Bayesian social learning in linear networks of agents with random behavior

In this paper, we consider the problem of social learning in a network of agents where the agents make decisions on K hypotheses sequentially and broadcast their decisions to others. Each agent in the system has a private observation that is generated by one of the hypotheses. All the observations are independently generated from the same hypothesis. We study a setting where the agents randomly choose to make decisions prudently or non-prudently. A prudent decision is based on the private observation of the agent and all the previous decisions, whereas a non-prudent decision relies only on the private observation of the agent. We present a Bayesian learning method for the agents that exploits the information from other decisions. We analyze the asymptotical property of this system. A proof is presented that with the proposed decision policy, the posterior probability of the true hypothesis converges to one in probability. Simulation results are also provided.