Bayesian Heuristics for Group Decisions

We propose a model of inference and heuristic decision making in groups that is rooted in the Bayes rule but avoids the complexities of rational inference in partially observed environments with incomplete information. According to our model, the group members behave rationally at the initiation of their interactions with each other; however, in the ensuing decision epochs, they rely on a heuristic that replicates their experiences from the first stage. Subsequently, the agents use their time-one Bayesian update and repeat it for all future time-steps; hence, updating their actions using a so-called Bayesian heuristic. This model avoids the complexities of fully rational inference and also provides a behavioral and normative foundation for non-Bayesian updating. It is also consistent with a dual-process psychological theory of decision making, where a controlled (conscious/slow) system develops the Bayesian heuristic at the beginning, and an automatic (unconscious/fast) system takes over the task of heuristic decision-making in the sequel. 

We specialize this model to a group decision scenario where private observations are received at the beginning, and agents aim to take the best action given the aggregate observations of all group members. We present the implications of the choices of signal structure and action space for such agents. We show that for a wide class of distributions from the exponential family the Bayesian heuristics take the form of an affine update in the self and neighboring actions. Furthermore, if the priors are non-informative (and possibly improper), then these action updates become a linear combination. We investigate the requirements on the modeling parameters for the action updates to constitute a convex combination as in the DeGroot model. The results reveal the nature of assumptions that are implicit in the DeGroot updating and highlights the fragility and restrictions of such assumptions; in particular, we show that for a linear action update to constitute a convex combination the precision or accuracy of private observations should be balanced among all neighboring agents, requiring a notion of social harmony or homogeneity in their observational abilities. Following the DeGroot model, agents reach a consensus asymptotically. We derive the requirements on the signal structure and network topology such that the consensus action aggregates information efficiently. This involves additional restrictions on the signal likelihoods and network structure. In the particular case that all agents observe the same number of i.i.d. samples from the same distribution, then efficiency arise in degree-regular balanced structures, where all nodes listen to and hear from the same number of neighbors. 

We next shift attention to a finite state model, in which agents take actions over the probability simplex; thus revealing their beliefs to each other. We show that the Bayesian heuristics, in this case, prescribe a log-linear update rule, where each agent’s belief is set proportionally to the product of her own and neighboring beliefs. We analyze the evolution of beliefs under this rule and show that agents reach a consensus. The consensus belief is supported over the maximizers of a weighted sum of the log-likelihoods of the initial observations. Since the weights of the signal likelihoods coincide with the network centralities of their respective agents, these weights can be equalized in degree-regular and balanced topologies, where all nodes have the same in and out degrees. Therefore, in such highly symmetric structures the support of the consensus belief coincides with the maximum likelihood estimators (MLE) of the truth state; and here again, balanced regular structures demonstrate a measure of efficiency. Nevertheless, the asymptotic beliefs systematically reject the less probable alternatives in spite of the limited initial data, and in contrast with the optimal (Bayesian) belief of an observer with complete information of the environment and private signals. The latter would assign probabilities proportionally to the likelihood of every state, without rejecting any of the possible alternatives. The asymptotic rejection of less probable alternatives indicates a case of group polarization, i.e. overconfidence in the group aggregate that emerges as a result of the group interactions. Unlike the linear action updates and the DeGroot model which entail a host of knife-edge conditions on the signal structure and model parameters, we observe that the belief updates are unweighted; not only they effectively internalize the heterogeneity of the private observations, but also they compensate for the individual priors. Thence, we are led to the conclusion that multiplicative belief updates, when applicable provide a relatively robust description of the decision-making behavior.