I calculated the piece values in terms of Cavaliers and not Pawns, because there are no Pawns in the game. In Infantry and Cavalry I, a variation which replaces Cavaliers with Pawns, Queens are worth about 8 Pawns, Marshalls about 7 Pawns, Paladins about 6 Pawns, and Nightriders about 4. As a check on the accuracy of this method, I also used it to calculate values for pieces in Chess. I got values of approximately 8 Pawns for a Queen, 5 for a Rook, and 3 for the Bishop and Knight, with the Bishop being worth almost half a Pawn more than a Knight. This matches the traditional values in Chess with enough accuracy to trust the values I got for Cavalier Chess.
In Chess, pieces besides the Pawns and Kings are distinguished as minor pieces and major pieces. Rooks and Queens are major pieces. Bishops and Knights are minor pieces. A major piece, with help only from the King, can checkmate an opponent's lone King. A minor piece cannot. The same distinction holds among pieces in Cavalier Chess. A piece which can checkmate an opponent's long Knight King with help only from the allied Knight King is a major piece, and a piece which cannot is a minor piece. Queens and Marshalls are major pieces, and Paladins and Nightriders are minor pieces. Furthermore, in Chess two Bishops equal a major piece, but two Knights do not. The same holds for their counterparts in Cavalier Chess. Two Paladins can checkmate a lone Knight King, but two Nightriders need more help.
I have introduced a couple fairy pieces for use in variations. These are the Lighthorse and the Lightrider. Each can move like a Queen without capturing. The Lighthorse can also move and capture like a Knight, and the Lightrider can also move and capture like a Nightrider. Zillions thinks each of these are worth more than a Queen. This seemed wrong to me, and I did experiements with these pieces, using them in games that I had Zillions play against itself. My conclusions are that a Lightrider is worth about the same as a Paladin, and a Lighthorse is worth about the same as a Nightrider. Both are minor pieces, because neither can checkmate a lone Knight King with only the allied Knight King as backup. Two Lightriders can checkmate the lone Knight King, but it seems that two Lighthorses cannot.
Although Zillions gets the values for these two pieces wrong, I don't think this provides sufficient grounds for distrusting the values I got for the other pieces. These two pieces are distinguished by moving and capturing differently. Each has great freedom of mobility and limited capturing powers. I believe that Zillions overvalued their mobility and so overvalued the pieces. The other pieces all move and capture in the same way, which makes their mobility equal with their capturing power. Thus, even if Zillions places too much value on the mobility of a piece, this does not affect how it would rank the relative values of pieces who all move and capture in the same way. Therefore, the values given for the other pieces should still be regarded as trustworthy.
You may want to avoid stripping your opponent down to a lone Knight King. Doing this could increase the odds of a stalemate. If your opponent has one other piece remaining, you can trap his Knight King with yours and check with any other piece, making checkmate. If your opponent has only a Knight King left, you will need at least a major piece or a Paladin with another minor piece to make checkmate. Without that much, you may as well call a draw.
Each method will work in some situations and not others. The corral and check method works so long as you and your opponent each have at least two pieces. You will need one piece for corralling and one for checking. And you will need your opponent to have another piece left on the board, so that your corral doesn't lead straight to stalemate. The advantage to this method is that it doesn't matter what your second piece is. All it has to do is check a corralled Knight King, and any piece can do that.
The partial corral and close method works so long as there are at least three pieces between you and your opponent, but if there are only three pieces and none are major pieces, it will not work. A partial corral limits a Knight King's movement without causing stalemate, because it leaves him some room for movement. This is why a partial corral can work when your opponent is down to a bare Knight King. You can create a partial corral with multiple pieces or with a major piece backed up by the Knight King.
The suprise trap method has the advantage of being quick and unexpected, and with multiple pieces on the board, it may sometimes be easier to pull off. The disadvantage to this method is that it requires the presence of multiple pieces. In an end game with few pieces left, the other methods may be the only options you have available.
When there are many pieces on the board, there is more to checkmate than preventing escape and checking. With other pieces on the board, a check can be broken by a capture or a block. The methods I've described remain components of checkmating with other pieces around, but it also becomes important to prevent blocks and captures.
One method for doing this is the double check. With a double check, you reveal a check and also make another check. With two pieces checking the King, capturing one checking piece will not end the check, nor will blocking one and not the other. This method normally leaves your opponent with no option but flight, and if you have successfully cut off his King's freedom of movement, it is checkmate.
A good method for preventing your opponent from blocking your check is to check with a piece's ability to move as a Knight. Knights can jump over pieces and cannot be blocked. This leaves flight and capture as your opponent's only options for ending the check. Since the King has a Knight move, it can capture the checking piece if it is left unprotected. So this method will work only if you cover the square you're moving to with another piece. If you check from a square your opponent can't move anything to, his only option will be flight, and if you have successfully cut off his King's freedom of movement, it is checkmate.