If we define "P" as the above 4-element strategy vector of "X" and formula_4 as the 4-element strategy vector of "Y", a transition matrix "M" may be defined for "X" whose "ij" th entry is the probability that the outcome of a particular encounter between "X" and "Y" will be "j" given that the previous encounter was "i", where "i" and "j" are one of the four outcome indices: "cc", "cd", "dc", or "dd". For example, from "X" 's point of view, the probability that the outcome of the present encounter is "cd" given that the previous encounter was "cd" is equal to formula_5. (The indices for "Q" are from "Y" 's point of view: a "cd" outcome for "X" is a "dc" outcome for "Y".) Under these definitions, the iterated prisoner's dilemma qualifies as a stochastic process and "M" is a stochastic matrix, allowing all of the theory of stochastic processes to be applied.