As anthropomorphic agents (AI and robots) are increasingly used in society, empathy and trust between people and agents are becoming increasingly important. A better understanding of agents by people will help to improve the problems caused by the future use of agents in society. In the past, there has been a focus on the importance of self-disclosure and the relationship between agents and humans in their interactions. In this study, we focused on the attributes of self-disclosure and the relationship between agents and people. An experiment was conducted to investigate hypotheses on trust and empathy with agents through six attributes of self-disclosure (opinions and attitudes, hobbies, work, money, personality, and body) and through competitive and cooperative relationships before a robotic agent performs a joint task. The experiment consisted of two between-participant factors: six levels of self-disclosure attributes and two levels of relationship with the agent. The results showed that the two factors had no effect on trust in the agent, but there was statistical significance for the attribute of self-disclosure regarding a person's empathy toward the agent. In addition, statistical significance was found regarding the agent's ability to empathize with a person as perceived by the person only in the case where the type of relationship, competitive or cooperative, was presented. The results of this study could lead to an effective method for building relationships with agents, which are increasingly used in society.