Mathematics in everyday life

Decision making in groups


I find it fascinating how animal groups (including humans) make decisions. If collective action is to take place, groups of animals need to reach agreement on a decision. However, communication is frequently only local (you talk only to “neighbors” in your group), and there is frequently no exterior intelligence that polls and integrates the information from all members of the population. How is it possible to reach a global decision, given that information is exchanged only locally?

There has been quite a bit of work on this question. For instance, when choosing a new place for a hive, honeybees will communicate to each other the location of a promising spot using a waggle dance (Seeley, et al. Science, vol. 335 (6064) pp. 108-11). Scouts will head-butt a dancing bee who is signaling a competing location. Competing factions thus mutually inhibit each other. The winning faction is the one that out-butts the others. Presumably, this faction was also larger, or more vigorous in promoting a new spot because the spot was actually better.

In a related study, a number of models have been used to examine the impact of a vocal, opinionated minority in making collective decisions (Couzin, et al. Science, vol. 334 (6062) pp. 1578-80). In a head-to-head competition with a less opinionated majority, the smaller group can carry the day. This advantage disappears in the presence of a population of individuals that do not belong to either group. This uninformed group seems to temper the influence of a strident minority. Experiments with schooling fish provided nice support for the theory.

However, there is something quite puzzling about the flow of information in groups. Consider the case of a jury in a trial. Each juror forms an independent opinion about the guilt or innocence of a defendant during the trial. During the deliberation information is exchanged, and opinions become more correlated. In fact, in the absence of complete agreement a mistrial is declared (think of the Twelve Angry Men as an example).

Would we be more likely to reach the right decision if jury members were barred from discussion between themselves, and we simply followed the decision of the majority? This may sound counterintuitive, but it is not at all clear that deliberation increases the collective wisdom of the group. A shrill minority may effectively promulgate its views in such small groups. Indeed, Daniel Kahneman suggests that the increased correlation that results from discussion decreases the quality of decisions. He urges that all participants in a meeting write down their opinions before any discussion takes place, and that these initial opinions carry significant weight in the eventual decision.

I’m somewhat new to this, but I am interested in understanding what allows for good information transfer in groups of individuals. The information available to an individual may increase, and individual decisions may become better. However, the data processing inequality suggests that however the exchange occurs, the information of the collective as a whole will only decrease. Again, this assumes that some external intelligence has access to all the information, and can make decisions based on this complete information. The problem is that groups are constrained in how they make decisions – for instance, going with the majority of a group may be suboptimal for somebody who can gather and assess information from all individuals in a group. However, it may be close to optimal in other circumstances.

Surprisingly, the process of sharing information is not always detrimental. In a recent paper, Mossel and Tamuz show that individuals in a network can reach an optimal decision (or estimate) even though they are restricted to exchanging information only with their neighbors. Surprisingly, it seems essential that the network is recurrent, i.e. that there is a path of communication between any two individuals in the network. The model is somewhat idealized, as all individuals know the exact structure of the network, as well as the belief or quality of the initial estimate of everyone else in the network. Still I find it amazing that as the estimate becomes more correlated among individuals in the group, it also becomes better. Each individual exchanges information locally. Yet, in finite time the estimate of each individual equals that of an intelligence that is able to gather complete information from all individual in the network.

This is an example of an essentially perfect exchange of information in a network. Are there other mechanisms to achieve this? Maybe more importantly, can we test whether groups of animals or social groups can integrate information locally so as to make near optimal collective decisions? I think these are very important questions, that will be examined extensively in the near future.