The pushy person's game? (5)
I believe the answer is:
poker
I believe this is a double definition.
'the pushy person's' is the first definition.
(I've seen this before)
'game?' is the second definition.
(poker is a kind of game)
(Other definitions for poker that I've seen before include "Fireside implement - card game" , "Gambling card game" , "Rod; betting game" , "Jabber" , "Fire-iron" .)