In the story, A Game of Thrones, Tyrion Lannister demands a trial by combat after being wrongly accused of trying to kill Bran Stark and conspiring to kill Lord Jon Arryn.
The halls of the Eyrie remain silent for a moment when Tyrion asks for a champion for nobody wants to fight on his behalf until Bronn offers.
Tyrion: Do I have a volunteer?
Laughter and gigges and awkward silence.
Tyrion: No one?
Robert Arryn giggles hoping to see Tyrion defeated and thrown threw the Moon Door.
Lady Lysa: I think we can assume no one is willing to ...
Bronn: I will stand for the dwarf.
Bronn is an amoral and selfish sellsword. When is introduced to the story, he is a freerider more interested in his own comfort and gold than whether others simply live. Much later in the story, he admitted the only reason he saved Jaime Lannister from a dragon's breath was so that Jaime could stay alive until Bronn got his castle from the Lannisters.
He is not moved by claims of honor, so a trial by combat over the rights and honors of hated dwarf of a hated family should hold no appeal to him. I could see him amused to watch Tyrion face combat, but that would be short lived entertainment. I could not see him wishing to participate in that trial.
If he loses, he might die. Defeated but still alive, he might be maimed for life and reviled by the knights of the Vale. If he is maimed, his career as a freeriding sellsword will be cut short.
If he wins, he gains nothing. No gold or promises come to those who participate in a trial by combat. Even if he wins, he might be maimed in the duel. If he wins by killing the favorite knight of Lady Lysa, he will certainly be reviled throughout the Vale.
Both outcomes have very negative outcomes for him. Neither outcome of the trial has a positive outcome.
Why would Bronn offer to fight when either outcome almost certainly has a very high cost for him and no benefit to him?
Please provide answers from the books, but I will accept answers from the TV series.