Axelrod and Hamilton (1981) write that there are two characteristics to the payoff structure of the prisoner's dilemma:
- $T>R>P>S$ and
- $R>(S+T)/2$
Has someone published a comprehensive list of such characteristics for all possible games with Bernoulli payoffs?
EDIT: By Bernoulli payoffs I mean ordinal payoffs. By all possible games, I mean all possible 2x2 games.
EDIT: I was hoping to find a comprehensive list of inequalities and other constraints for all possible 2x2 games. The typology I'm looking for will give a set of constraints that define each game. For example $T>R>P>S$ and $R>(S+T)/2$ are the two constraints that define a PD game. Obviously each 2x2 game has at least one constraint, which is the order of preferences, but I would like to know which games have which other constraints.
The first contribution for payoffs on an ordinal scale is Rapoport and Guyer (1966), "A taxonomy of $2 \times 2$ games", General Systems 11, 203–214. This deals with the 78 strict (no ties allowed) ordinal $2 \times 2$ games. See also Rapoport, Guyer and Gordon (1976), The $2 \times 2$ game, University of Michigan Press.
A full taxonomy (including games with ties) is in Fraser and Kilgour (1986), "Non-strict ordinal $2 \times 2$ games: A comprehensive computer-assisted analysis of the 726 possibilities'', Theory and Decision 20, 99–121.
An accessibile summary is Kilgour and Fraser (1988).
A bit of reverse bibliographic search starting from these references should point you to discover other approaches.