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Okay, I've done some additional thinking on how to pair people up. To
maximize how many people get their full preferences met, it is important
to pair people up for less popular games before pairing them up for more
popular games. Also, if the tournament is played in multiple rounds, it
will be okay to let people play the same game twice, so long as it is
played for a second time in a subsequent round between two people who each
won the game against other opponents in a prior round. This will help
allow everyone to play only his preferred games. To better enable this
option, it will also help to play the most widely preferred games during
the first round. With these things in mind, here is how I propose handling
this.

First, make a list of how many times each game is included among
someone's top n games. Next, make a table of which games are among the
top n for each pair of players. Make an ordered list of all pairs of
players, using the following criteria:

Give precedence to the pair that prefers fewer games in common.
When two pairs prefer an equal number of games in common, give precedence
to the pair whose commonly preferred games includes the least popular game
among both sets of commonly preferred games. In case of a tie, appeal to
the second least popular game in either set, and so on.

Go through the sorted list from the beginning, pairing each set of
entrants together on the game that is least popular among all the
entrants, for which neither entrant has already been paired up with
someone else. In the event of a tie on this score, pair them up on the
game most preferred by both. If there is no game most preferred by both,
pair them up on the game most preferred by the entrant who has had fewer
of his preferences met so far.

If this process does not pair everyone up with everyone else, additional
pairing may wait until the second round as long as each player has been
paired up on enough games for the first round. On the next round, anyone
who still needed to be paired up for some games could be paired up on
games he and someone else each won in the first round. In some cases, two
entrants who had been paired up for one game could be allowed to switch to
a game both won in a previous round. This could free them up to play the
game they had been paired up for against other opponents. In either case,
this would help all players play only their preferred games.

In the event that people still had to be paired up for the present round,
the previous procedure could be repeated with everyone's top n+1 games.
As needed, it could be repeated again with everyone's n+2 games and
finally with everyone's n+3 games. At n+3, all games would be tied for
overall popularity, and pairing would be based on overall preference
between both players. In each case, someone would get to play one of his
preferred games.