Intoxicated by British game (6)
I believe the answer is:
blotto
'intoxicated' is the definition.
(similar in meaning)
'by british game' is the wordplay.
'by' says to put letters next to each other.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'game' becomes 'lotto' (the UK National Lottery).
'b' put next to 'lotto' is 'BLOTTO'.
(Other definitions for blotto that I've seen before include "More than happy" , "Unconscious from drink" , "To bolt away slangily drunk" , "Plastered, hammered, sloshed" , "Dead drunk" .)