For 2-player chess Zermelo's theorem states that there is either a winning strategy for one of the players or a way to force a draw for both players. However, this result does not extend to four-player chess. It is possible that no player can force a win and no player can force a draw in this game.
Let's modify the game by introducing non-determinism. Before every move, a player rolls a dice and skips their turn with some constant probability $p$. Placing your king under check is allowed because there is always a chance it will not be captured successfully, and winning requires that kings of all other players are captured. This way there is always a chance for any player to win at any point in game and therefore an incentive to choose the move maximizing their chance to win.
The question is, is there an optimal strategy for every player in this game, maximizing their chance to win under assumption that every other player is trying to maximize their chance to win?
There is a related question asking for an optimal strategy for deterministic N-player chess at Is perfect play always possible in a game with perfect information and more than 2 players?
There is also a related algorithmic question about extending MiniMax algorithm to N-player games: https://stackoverflow.com/questions/14826451/extending-minimax-algorithm-for-multiple-opponents However, it is not clear what happens if some players face choices which are equal for them but affect other players differently.
For background, more practical question I am interested in is whether it makes sense to search for optimal AI in tactical games like The Battle for Wesnoth assuming fog of war is disabled so it is a game with perfect information. Stochastic chess described above seems to capture the essence of such tactical games.
There are similar ideas in game theory introducing forced non-determinism into player strategy like Trembling hand perfect equilibrium, but there are multiple ways to do it and I am rather interested in existence of optimal strategy for a game that has already non-deterministic rules.