How many people in your company have to be convinced a book can sell before you offer a contract?14 1/2.
I'm not sure what the purpose behind this question is. Knowing the answer won't really make a difference in the process as far as the writer is concerned—unless you want to try to bribe everyone who is involved in the decision making process. (That probably won't help much either.)
But to answer your curiosity, this varies a lot between companies, depending on their size. Job positions that have to be on board are Editor, Readers, Marketing, Management. Each of these positions may be held by one single individual in a very small company or there may be committees of editors, readers, marketing people, and managers that have to be in agreement in a large company. Some companies may require a consensus, some only a majority. The opinions of some people on the committees may carry more weight than others. There may be one person at the top of the chain who has veto power, regardless of how many others think it's a good idea; or one person may have the power to push a project through, regardless of how many people think the book won't sell.
Bottom line: the answer to this question won't help you get published. Write a good book, send it to the publisher or editor you think will love it and move on.