I assume that in a 3P game, there are three "matchups" taking place: A vs. B, B vs. C, and A vs. C. Each "matchup" is given a result based on final placement in the game, so if it goes:
first place: A
second place: B
third place: C
...then A wins both of her matchups and B wins his matchup against C.
Let's say C resigns. There aren't really ways to try and evaluate the A vs. B matchup. Each one will be flawed -- this includes just letting the game go as a 2P game, or subbing in a bot for the resigned player. These are all good options for fun, but in terms of a result on the leaderboard that has any integrity, that just can't happen unless C finishes out the game.
The only result I can think of that would be fair is to have C lose both of his matchups, and have the A vs. B matchup not exist -- not a tie, but have it not even count for anything.