@jan This sounds like it could get complicated. However, it doesn't need to be. If everyone is bringing 10 games each, I don't think there needs to be a vote on that at all. People just bring the games that they want to. Essentially everyone gets 10 nominations into the pool that you vote on.

It might be the case that one person's favourite game is owned by someone else, but they can easily ask to borrow it for their own 10.

As for what to play, if you are playing a lot of games, I would prefer some sort of proportional system rather than a simple single-winner type vote each time. E.g. If 3 out of 5 players generally prefer one type of game and 2 prefer another type, it makes sense for there to be a 3:2 split rather than the 3 choosing every time.

You could simply take it in turns to choose. Alternatively I would recommend a lottery method, so that you would get proportional representation over the long haul without having to keep track of what has already happened. I think simple random ballot, where you randomly pick who chooses each game, would be inferior to taking turns. Some people would get to choose more, just by luck.

But one alternative, which has a balance between randomness and consensus is COWPEA Lottery. Each person approves as many games as they want. Then you pick a ballot at random and if that ballot lists more than one game, you just pick other ballots at random as a tie-break. So if the next ballot approves all or none of the remaining games in contention, you ignore it. Otherwise you eliminate games not approved. Continue until one game is left. If there is still a tie after all ballots have been picked, just pick at random among the remaining games.