Bilateral oligopoly is a market game where two sides compete strategically. When there are many buyers, it becomes a game of quantity competition for sellers. This game is not usually like the Cournot game. Instead, a different game called the market share game gives the same results. The Cournot game and market share game have the same outcomes only if the price elasticity is one. These findings help us understand when the Cournot game is a good approximation for bilateral oligopoly with many buyers and how total output is ordered when certain conditions are not met.