuable?