Decision making is a dynamical phenomenon and thus the transition from indecision to decision can be described well by bifurcation theory. Agreement, or consensus, is only one of many ways in which collective decisions can be made. However, theories of multi-agent multi-option decision making either focus on agreement decisions or average at the agent level to achieve a mean-field description of the decision dynamics. Introducing the agent level back into collective decision-making models uncovers a plethora of novel collective behaviors beyond agreement and mean-field descriptions of disagreement and polarization. These include uniform and moderate-extremist disagreement, and switchy-and-fast versus continuous-and-slow transitions from indecision to decision. Which of these behaviors emerge depends on the model parameters, in particular, the number of agents and the number of options.Our study is grounded in Equivariant Bifurcation Theory, which allows us to formulate model-independent predictions and to develop a constructive sensitivity analysis of the decision dynamics at URLanizing equivariant singularity. The localized sensitivity analysis reveals how collective decision making can be both flexible and robust in response to subtle changes in the environment or in the deciding agent interactions. Equivariant Bifurcation Theory also guides the construction of new multi-agent multi-option decision-making dynamics, for an arbitrary number of agents and an arbitrary number of options, and with fully controllable dynamical behaviors\ES\ES\R\otimes}\R. We also generalize this list to the action of }\ES\ES\R\otimes }\R .
We develop a model-independent framework to study the dynamics of decision-making in opinion networks for an arbitrary number of agents and an arbitrary number of options. Model-independence means that the analysis is not performed on a specific set of equations, in contrast to classical approaches to decision making that fix a specific model and analyze it. Rather, the general features of decision making in dynamical opinion networks can be derived starting from empirically testable hypotheses about the deciding agents, the available options, and the interactions among them. After translating these empirical hypotheses into algebraic ones, we use the tools of equivariant bifurcation theory to uncover model-independent properties of dynamical opinion networks. The model-independent results are illustrated on a novel analytical model that is constructed by plugging a generic sigmoidal nonlinearity, modeling boundedness of opinions and opinion perception, into the model-independent equivariant structure. Our analysis reveals richer and more flexible opinion-formation behavior as compared to model-dependent approaches. For instance, analysis reveals the possibility of switching between consensus and various forms of dissensus by modulation of the level of agent cooperativity and without requiring any particular ad-hoc interaction topology (e.g., structural balance). From a theoretical viewpoint, we prove new results in equivariant bifurcation theory. We construct an exhaustive list of axial subgroups for the action of\ES_n \times\ES_3 on\R^{n-1\otimes}\R^{2. We also generalize this list to the action of }\ES_n \times\ES_k on\R^{n-1\otimes }\R^{k-1 .
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Decision making is a dynamical phenomenon and thus the transition from indecision to decision can be described well by bifurcation theory. Agreement, or consensus, is only one of many ways in which collective decisions can be made. However, theories of multi-agent multi-option decision making either focus on agreement decisions or average at the agent level to achieve a mean-field description of the decision dynamics. Introducing the agent level back into collective decision-making models uncovers a plethora of novel collective behaviors beyond agreement and mean-field descriptions of disagreement and polarization. These include uniform and moderate-extremist disagreement, and switchy-and-fast versus continuous-and-slow transitions from indecision to decision. Which of these behaviors emerge depends on the model parameters, in particular, the number of agents and the number of options.Our study is grounded in Equivariant Bifurcation Theory, which allows us to formulate
We develop a
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predictions and to develop a constructive sensitivity analysis of the decision dynamics at URLanizing equivariant singularity. The localized sensitivity analysis reveals how collective decision making can be both flexible and robust in response to subtle changes in the environment or in the deciding agent interactions. Equivariant Bifurcation Theory also guides the construction of new multi-agent multi-option
framework to study the dynamics of
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dynamics, for an arbitrary number of agents and an arbitrary number of options, and with fully controllable dynamical behaviors
in opinion networks for an arbitrary number of agents and an arbitrary number of options. Model-independence means that the analysis is not performed on a specific set of equations, in contrast to classical approaches to decision making that fix a specific model and analyze it. Rather, the general features of decision making in dynamical opinion networks can be derived starting from empirically testable hypotheses about the deciding agents, the available options, and the interactions among them. After translating these empirical hypotheses into algebraic ones, we use the tools of equivariant bifurcation theory to uncover model-independent properties of dynamical opinion networks. The model-independent results are illustrated on a novel analytical model that is constructed by plugging a generic sigmoidal nonlinearity, modeling boundedness of opinions and opinion perception, into the model-independent equivariant structure. Our analysis reveals richer and more flexible opinion-formation behavior as compared to model-dependent approaches. For instance, analysis reveals the possibility of switching between consensus and various forms of dissensus by modulation of the level of agent cooperativity and without requiring any particular ad-hoc interaction topology (e.g., structural balance). From a theoretical viewpoint, we prove new results in equivariant bifurcation theory. We construct an exhaustive list of axial subgroups for the action of
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