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Price Competition in Spectrum Markets: How Accurate Is the Continuous Prices Approximation?

Dynamic Spectrum Access technology enables two types of users to operate on a channel- primary users, which have prioritized access to the channel and secondary users, which can use the channel when it is not in use by the primaries. We consider a scenario in which multiple primaries own bandwidth in a large region (e.g., a state), which is divided into smaller locations (e.g., towns). A primary that has unused bandwidth in a time slot would like to lease it out to secondaries at a set of mutually non-interfering locations in return for a fee. This results in price competition among the primaries. In prior work, this price competition has only been studied under the approximation, made for analytical tractability, that the price of each primary takes values from a continuous set. However, in practice, the set of available prices is discrete. In this paper, we investigate the fundamental question of how the behaviour of the players involved in the price competition changes when this continuity assumption is removed. Our analysis reveals several important differences between the games with continuous and discrete price sets. For example, in the game at a single location, no pure strategy Nash equilibrium (NE) exists in the game with continuous price sets, whereas a pure strategy NE may exist in the game with discrete price sets. Also, multiple symmetric NE exist in the game with discrete price sets in contrast to the game with continuous price sets, where a unique NE exists. However, we show that as the number of available prices becomes large in the discrete prices game, the strategies of the primaries under every symmetric NE converge to the unique NE strategy of the game with continuous price sets.