The sharing and collective processing of information by certain insect societies is one of the reasons that they warrant the superlative epithet 'super-organisms' (Franks 1989, Am. Sci. 77, 138-145). We describe a detailed experimental and mathematical analysis of information exchange and decision-making in, arguably, the most difficult collective choices that social insects face: namely, house hunting by complete societies. The key issue is how can a complete colony select the single best nest-site among several alternatives? Individual scouts respond to the diverse information they have personally obtained about the quality of a potential nest-site by producing a recruitment signal. The colony then deliberates over (i.e. integrates) different incoming recruitment signals associated with different potential nest-sites to achieve a well-informed collective decision. We compare this process in honeybees and in the ant Leptothorax albipennis. Notwithstanding many differences - for example, honeybee colonies have 100 times more individuals than L. albipennis colonies - there are certain similarities in the fundamental algorithms these societies appear to employ when they are house hunting. Scout honeybees use the full power of the waggle dance to inform their nest-mates about the distance and direction of a potential nest-site (and they indicate the quality of a nest-site indirectly through the vigour of their dance), and yet individual bees perhaps only rarely make direct comparisons of such sites. By contrast, scouts from L. albipennis colonies often compare nest-sites, but they cannot directly inform one another of their estimation of the quality of a potential site. Instead, they discriminate between sites by initiating recruitment sooner to better ones. Nevertheless, both species do make use of forms of opinion polling. For example, scout bees that have formerly danced for a certain site cease such advertising and monitor the dances of others at random. That is, they act without prejudice. They neither favour nor disdain dancers that advocate the site they had formerly advertised or the alternatives. Thus, in general the bees are less well informed than they would be if they systematically monitored dances for alternative sites rather than spending their time reprocessing information they already have. However, as a result of their lack of prejudice, less time overall will be wasted in endless debate among stubborn and potentially biased bees. Among the ants, the opinions of nest-mates are also pooled effectively when scouts use a threshold population of their nest-mates present in a new nest-site as a cue to switch to more rapid recruitment. Furthermore, the ants' reluctance to begin recruiting to poor nest-sites means that more time is available for the discovery and direct comparison of alternatives. Likewise, the retirement of honeybee scouts from dancing for a given site allows more time for other scouts to find potentially better sites. Thus, both the ants and the bees have time-lags built into their decision-making systems that should facilitate a compromise between thorough surveys for good nest-sites and relatively rapid decisions. We have also been able to show that classical mathematical models can illuminate the processes by which colonies are able to achieve decisions that are relatively swift and very well informed.