Basically, anyone who trades that product plays a role in the market width. However, there are rules on each exchange regarding the maximum width that quotes may be. The maximum bid-ask differentials are the same at exchanges that trade options. There are occasions and market situations at the exchanges that may necessitate modification or waiver of the maximum bid-ask differentials.
The U.S. exchanges that list options have rules that specify the maximum bid-ask differentials in option contracts. The members of these exchanges are obligated, under normal circumstances, to honor their displayed quote for a minimum number of contracts. The number of contracts can vary, depending on the stock or index in question, and could be 1, 10, 20, 50 or even 250 contracts.
In general, the market or displayed bid/ask spread could be composed of any variety of participants including market makers, institutional investors or private investors.