What is the one-word antonym of monopoly?
Examples:
The postal service has a monopoly on delivering letters.
AT&T used to have a monopoly on telephone service.
The Smith gang has a monopoly on drug dealing on the south side.
The government has a monopoly on the legal use of violence.
I was thinking about multipoly or polypoly, such as in monotheistic versus polytheistic.
Polypoly (also spelled polyopoly) comes somewhat close to the meaning you seek, but it is much less widely used than monopoly. It occurs mainly in the context of economics, where it describes a situation in which there are many small sellers but no large ones. Polypoly is also sometimes used as a synonym for oligopoly (see Jay's comment on your question).
Because its meaning is liable to be confused in this way, polypoly is not a good general antonym for monopoly. (Nor is multipoly, which native English speakers would probably regard as ‘not a real word.’) In addition to Jay's suggestions given above, you might consider describing the situation as competitive or as one of free competition.
As a side note, you can often get a sense of a word's scarcity by using Google Ngrams, which chart the frequency of a word's use in books over a period of time. This one shows how often monopoly, polypoly, and multipoly occur in books written between 1900 and 2000. You can see that polypoly is extremely rare compared to monopoly, while multipoly is rarer still.
