Last night I won a game of four player chess by accident. I know that any of the other players could have crushed me in a two-player chess game, but the four-player version is a different game in interesting ways. For example, it is rarely a good idea to trade pieces, even if you can take a more valuable piece, because that means that your two other opponents are both in a better position.
The game is fascinating. Although the pieces move the same way, the strategic dynamics are very different. It is not possible to play 4-player chess the same way you play 2-player chess. If you try to think through future moves, you will be exhausted from the mental effort and your work will be for naught, because the board can change dramatically based on the actions of your opponents moving against each other. One player was paralyzed with indecision, not wanting to move anything, before giving up and turning his side over to someone else.
You have to play 4-player chess like you play Go. Instead of trying to be a chess computer, go with intuitions of life, breathing, influence, and position. From the start I did not try to think through things; I made bold moves to shake up the game and cause interesting things to happen. Very early on in the game I 'opened Pandora's Box' by moving a piece where it could have been costlessly captured by another player, pointing out that he had no incentive to capture the piece because it was threatening his opponents more than him.
Later, there were many times in the game where I had no obvious move, so I just shifted my pieces so that they had more options, threatened more squares, or were defended by more pieces. Someone else was asking me a question about fitting Vibram Fivefingers, so I was not paying much attention, and for a couple turns I was randomly putting pressure on the leading player by putting him in check. Then he congratulated me for a checkmate and game win. When I asked how that happened, he told me that I could move my queen right next to his king, where it was covered by my knight, and I had a bishop blocking the escape routes.
When you take someone's king, you control all the pieces left on that side. The player I could checkmate had already done this to another player. It had been a very swift reversal; it had appeared that he was aiming for one player but then he saw an opportunity to checkmate another player instead.
I suspect that the game could degenerate very quickly if you played to win rather than have fun. The optimal strategy is probably to 'turtle up' and build a fortress. Once the game is shaken up and moving quickly, it is fun, but you need a kind of cooperative 'play for the lulz' attitude to make that happen. It may not be a coincidence that I had been playing to have fun and not win, and ended up both winning and making the game fun.