Heads-up: nowadays, when people talk about neural networks for games, they really mean deep learning combined with reinforcement learning.
Back to your question: When you don't have a log of games, you typically have some other way of assessing performance, e.g. assigning a "score" to the state of the game, which you can quantify and optimize.
For a specific well-known example, I think this paper on training to play Atari games with deep reinforcement learning goes over a lot of the actual math / implementation details.
Recently, I've read tutorial about neural network in a snake game. (you can see it here - https://towardsdatascience.com/today-im-going-to-talk-about-a-small-practical-example-of-using-neural-networks-training-one-to-6b2cbd6efdb3) But, the way it train the neural network it bit funny - in practice, it depends on knowing what the snake should do in every step, and when he didn't, train him. Not very useful(cause, you know, I could just program him to do that).
The other way, I guess, is to use log of the game, assume that the players doing the optimal steps (lets say, exept of the end of there game😉) and then train the network on this situations.
But, when I haven't log of the game, is there a way I can use neural networks for a game? And if there is, how?