Why did Balram, Vidur, and Rukmi not participate in the Mahabharata War?

Why did Balram, Vidur, and Rukmi not participate in the Mahabharata War? Balarama: Balarama decided to stay neutral and not get involved in the war. Balarama's relationship with Pandavas and Kouravas were equal. He left for pilgrimage a few days later and returned on 18th day of war. “O tiger among men! It is appropriate that you should know everything that I have said earlier in the kingdom of Virata, on the occasion of the wedding. O descendant of the Kuru lineage! It is for your sake that I sought to restrain Hrishikesha. O king! I said repeatedly that my relationship with both of you was equal. But Keshava did not accept the words that I had spoken. I cannot exist for an instant without Krishna. I will come to the aid of neither Partha, nor Duryodhana. After looking towards Vasudeva, this is the decision I have arrived at. You have been born in the Bharata lineage, one that is honoured by all the lords of the earth. O bull among the Bharata lineage! Go and fight in accordance with the dharma of the kshatriyas.” Source: Mahabharata translated by Bibek Debroy Chapter 670(7) Vidur: Vidur is not a trained fighter to participate in war. Vidur was a politician and was giving good advices to Dhritarashtra time to time. Vidur never participated in any war in his life time. Rukmi: Rukmi was rejected by both sides due to his arrogance. First he arrived at the camp of Pandavas and said that if they are afraid of Kouravas then they can take his help. “O Pandava! If you are afraid, I will stand as your aide on the field of battle. In the course of the war, I will help you so that your enemies will not be able to bear it. There is no man on this earth who is equal to me in valour. O Phalguna! When I have killed your enemies in battle, I will hand them over to you.” Arjuna was not happy with arrogant nature of Rukmi and rejected him. After getting rejected by Arjuna, Rukmi went to Duryodhana and boosted about himself in the same manner and again got rejected by Duryodhana. Rukmi then returned with that army, which was like an ocean. O bull among the Bharata lineage! In a similar fashion, he approached Duryodhana. Arriving there, that lord of the earth spoke in the same way. But since he prided himself on his valour, he was rebuffed there too. Source: Mahabharata by Bibek Debroy Udyoga Parva Chapter 818(155)