3 players game theory formulation

Question

I really confused about the formal representation of three players game theory formulation. I read several references and find a big difference in these references. I want some help to write the right formulation for pure Nash equilibrium using 3 players game theory. the second question is when I find more than one Nash, how can I select the best solution?

Answer 1

Let's consider the game below with 3 players. Analyzing this game with three players is quite interesting. A three-dimensional array of cells, each with three payoffs, would be the best method to present them. But writing this down on a two-dimensional piece of paper is challenging. Consequently, a game of dimensions n×m×l is often presented as l distinct n×m matrices. In the given scenario, consider the following perspective:

• Player 1 selects a row (upper row A or lower row B).

• Player 2 chooses for a column (left column A or right column B).

• Player 3 makes a choice of matrix (upper matrix A or lower matrix B).

To emphasize the best payoff for Player 3, you need to compare the player-3-payoffs for each of the 4 combinations of choices made by Players 1 and 2 (each corresponding to a cell in the matrices). For instance, in the (A, A) cell, highlight the 70 from the upper matrix, surpassing the 60 in the lower matrix. In the (A, B)-cell, highlight 23, which is greater than 0, and so forth. Ultimately, the (A, A)-cell in the upper matrix (related to the choice of A by Player 3) will feature all three payoffs highlighted. The same holds for the (B, B)-cell in the upper matrix (again tied to the choice of A by Player 3). Consequently, the identified pure Nash Equilibria are (A, A, A) and (B, B, A).

Photo of the game