One result of stochastic theory is that there exists a stationary vector "v" for the matrix "M" such that formula_6. Without loss of generality, it may be specified that "v" is normalized so that the sum of its four components is unity. The "ij" th entry in formula_7 will give the probability that the outcome of an encounter between "X" and "Y" will be "j" given that the encounter "n" steps previous is "i". In the limit as "n" approaches infinity, "M" will converge to a matrix with fixed values, giving the long-term probabilities of an encounter producing "j" which will be independent of "i". In other words, the rows of formula_8 will be identical, giving the long-term equilibrium result probabilities of the iterated prisoners dilemma without the need to explicitly evaluate a large number of interactions. It can be seen that "v" is a stationary vector for formula_7 and particularly formula_8, so that each row of formula_8 will be equal to "v". Thus the stationary vector specifies the equilibrium outcome probabilities for "X". Defining formula_12 and formula_13 as the short-term payoff vectors for the {cc,cd,dc,dd} outcomes (From "X" 's point of view), the equilibrium payoffs for "X" and "Y" can now be specified as formula_14 and formula_15, allowing the two strategies "P" and "Q" to be compared for their long term payoffs.