The boaster's game (4)
I believe the answer is:
brag
This clue is a double definition.
'the boaster's' is the first definition.
(I've seen this in another clue)
'game' is the second definition.
(I've seen this before)
(Other definitions for brag that I've seen before include "popular with 15 [EGOTIST] , no doubt" , "game of bluff" , "Boast ? card-game" , "Show off, crow" , "boast; form of poker" .)