There's a 5 year age gap between me and my brother, so even though he didn't win all the time (we played games based on luck quite a lot), he did win a lot. Combine that with being generally very spoiled and he is a very bad loser. I remember when I was in my teens (last time I actually played a game with him, before he nicked them all, even the ones that were mine (see spoiled)) and we had my Grandpa visiting, so obviously we were supposed to be on our best behaviour. We were playing a board game a bit like Ludo, but slower. Anyway, we play, I win, my brother (who was 9 or so) throws the entire board up in the air and scatters pieces everywhere. No more board game. He might actually have come second, especially as I wasn't trying hugely hard to win and my Grandpa hasn't played before the game before, but no, he ruins it instead.
My parents didn't even make him tidy it up. I had to do that (which is probably why I remember it - the little sod storms off and I get left to clear up the mess).
Yes that was 20 or so years ago, but my point is, he hasn't changed, and he won't. I know the miserable times of making your four year old follow the rules might seem rubbish now OP (I think you've had some great suggestions on how to minimise this) but please put up with it so your 4yo doesn't turn into my brother! (My brother used to cheat too, which made it incredibly rubbish for me. Consequently, one of my favourite games was one where you mixed up the head, body and legs of different characters to see what silly options you could come up with. There is no way of winning or losing, you just see what each other find funny.)