Im reading a textbook on game theory and the last example in the book is a three player game, where the first and second players choose left/right and the third player chooses up/down.
Similar to the example in this answer: Does there exist MATRIX games for more than two players
My question is that does there exist well-known matrix games with more than 3 players?
I mean it is not hard to generalize to three players, but I wonder if there is a game that makes sense in some contexts.
Note that the interaction must involve more than 3 players simultaneously, rather than having them in a graph, as in polymatrix game, where interaction is between groups of two players.
Fundamentally, a game matrix is exists for a 2-player game because the result of the game is determined by two parameters, namely, the strategies chosen by each player. When there are more than two players, the lookup table becomes more than 2-dimensional, because the outcome depends on more than two strategies. It is probably more conventional to refer to a higher-than-2D table as a game tensor or hypermatrix, since matrices are conventionally 2D.
However, the core principle motivating game matrices in the 2-player case is just that the outcome of the game is a function of the tuple of strategies chosen by all players. This is still true with more players, even if there is no longer a good 2D visualization of that function.