We propose a theory of strategic voting in multi-winner elections with approval balloting: A fixed number M of candidates are to be elected; each voter votes for as many candidates as she wants; the M candidates with the most votes are elected. We assume that voter preferences are separable and that there exists a tiny probability that any vote might be misrecorded. Best responses involve voting by pairwise comparisons. Two candidates play a critical role: the weakest expected winner and the strongest expected loser. Expected winners are approved if and only if they are preferred to the strongest expected loser and expected losers are approved if and only if they are preferred to the weakest expected winner. At equilibrium, if any, a candidate is elected if and only if he is approved by at least half of the voters. With single-peaked preferences, an equilibrium always exists, in which the first M candidates according to the majority tournament relation are elected. The theory is tested on individual data from the 2011 Regional Government election in Zurich.