Fines perhaps used to resolve stalemates (9)
I believe the answer is:
penalties
'fines perhaps' is the definition.
(fine is a kind of penalty)
'to resolve stalemates' is the wordplay.
'to resolve' becomes 'penal' (I can't explain this - if you can you should believe this answer much more).
'stalemates' becomes 'ties' (I've seen this in another clue).
'penal'+'ties'='PENALTIES'
'used' acts as a link.
Can you help me to learn more?
(Other definitions for penalties that I've seen before include "parking fines" , "Fines or forfeitures in Palestine" , "Punitive impositions in Palestine" .)