THROW CAUTION TO THE WIND means to take a big risk.
They threw caution to the wind and backed his risky project.
Take a risk for once. Throw caution to the wind.
THROW GOOD MONEY AFTER BAD means to invest further money in a project that has done badly.
We have already lost a million dollars on this project. If we invest any more money, it will be throwing good money after bad.
Don’t throw good money after bad. Accept your losses and move on.
THROW IN THE TOWEL means to give up.
After seven unsuccessful attempts, he decided to throw in the towel.
Why don’t you throw in the towel? You have tried and failed so many times.