Presumably you general intuition is that
- enjoyment and willingness to play a game again have a lot to do with each other,
- that some people perhaps just generally enjoy games and/or are generally more willing/want to play a game a few more times, and
- that maybe one of the games is just better/more popular/more attractive for playing again?
If so, a logical model set-up would be to hypothesize
- underlying latent two-dimensional (correlated) multivariate normal random effect (for generally liking games and for wanting to play them again) for each person (you could also hypothesize that this is just a single latent quantity to simplify the model - although in the manner described below it should be easy to fit this), and
- an underlying latent property of the game, which if you just have two games may have to be a fixed effect (while with many games this could also be a random effect - unless you go Bayesian and have some reasonable degree of prior information e.g. based on how much online rating for different games vary or something like that).
So, if your data consists of Yes/No answers, then a model could be a logistic regression with
- Main effect for the game (A vs. B)
- Main effect for the question (How good was the game? vs. Do you want to play it gain?)
- Random intercept effect that varies across person (can be centered on zero, because the main effect already models the mean for the random effect)
- Random question effect that varies across person (can be centered on zero, because the main effect already models the mean for the random effect, may want to code this as -0.5 for question 1 and +0.5 for question 2)
- Game main effect
- Game by question interaction
What this is not yet capturing is that different people might like different games. If that is something you want to look into, then e.g. adding a random game effect that varies across person would be an option
If the rating scale is ordinal (e.g. "Bad", "Indifferent", "Good"), multinomial models could be an option. Correspondingly, a linear mixed effects model, if it's a continuous rating scale.