In this paper, we present a framework for facilitation robots that regulate imbalanced engagement density in a four-participant conversation as the forth participant with proper procedures for obtaining initiatives. Four is the special number in multiparty conversations. In three-participant conversations, the minimum unit for multiparty conversations, social imbalance, in which a participant is left behind in the current conversation, sometimes occurs. In such scenarios, a conversational robot has the potential to objectively observe and control situations as the fourth participant. Consequently, we present model procedures for obtaining conversational initiatives in incremental steps to harmonize such four-participant conversations. During the procedures, a facilitator must be aware of both the presence of dominant participants leading the current conversation and the status of any participant that is left behind. We model and optimize these situations and procedures as a partially observable Markov decision process (POMDP), which is suitable for real-world sequential decision processes. The results of experiments conducted to evaluate the proposed procedures show evidence of their acceptability and feeling of groupness. 
Four-participant group conversation: A facilitation robot controlling engagement density as the fourth participant