The solution of paternity disputes using results from scientific analyses is studied from a decision-theoretical viewpoint. Two alternative approaches to decision making, the so-called 'Bayes' and 'Minimax' strategies, are described and discussed. If prior probabilities of paternity are exactly known, then Bayes decisions are (a) independent of the source of evidence and (b) optimal with respect to average losses caused by wrong decisions. However, it is concluded that Minimax decisions, which depend upon the employed test system but not upon prior probabilities, are more appropriate in paternity cases if equal prior good will towards disclaimed children and alleged fathers is demanded. It is further demonstrated that, when major evidence about paternity comes from multilocus DNA fingerprinting, prior probabilities must be known quite accurately for Bayes decisions to be superior with respect to average losses. Finally, we are able to show that 'quasi' Bayes decision making, that is, adopting a neutral prior probability of 0.5 but leaving thresholds for decision making unchanged, coincides with Minimax decision making if multilocus DNA fingerprinting is employed.