In the Mahabharata, the episode of Draupadi Swayamvara is an extremely interesting one. Depending upon the version of Mahabharata one is reading, the same story can be interpreted differently. Also, many of the incidents that occur in this episode are difficult to comprehend if one sticks to just the original. This is where commentaries such as the Mahabharata Tatparya Nirnaya by Sri Madhvacharya come in very handy.
One of the questions that is constantly asked with regard to the Swayamvara is – why did Bhimasena not attempt to lift and string the bow?
One naive conclusion drawn usually is that it would not have been possible for him to lift or string the bow. It is established beyond a doubt, in the original Mahabharata itself, that Bhimasena is incomparably stronger than Arjuna. Further, Draupadi herself declares in the Mahabharata that the three people who can string the Gandiva are Krishna, Arjuna and Bhima.
http://www.mahabharataonline.com/translation/mahabharata_03012.php Page 31
Therefore, the bow that Drupada had arranged for the contest would not have been a challenge for Bhimasena.
The other question – Did the Pandavas want Draupadi to marry Arjuna only? Is that the reason why Bhimasena did not attempt stringing the bow? That is also not true. Even before the Pandavas enter the city of the Panchalas, Lord Veda Vyasa visits them in the forest and informs them that the daughter of Draupadi will become the wives of all five Pandavas.
http://www.mahabharataonline.com/translation/mahabharata_01172.php Page 344
So all the Pandavas, especially Bhimasena, knew very well in advance that Draupadi would marry all of them. So it was a question of who would string the bow. And Arjuna did it.
One reason for Arjuna getting up ahead of Bhimasena is that, according to the Shastras, an achievement by the younger brother is also considered as an achievement of the older ones. The reverse is not true. Hence one could say that Arjuna stringing the bow is equivalent to Bhimasena stringing it as well. But that is just a less important reason. The real reason has been brought out beautifully by Sri Madhvacharya in the MBTN.
To understand that reason, we need to briefly recall the story associated with Drupada and the birth of Draupadi.
When Bhishma brought Amba, Ambika and Ambalika to Hastinapur as wives for Vichitravirya, Amba reveals that she likes Salva. Bhishma releases her and she approaches Salva. Salva refuses to accept her as he was defeated by Bhishma. She returns to Hastinapur and there she is rejected by Bhishma also. Dejected, she performs a penance. Lord Shiva appears in front of her and grants her two things. Firstly, he grants her a boon saying that she would, in her next life, be a important part of the plot in killing Bhishma. The same Amba is born as Shikhandini in her next life and later turns into a man called Shikhandi. Shikhandi is the reason why Bhishma lays down arms on the tenth day of the great war!
Secondly, Lord Shiva grants Amba a celestial garland and tells her that the person who wears the garland would kill Bhishma. Amba goes around many kingdoms asking people to wear the same. Knowing the boon of Shiva, and utterly scared of Bhishma, no one dares to accept the garland. Amba finally places the garland on the gates of Drupada’s palace and ends her life.
Drupada keeps the garland with him. During the early years of Shikhandi, the garland is worn by him once. Now, the same garland is given to Draupadi during the Swayamvara. Which means the person whom Draupadi garlands in the Swayamvara will (have to) kill Bhishma. Bhimasena, a true follower of the Bhagavata Dharma, would never kill a devotee of Krishna (Vishnu). And it is well known that Bhishma is a great devotee of Krishna (he gave us the Vishnu Sahasra Nama).
Therefore, since Bhimasena would not kill Bhishma at any cost, he would not accept that particular garland from Draupadi. Another thing to note is that Bhimasena in this case, decides to honor the boon, unlike in the cases of other Rakshasas who had boons from various Devatas. In those cases, Bhimasena simply disregards the boon and proceeds to destroy them. Amba, in her original form, is the wife of the Vasu Dyaus, who incarnates as Bhishma. The fulfillment of Amba’s vow would only unite them back and hence Lord Shiva’s boon was to be honored.
So, the real reason for Arjuna stringing the bow ahead of Bhimasena is because it is Arjuna who has to kill Bhishma and not Bhimasena.