"I have created a field in the table (BIDS) to mark if the bid that won the contract."
Include that field in your relationship to filter out all losing bids.
Let's say this field is called "BidWinner" and you've set it to store "Winner" if the bid wins.
Let's say you have a field called ContractID to link Contracts to all the bids on your contract.
Go to Manage | Database | Relationships and use the button with two plus signs to create a new table occurrence of Bids. Name it "WinningBIds".
Define a new field, cBidWInner, in Contracts as a calculation field with "Winner" as the value it returns.
Set up your portal relationship like this:
Contracts::ContractID = WinningBids::ContractID AND
Contracts::cBidWinner = WinningBids::BidWinner