# On the Efficiency of the Simplest Pricing Mechanisms in Two-Sided Markets

## Abstract

We study the price of anarchy of a trading mechanism for divisible goods in markets containing both producers and consumers (i.e. in two-sided markets). Each producer is asked to submit a linear pricing function (or, equivalently, a linear supply function) that specifies a per-unit price *p*(*d*) as a function of the demand *d* that they face. Consumers then buy their preferred resource amounts at these prices.

We prove that having three producers for every resource guarantees the price of anarchy is bounded. In general, the price of anarchy depends heavily on the level of horizontal and vertical competition in the market, on the producers’ cost functions, and on the elasticity of consumer demand. We show how these characteristics affect economic efficiency and in particular, we find that the price of anarchy equals 2/3 in a perfectly competitive market, 3/4 in a monopsony, and 2*ε*(2 − *ε*)/(4 − *ε*) in a monopoly where consumer valuations have a fixed elasticity of *ε*. These results hold in markets with multiple goods, particularly in bandwidth markets over arbitrary graphs.

Pricing mechanisms are used in several real-world applications; our results suggest how to add formal efficiency guarantees to these mechanisms. On the theory side, we show that near-optimal efficiency can be achieved within two-sided markets by simple mechanisms in the spirit of Bertrand and Cournot. This result extends to the two-sided setting the analyses for fixed-supply and fixed-demand markets of Johari and Tsitsiklis (2005), Acemoglu and Ozdaglar (2007), and Correa et al. (2010).

## Keywords

Nash Equilibrium Electricity Market Price Mechanism Price Function Arbitrary Graph## Preview

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