Answer:
Me
Step-by-step explanation:
Me fits better then 'I'. If you read it, 'The principle gave Patti as large a prize as [he did] I.' that doesn't make any sense. However if you rewrite it with 'me' then it makes more sense, 'The principle gave Patti as large a prize as [he did] me.' it just sounds better and it comes out more fluent.