placing station tokens, operating trains, withholding income or paying dividends, and buying trains. "Certificate limit": there is usually a limit to how many corporate shares and private companies a player may own, to keep the game competitive by preventing snowball effects resulting from early leads by some players. The "President's certificate" (or "Director's certificate") represents control of a railroad corporation, usually represents a greater percentage of corporate stock than other certificates (e.g., 20% as opposed to 10%), and is usually the first one purchased for a company (with its purchaser setting the price, or "par value", for regular shares of stock in many titles in the series). If another player accumulates more shares in a company than the current president, they acquire the President's certificate (with attendant side-effects for both players regarding certificate limits) in exchange for their own lesser shares, and becomes the new controller of the corporation. Certain games may impose restrictions on the order in which companies may be started (generally to impose a historical context upon the game), and they vary in how many shares must be purchased before the company may operate ("floats"). The map is usually a hex grid that depicts cities and terrain features. Hexagonal "track" tiles (representing available land-rights) are laid on top of this map to represent the growth of railroad networks, and tokens are placed on the board to represent stations (as well as special abilities from private companies). Cities have values which can vary based on which tiles have been laid on the city, the phase of the game, their historical importance to a corporation, or type of train used to reach them. Different color tiles are available in succession, and in phases that are typically determined by the first purchase of a more advanced type of train. A company's stock price is adjusted based on the revenues earned and whether the president chose to pay dividends or to withhold the earnings in the company treasury. Stock prices are usually also affected by actions in the stock round, and some games have other mechanisms that affect the stock price. Scarcity (forcing future-turn planning by players) of available corporations, shares thereof, train types, and track tiles. Trains eventually become obsolete and must be replaced by more-expensive trains that also have greater capacity for earning revenue. Purchase of a new type of train usually triggers other phase-changes in the game, such