In 2012, William H. Press and Freeman Dyson published a new class of strategies for the stochastic iterated prisoner's dilemma called "zero-determinant" (ZD) strategies. The long term payoffs for encounters between "X" and "Y" can be expressed as the determinant of a matrix which is a function of the two strategies and the short term payoff vectors: formula_16 and formula_17, which do not involve the stationary vector "v". Since the determinant function formula_18 is linear in "f", it follows that formula_19 (where "U"={1,1,1,1}). Any strategies for which formula_20 is by definition a ZD strategy, and the long term payoffs obey the relation formula_21.