  This paper introduces a Markov chain approach that allows a rigorous analysis
of agent based opinion dynamics as well as other related agent based models
(ABM). By viewing the ABM dynamics as a micro description of the process, we
show how the corresponding macro description is obtained by a projection
construction. Then, well known conditions for lumpability make it possible to
establish the cases where the macro model is still Markov. In this case we
obtain a complete picture of the dynamics including the transient stage, the
most interesting phase in applications. For such a purpose a crucial role is
played by the type of probability distribution used to implement the stochastic
part of the model which defines the updating rule and governs the dynamics. In
addition, we show how restrictions in communication leading to the co-existence
of different opinions follow from the emergence of new absorbing states. We
describe our analysis in detail with some specific models of opinion dynamics.
Generalizations concerning different opinion representations as well as opinion
models with other interaction mechanisms are also discussed. We find that our
method may be an attractive alternative to mean-field approaches and that this
approach provides new perspectives on the modeling of opinion exchange
dynamics, and more generally of other ABM.
