  How does a group of agents break indecision when deciding about options with
qualities that are hard to distinguish? Biological and artificial multi-agent
systems, from honeybees and bird flocks to bacteria, robots, and humans, often
need to overcome indecision when choosing among options in situations in which
the performance or even the survival of the group are at stake. Breaking
indecision is also important because in a fully indecisive state agents are not
biased toward any specific option and therefore the agent group is maximally
sensitive and prone to adapt to inputs and changes in its environment. Here, we
develop a mathematical theory to study how decisions arise from the breaking of
indecision. Our approach is grounded in both equivariant and network
bifurcation theory. We model decision from indecision as synchrony-breaking in
influence networks in which each node is the value assigned by an agent to an
option. First, we show that three universal decision behaviors, namely,
deadlock, consensus, and dissensus, are the generic outcomes of
synchrony-breaking bifurcations from a fully synchronous state of indecision in
influence networks. Second, we show that all deadlock and consensus value
patterns and some dissensus value patterns are predicted by the symmetry of the
influence networks. Third, we show that there are also many `exotic' dissensus
value patterns. These patterns are predicted by network architecture, but not
by network symmetries, through a new synchrony-breaking branching lemma. This
is the first example of exotic solutions in an application. Numerical
simulations of a novel influence network model illustrate our theoretical
results.
