Decomposed 2*2 games - a conceptual review

2*2 games, such as the Prisoner's Dilemma, are a common tool for studying cooperation and social decision-making. In experiments, 2*2 games are usually presented in matrix form, such that participants see only the possible outcomes. Some 2*2 games can be decomposed into payoffs for self and other, such that participants see the direct consequences of two actions. While the final outcomes of the decomposed form and the matrix-form can be identical, the framing differs: the matrix form emphasises the outcome, the decomposed form emphasises the action. This allows decomposed games to address questions that could not be answered with matrix games. Here, we provide a conceptual overview of decomposed games that is accessible without knowing the underlying mathematics. We explain which 2*2 games can be decomposed, why the same payoff matrix can be decomposed into infinitely many decompositions, and we apply this to (a)symmetric games, (a)symmetric decompositions, and games with ties. Finally, we show how to calculate all decompositions for a given game and we suggest when the decomposed form might be more appropriate than the matrix form for an experimental design.