# Do there exist matrix games with more than two players? Can we extend existent models that have already used in signaling games?

In game theory, the simplest way to represent a two player games is with the help of matrix games. For example, assume that we want to model a market of traders, where one is a speculator and the rest of them traders in the market are short term or long term investors and small traders (in the sense of budget constraints and hence low market impact).

1. Do there exist matrix games for more than two players that could model such a situation like $$N$$-player matrix games?

2. Could we use some specific model from signaling games like the Spence job market signaling game, extending the the types of players for more than two for this case?

The term "matrix game" is not a standard term of game theory, and where it is actually used, it is often used in different ways. Originally, it refered to finite 2-player zero-sum games. Later is was sometimes used for the representation of the normal form of a finite 2-player game, usually more specifically called a bimatrix game, by a payoff matrix with 2 entries in each cell (the payoffs for player 1 and 2, respectively). In some texts, the term matrix game is reserved for a symmetric bimatrix game, since for these it is enough to specify a single payoff in each cell of the payoff matrix. Some allow a countably infinite number of strategies and call this an infinite (bi)matrix game.

All these usages are, however, restricted to 2-player games. Finite 3-player games are sometimes represented by a pair of trimatrices (matrices with three payoff entries in each cell). E.g., a three-player game with two strategies per player is sometimes called a 2x2x2 trimatrix game. As you can see, the terminology is just not standardized. That said, since a matrix is a finite, rectangular and thereby 2-dimensional array of elements, the average game theorist will most likely associate some finite 2-player game with the term matrix game. In this sense, the answer to your first question is NO.

I don't really understand your second question. I'd say it's always possible to "extend" a 2-player model to a more-than-2-player model (or a 2-types model to a more-than-2-types model) in some way, but this is quite obvious and most likely not what you meant to ask.