The extant literature on matching markets assumes ordinal preferences for matches, while bargaining within matches is mostly excluded. Central for this paper, however, is the bargaining over joint profits from potential matches. We investigate, both theoretically and experimentally, a seemingly simple allocation task in a 2x2 market with repeated negotiations. More than 75% of the experimental allocations are unstable, and 40% of the matches are inefficient (in cases where inefficiency is possible). By defining the novel concept 'altruistic core', we can explain the occurrence of inefficient matches as well as the significant behavioral differences among our six treatments.