My Word can beat up your Word

J, 100

z=:"."0@":@(+/)^:9@(64-~a.i.])@(#~' '&i.)"1
f=:*@-/"2@(z@((]#~]i.~{.@\:~)"1^:([:=/z))){'STALEMATE'&,

runs like this:

f 'NO',:'ZOO'
NO       
f 'CAN',:'BAT'
CAN      
f 'FAT',:'BANANA'
FAT      
f 'ONE',:'ONE'
STALEMATE

it doesn't yet accept input exactly as asked.

APL (Dyalog) (91 86)

⎕ML←3⋄{Z≡∪Z←{2>⍴⍕⍵:⍵⋄∇+/⍎¨⍕⍵}¨+/¨⎕A∘⍳¨⍵:G[↑⍒Z]⋄1∊↑¨⍴¨⍵:'STALEMATE'⋄∇1∘↓¨⍵}G←Z⊂⍨','≠Z←⍞

Explanation (in order of execution):

• ⎕ML←3: set ML to 3 (this makes ⊂ mean partition, among other things).
• G←Z⊂⍨','≠Z←⍞: read input, separate by commas, store in G and pass to the function.
• +/¨⎕A∘⍳¨⍵: calculate the score for each word. (⎕A is a list containing the alphabet.)
• Z←{2>⍴⍕⍵:⍵⋄∇+/⍎¨⍕⍵}¨: calculate the digital root for each score (by summing all digits as long as there is still more than one digit) and store them in Z.
• Z≡∪Z: if all scores are unique...
• :G[↑⍒Z]: ...then output the word with the highest score (from the original list).
• ⋄1∊↑¨⍴¨⍵:'STALEMATE': otherwise (if there's a tie), if one of the words is of length 1, output STALEMATE.
• ⋄∇1∘↓¨⍵: otherwise, take the first letter off each word and run the function again.

Ruby - 210

d,e=(a,b=$<.read.chop.split(/,/)).map{|w|w.bytes.sort}
r=->w,o=65{n=0;w.map{|c|n+=c-o};n>9?r[n.to_s.bytes,48]:n}
d.pop&e.pop while r[d]==r[e]&&d[1]&&e[1]
$><<[[:STALEMATE,a,b][a.size<=>b.size],a,b][r[d]<=>r[e]]

Tests:

$ ruby1.9 1128.rb <<< CAN,BAT
CAN

$ ruby1.9 1128.rb <<< ZOO,NO
NO

$ ruby1.9 1128.rb <<< ZOO,ZOO
STALEMATE