Abstract: A stochastic evolutionary dynamics of two strategies given by 2 × 2 matrix games is studied in ﬁnite populations. We focus on stochastic properties of ﬁxation: how a strategy represented by a single individual wins over the entire population. The process is discussed in the framework of a random walk with site dependent hopping rates. The time of ﬁxation is found to be identical for both strategies in any particular game. The asymptotic behavior of the ﬁxation time and ﬁxation probabilities in the large population size limit is also discussed. We show that ﬁxation is fast when there is at least one pure evolutionary stable strategy (ESS) in the inﬁnite population size limit, while ﬁxation is slow when the ESS is the coexistence of the two strategies.