The prisoner's dilemma and chicken are 2 x 2 games, of course. There are seventy-eight distinct 2 x 2 games when payoffs are simply ranked rather than assigned numerical values. A symmetric game is one where the payoffs are the same for each player under comparable circumstances - where neither player is in a privileged position. If player A is the lone cooperator, his payoff is the same as player B's would be if he was the lone cooperator, and so forth. Symmetric games are the easiest to understand, and probably the most important in social interactions.
People are much the same the world over.
No conflicts are so common or so bitter as those between people wanting the same thing. So let's look at the symmetric games. There are just four payoffs to worry about in a symmetric 2 x 2 game. Any given preference ordering of the four payoffs defines a game. Not all are dilemmas.