In spite of being a price-setter (a company that can set its own price), why would a monopoly player choose to

Answer 1

choose to what..? A monopoly player by definition has control
of the market they are in. Do you mean open it up to competition?
I'm not sure where you're going with this question. If a monopoly player chooses to open it up to competition, chances are you are
referring to a government-owned monopoly, and this because of legislation introduced to allow private sector entrants to compete for the best service or goods at the best price the market will bear.