Several companies are competing (against/with each other) for the contract/to gain the contract.
几家公司正为争取一项合同而互相竞争.
We're in competition with (ie competing against) several other companies for the contract.
我们与另几家公司角逐争取这项合同.
Several publishers are competing in the same market.
几家出版公司正在同一市场上进行竞争。