Bilateral monopoly is a market structure in which there is only a single buyer (monopsony) and a single seller (monopoly). Game theory is frequently used when analysing this kind of market structure. Analysing bilateral monopolies becomes relevant when analysing factor markets, specially when analysing the labour market.
Depending on which side has greater negotiation power there can be different outcomes that resemble more or less its two extreme possibilities, as shown in the adjacent figure. The obvious and antagonistic scenarios that can result in this market are monopsony and monopoly. When the demand side holds all the negotiation power we will be dealing with a monopsony-like situation such as m, where price Pm is lower than the monopolist price (PM) and the price of a perfectly competitive market (PPC). When negotiation power is held by the supply side, the monopolistic firm will sell less quantity (QM) than the monopsonist would buy if it had more power (Qm). However, when the bargain power is equally shared between the two sides there may be joint profit maximisation, which can be done by colluding, or even vertical integration may occur if both firms merge, which would get both firms to jointly get an equilibrium corresponding to perfect competition (C). A bilateral monopoly, however, will have worse results for both firms. The quantity sold will be very low (QBC) at a rather low price (PBC).
A bilateral monopoly can also be considered as a firm that has high negotiation power with its clients, which would get the firm to be considered as a monopoly, and high negotiation powers with its suppliers, which would mean the firm is also a monopsony. Consider as an example Standard Oil, back in the days before its breakup. In 1911, the United States Supreme Court ruled that the company was an illegal monopoly. However, Standard Oil could also be considered as a monopsony: being the biggest oil corporation in the US, it had incredible power when negotiating prices with its suppliers when acquiring parts for its refining factories.