We study a problem of multi-agent exploration with behaviorally heterogeneous robots. Each robot maps its surroundings using SLAM and identifies a set of areas of interest (AoIs) or frontiers that are the most informative to explore next. The robots assess the utility of going to a frontier using Behavioral Entropy (BE) and then determine which frontier to go to via a distributed task assignment scheme. We convert the task assignment problem into a non-cooperative game and use a distributed algorithm (d-PBRAG) to converge to the Nash equilibrium (which we show is the optimal task allocation solution). For unknown utility cases, we provide robust bounds using approximate rewards. We test our algorithm (which has less communication cost and fast convergence) in simulation, where we explore the effect of sensing radii, sensing accuracy, and heterogeneity among robotic teams with respect to the time taken to complete exploration and path traveled. We observe that having a team of agents with heterogeneous behaviors is beneficial.