Short answers, pulled from the story.
A natural monopoly is an industry in which a single firm can supply the entire market at a lower long-run average cost than multiple competing firms could. This typically occurs where fixed infrastructure costs are so large relative to the market that building a second competing network is not economically viable. William Baumol formally defined it in 1977 as an industry where multi-firm production is more costly than production by a monopoly.
John Stuart Mill is credited with developing the concept of natural monopoly, writing before the marginalist revolution in economics. His initial use of the term in Principles of Political Economy referred to skilled labour such as jewellers, physicians, and lawyers, whose higher rewards came from the absence of competition rather than from legal protection. He later applied the term to network industries like water, gas, rail, and electricity.
Railways, electricity, water services, telecommunications, and gas supply are the most common examples of natural monopolies. In each case, the fixed cost of constructing the network (tracks, grids, pipes, poles) is so high that a second firm cannot profitably build a competing system. Mail and mass transportation are also frequently cited.
Governments regulate natural monopolies because an unregulated monopolist can raise prices far above cost for goods consumers cannot refuse, such as water or electricity. The 2000 Cochabamba protests in Bolivia demonstrated the harm: a firm with a monopoly on water supply raised rates sharply to fund dam construction, leaving many residents unable to afford the essential good. Regulation aims to limit abusive pricing and promote investment in the network.
Subadditivity of a cost function means that one firm can produce any given output at lower total cost than any combination of smaller firms producing the same total output. William Baumol tied his 1977 formal definition of natural monopoly to this mathematical condition. It is a more precise test than economies of scale alone, because a firm can satisfy subadditivity even without conventional economies of scale throughout its entire output range.
Cooperative ownership and open-source technology are two alternatives to direct state control. The Depository Trust and Clearing Corporation is an American cooperative that handles the majority of clearing and settlement in the securities industry; its user-owned structure prevents it from exploiting its dominant position. The web's open-source architecture is another example, stimulating large-scale growth while preventing any single company from controlling the entire network.