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The voter model is a simple agent-based model to mimic opinion dynamics insocial networks: a randomly chosen agent adopts the opinion of a randomlychosen neighbour. This process is repeated until a consensus emerges. Althoughthe basic voter model is theoretically intriguing, it misses an importantfeature of real opinion dynamics: it does not distinguish between an agent'spublicly expressed opinion and her inner conviction. A person may not feelcomfortable declaring her conviction if her social circle appears to hold anopposing view. Here we introduce the Concealed Voter Model where we add asecond, concealed layer of opinions to the public layer. If an agent's publicand concealed opinions disagree, she can reconcile them by either publiclydisclosing her previously secret point of view or by accepting her publicopinion as inner conviction. We study a complete graph of agents who can choosefrom two opinions. We define a martingale $M$ that determines the probabilityof all agents eventually agreeing on a particular opinion. By analyzing theevolution of $M$ in the limit of a large number of agents, we derive theleading-order terms for the mean and standard deviation of the consensus time(i.e. the time needed until all opinions are identical). We thereby give aprecise prediction by how much concealed opinions slow down a consensus.

Consensus time in a voter model with concealed and publicly expressed opinions