  To make decisions we are guided by the evidence we collect, as well as the
opinions of friends and neighbors. How do we integrate our private beliefs with
information we obtain from our social network? To understand the strategies
humans use to do so it is useful to compare them to observers that optimally
integrate all evidence. Here we derive network models of rational (Bayes
optimal) agents who accumulate private measurements and observe decisions of
their neighbors to make an irreversible choice between two options. The
resulting information exchange dynamics has interesting properties: When one
option is preferred, the absence of a decision can be increasingly informative
over time. In recurrent networks an absence of a decision can lead to a
sequence of belief updates akin to those in the literature on common knowledge.
Information obtained from observing repeated non-decisions is independent of
realization, unless the private information of agents is redundant. On the
other hand, in larger networks a single decision can trigger a cascade of
agreements and disagreements that depend on the private information agents have
gathered. Our approach provides a bridge between social decision making models
in the economics literature, which largely ignore the temporal dynamics of
decisions, and the single-observer evidence accumulator models used widely in
neuroscience and psychology.
