What is the phrase ending in “game” that means a fool's errand?

Question

I am trying to remember the phrase that describes a particular kind of folly - trying to "beat the house", or willingness to play a rigged game. As I recall it has the form of "a fool's errand" - that is, the game is referred to as belonging to the person who is silly enough to play it. I'd appreciate knowing the origin of the phrase, too.

Answer 1

A little reflection brought to mind the phrase I was seeking: A Mug's Game. Originated 1910.

Answer 2

The phrase "it is a fool's game" seems common and idiomatic enough.