We consider a multistage version of the Perfect Matching problem which models the 15 scenario where the costs of edges change over time and we seek to obtain a solution that achieves 16 low total cost, while minimizing the number of changes from one instance to the next. Formally, 17 we are given a sequence of edge-weighted graphs on the same set of vertices V , and are asked to 18 produce a perfect matching in each instance so that the total edge cost plus the transition cost 19 (the cost of exchanging edges), is minimized. This model was introduced by Gupta et al. (ICALP 20 2014), who posed as an open problem its approximability for bipartite instances. We completely 21 resolve this question by showing that Minimum Multistage Perfect Matching (Min-MPM) does 22 not admit an n 1−-approximation, even on bipartite instances with only two time steps. 23 Motivated by this negative result, we go on to consider two variations of the problem. In 24 Metric Minimum Multistage Perfect Matching problem (Metric-Min-MPM) we are promised 25 that edge weights in each time step satisfy the triangle inequality. We show that this problem 26 admits a 3-approximation when the number of time steps is 2 or 3. On the other hand, we 27 show that even the metric case is APX-hard already for 2 time steps. We then consider the 28 complementary maximization version of the problem, Maximum Multistage Perfect Matching 29 problem (Max-MPM), where we seek to maximize the total profit of all selected edges plus the 30 total number of non-exchanged edges. We show that Max-MPM is also APX-hard, but admits 31 a constant factor approximation algorithm for any number of time steps. 32 2012 ACM Subject Classification Theory of Computation → Design and Analysis of Algorithms 33 → Approximation Algorithms Analysis 34