When a company goes into an IPO wouldn't they try to make as much money as they can? Then how come the Greenshoe option says that some would try to not issue additional shares just so their share price doesn't raise more money than planned? I mean what kind of situations wouldn't you want to raise more money than keep the share price to the planned price?
You could compare a Greenshoe option to overbooking a plane: airlines tend to sell more tickets than there are seats in the plane in the expectation that some people will not show up. If they do not oversell then the plane will take off partially empty, which makes it more expensive. But if they oversell too much, then there will be many angry passengers because they can't travel.
The underwriters (=airline) oversell the company (=plane), i.e. make short sells such that if investors sell their shares in the aftermarket (=no-show passengers) the price won't drop because the underwriters can close their short sells at a profit. The Greenshoe option serves to protect the underwriters if the prices rise instead they can request additional shares rather than having to close their short sell at a loss at the market. But all this only serves to cover the initial volatility of the price right after the public offering.
Now when a company goes public it usually has plans for that money, e.g. expanding in other markets. Consider the case of a company that wants to get rid of a loan costing them 5% interest a year. Now if the IPO made them exactly the amount needed to pay off the debt they could show 5% extra earnings per share the next year. However, if they raised twice the needed amount from the IPO then they would have only 2.5% extra earnings per share. So raising more money than needed dilutes the shareholder value, because the extra earnings gets spread across more shares - and diluted shareholders are unhappy shareholders.
Furthermore most companies reserve the right to issue additional shares at a later time in case they need the money. Hence there is no need to make as much money as possible on an IPO (unless you're trying to con the investors) - the point is to make as much money as you need.