In the stable and cohesive social groups of primates (and other group-living mammals) individuals maintain long-lasting partnerships with a subset of their group companions. Such inter-individual relationships are based on the active exchange of a variety of affiliative, agonistic and cooperative behaviour patterns. Since these interactions influence the partners' biological fitness they are often regarded as 'services' or 'commodities' that are offered ('sold') and demanded ('bought') by individuals in order to cultivate those partnerships that best contribute to maximise their fitness. According to the biological markets theory, an individual's attractiveness as a social partner depends on the value of the services that it can provide and trade for, which depends on the levels of supply and demand, that is, a given service is more expensive to buy, the fewer are the individuals that offer it relative to the individuals that demand it. This paper uses data on grooming (i.e. investment and outbidding competition), female aggression (i.e. female contest competition), and male herding (i.e. male coercion) from a study of three differently-sized captive one-male, multi-female social systems of hamadryas baboons (Papio hamadryas) to test six biomarket-based predictions regarding the unit males' and females' allocation of grooming time and preferred grooming partnerships and the males' rate of coercion (Fig. 1). It was found that most core predictions were not borne out by the data. We propose an alternative model, namely, the 'constraint' model (Fig. 6), according to which the males' behaviour toward the females is expected to be constrained by the intensity of contest and outbidding competition with other males, by the number of relationships they have to service (i.e. their social time budget), and by how attractive (or valuable) they are to females. The latter should also influence the use of coercion by males. The females' behaviour toward the males is hypothesised to be constrained by the intensity of contest and outbidding competition with other females and by the extent of relationship conflict with males. Finally, female relationships are expected to be constrained by the males' behaviour, especially if they are prone to police female aggressive encounters, which contributes to reduce power differentials among them. Some of these predictions were supported by the data. Although both models share the view that an individual's availability, which is constrained by the social time budget already compromised, does influence his or her value as a social partner, the 'constraint' model does emphasise, in addition, that a male's accessibility, which is constrained by female competition, and quality (i.e. RHP), also contribute significantly to it and must be considered. Although market forces are likely to operate in the structuring of social relationships within primate groups, we suggest that their potential impact may be overriden in social systems in which contest and outbidding competition within trader classes, and conflict between trading partners, are strong. This effect should be even more exacerbated if power differentials among individuals are strong and if members of a trader class actively interfere the interactions of members of the other trader class, for example, by herding their affiliative interactions or by policing their agonistic encounters. In some kinds of animal social systems it appears that the main effect of demographic factors is to constrain the individuals' expression of their partner preferences and with it the possibility of detecting market forces.