Price oligopoly as a cooperative game

We consider an oligopolistic market as follows. In the market, one good is traded for money. Each oligopolist is a price setter and has the same linear cost function. Each buyer is a price taker and buys the good from oligopolists setting the lowest price. We formulate this market as a cooperative game, and consider two kinds of solution concepts, the core and a bargaining set of the game. First we show that in the monopolistic market, the core gives the monopoly price, but in the oligopolistic market, the core is empty. Second, we obtain the bargaining set of the oligopolistic market.