The cheapest good badger brush I know of is the Whipped Dog silvertip, which was $25 when I bought one a few years ago, so I reckon the advice to get a Plissoft synthetic is good if $20 is your absolute limit.
RazoRock Plissoft synthetic .... $ 10.
Way better than any badger or boar at your price point.
Badgers worth buying will cost considerably more than $ 20.
If you must have a boar, the Semogue 610 or 620 is an excellent brush at around your price.
Good Luck !
RazoRock Plissoft synthetic .... $ 10.