There are two main ways in which games can be represented in game theory. One is the normal form (also called strategic form) and the other is the extensive form.
This post will explain the normal form.
In a normal form representation the game is shown by grid or matrix. For a simple game of chicken it might look like this:
If both players choose to compromise then we read across from player 1’s choice of compromise and down from player 2’s choice of compromise and find the square in the grid where they intersect. Here it contains (0,0).
The two numbers in the square represent the outcomes for the two players. Player 1’s outcome is the first number in the pair, and player 2’s is the second number. In this case both players have compromised and got an outcome of zero. They didn’t gain but they didn’t have a car crash either.
If player 1 compromises and player 2 doesn’t then we get to the square with (-1,1) in it, at the top right. That means that player 1 is worse of, with an outcome of -1, because he chose to compromise when player 2 didn’t. Player 2 does better and gets a result of 1 (the second number in the pair)
Because the game is symmetrical we get the opposite result if player 2 compromises and player 1 doesn’t. This takes us to the (1,-1) square at the bottom left.
If both players choose to not compromise then we reach the (-10,-10) outcome. Both players get an outcome of -10. This is the situation where both sides don’t back down in a game of chicken and they are both much worse off.
This representation of games is useful for finding equilibria and also for other further analysis.