Multiple player's scenarios in game theory

Question

I am learning game theory in recent lectures and I am curious about the multiple player's scenarios. What if we have three players and two possible strategies. How would the payoff matrix look like? Could you give me an example to interpret the scenario?

Answer 1

In general most $m$ player games are set up by describing the strategy space for each player and their respective payoff functions. Normally only simple normal form games are represented in matrix form, but we can represent a three player normal form game in matrix form. For example we have three players with $n_1$, $n_2$, $n_3$ strategies each. We would represent this game using $n_3$ different $n_1 \times n_2$ matrices. The payoffs when player 1 plays their second strategy, player 2 plays their first strategy, and player 3 plays their first strategy would be in the second row and first column of the first matrix.

For example here is three players. Player one, two and three each have two strategies, $(A, B)$, $(L, R)$, and $(H, T)$ respectivly.

\begin{array}{|c|c|c|} \hline H & L & R \\ \hline A & (1, 0, 1) & (1, 1, 1) \\ \hline B & (0, 1, 1) & (1, 1, 3) \\ \hline \end{array}

\begin{array}{|c|c|c|} \hline T & L & R \\ \hline A & (1, 0, 1) & (1, 2, 0) \\ \hline B & (0, 0, 1) & (1, 0, 0) \\ \hline \end{array}