  A kidney exchange is a centrally-administered barter market where patients
swap their willing yet incompatible donors. Modern kidney exchanges use
2-cycles, 3-cycles, and chains initiated by non-directed donors (altruists who
are willing to give a kidney to anyone) as the means for swapping.
  We propose significant generalizations to kidney exchange. We allow more than
one donor to donate in exchange for their desired patient receiving a kidney.
We also allow for the possibility of a donor willing to donate if any of a
number of patients receive kidneys. Furthermore, we combine these notions and
generalize them. The generalization is to exchange among organ clubs, where a
club is willing to donate organs outside the club if and only if the club
receives organs from outside the club according to given specifications. We
prove that unlike in the standard model, the uncapped clearing problem is
NP-complete.
  We also present the notion of operation frames that can be used to sequence
the operations across batches, and present integer programming formulations for
the market clearing problems for these new types of organ exchanges.
  Experiments show that in the single-donation setting, operation frames
improve planning by 34%--51%. Allowing up to two donors to donate in exchange
for one kidney donated to their designated patient yields a further increase in
social welfare.
