Losing Game (also must-capture chess, or suicide chess, or Losing Chess, or Schlagschach in German) is a fairy variant of chess where captures are mandatory and where you win by losing.
From a retrograde analysis viewpoint, the most important element of Losing Game is that, when possible, captures are mandatory. Hence the must-capture alternative name. When several captures are possible, you pick the one you want.
Kings do not play any special role. They can (and must) be captured just like any other unit. No check or checkmate. You can promote into a King!
There are two different variants with respect to castling:
a) Castling is allowed as in orthodox games, i.e. if neither king or rook have moved so far, but there are no restrictions regarding check (e.g. the king may go into check or cross a square under check). This is the default rule.
b) Castling is not allowed at all.
A game is won when you have no unit left, or when you are stalemate. But this is usually not the point of retros in the losing game genre.
The must-capture rule turns out to be a very severe restriction to legal proof games. So that the losing-game genre allows very rich analysis in apparently innocent positions. Here is an example:
feenschach 91, May 1989
1st Prize feenschach 1989
14+15. (Losing Game) First move of wh. Pawn f2?