My apology if either this question has been asked elsewhere or it is well known (but not to a beginner in game theory like me).
Firstly, I haven't seen much work/literature on multiple player games comparing to 2-player games. Could I know what are reasons behind this?
Secondly, I'll greatly appreciate if someone could point me out a primer of this topic as well. I've just moved to this area. It would be nice to know a state of the art in this field since I'm looking for a solver of multiplayer zero-sum games perferably with some guarantees on the solution quality.
Many thanks!
Multi-player zero sum games are as hard as two-player non-zero games (i.e. hard) since one can trivially create a three-player zero-sum game from a two-player non-zero sum game by adding a dummy player with payoff equal to the negative of the sum of the other two players payoffs (for each strategy profile).
As such, multi-player zero-sum games are really not an independently studied interesting class of games.
If you actually have a zero-sum polymatrix game, then you can solve the game efficiently. See:
Cai Y, Candogan O, Daskalakis C, Papadimitriou C. Zero-sum polymatrix games: Zero-sum polymatrix games: A generalization of minmax. Mathematics of Operations Research 41.2 (2016): 648-655.