We study competition between wireless devices with incomplete information about their opponents. We model such interactions as Bayesian interference games. Each wireless device selects a power profile over the entire available bandwidth to maximize its data rate. Such competitive models represent situations in which several wireless devices share spectrum without any central authority or coordinated protocol. In contrast to games where devices have complete information about their opponents, we consider scenarios where the devices are unaware of the interference they cause to other devices. Such games, which are modeled as Bayesian games, can exhibit significantly different equilibria. We first consider a simple scenario of simultaneous move games, where we show that the unique Bayes-Nash equilibrium is where both devices spread their power equally across the entire bandwidth. We then extend this model to a two-tiered spectrum sharing case where users act sequentially. Here one of the devices, called the primary user, is the owner of the spectrum and it selects its power profile first. The second device (called the secondary user) then responds by choosing a power profile to maximize its Shannon capacity. In such sequential move games, we show that there exist equilibria in which the primary user obtains a higher data rate by using only a part of the bandwidth. In a repeated Bayesian interference game, we show the existence of reputation effects: an informed primary user can bluff to prevent spectrum usage by a secondary user who suffers from lack of information about the channel gains. The resulting equilibrium can be highly inefficient, suggesting that competitive spectrum sharing is highly suboptimal.
Deep Dive into Competition in Wireless Systems via Bayesian Interference Games.
We study competition between wireless devices with incomplete information about their opponents. We model such interactions as Bayesian interference games. Each wireless device selects a power profile over the entire available bandwidth to maximize its data rate. Such competitive models represent situations in which several wireless devices share spectrum without any central authority or coordinated protocol. In contrast to games where devices have complete information about their opponents, we consider scenarios where the devices are unaware of the interference they cause to other devices. Such games, which are modeled as Bayesian games, can exhibit significantly different equilibria. We first consider a simple scenario of simultaneous move games, where we show that the unique Bayes-Nash equilibrium is where both devices spread their power equally across the entire bandwidth. We then extend this model to a two-tiered spectrum sharing case where users act sequentially. Here one of th
primary user, is the owner of the spectrum and it selects its power profile first. The second device (called the secondary user) then responds by choosing a power profile to maximize its Shannon capacity. In such sequential move games, we show that there exist equilibria in which the primary user obtains a higher data rate by using only a part of the bandwidth.
Our paper is motivated by a scenario where several wireless devices share the same spectrum. Such scenarios are a common occurrence in unlicensed bands such as the ISM and UNII bands. In such bands, diverse technologies such as 802.11, Bluetooth, Wireless USB, and cordless phones compete with each other for the same bandwidth. Usually, these devices have different objectives, they follow different protocols, and they do not cooperate with each other. Indeed, although the FCC is considering wider implementation of “open” spectrum sharing models, one potential undesirable outcome of open spectrum could be a form of the “tragedy of the commons”: self-interested wireless devices destructively interfere with each other, and thus eliminate potential benefits of open spectrum.
Non-cooperative game theory offers a natural framework to model such interactions between competing devices. In [16], the authors studied competition between devices in a Gaussian noise environment as a Gaussian interference (GI) game. This work was extended in [2] for the case of spectrum allocation between wireless devices; the authors provided a non-cooperative game theoretic framework to study issues such as spectral efficiency and fairness. In [9], the authors derived channel gain regimes where cooperative schemes would perform better than non-cooperative schemes for the GI game.
The game theoretic models used in these previous works typically assume that the matrix of channel gains among all users is completely known to the players. This may not be realistic or practical in many scenarios, as competing technologies typically do not employ a coordinated information dissemination protocol. Even if information dissemination protocols were employed, incentive mechanisms would be required in a situation with competitive devices to ensure that channel states were truthfully exchanged. By contrast, our paper studies a range of non-cooperative games characterized by the feature that there is incomplete information about some or all channel gains between devices. Such scenarios are captured through static and dynamic Bayesian games [3].
We consider a simplistic model where two transmitter-receiver (TX-RX) pairs, or “users”, share a single band divided into K subchannels. 1 We assume both users face a total power constraint, and that the noise floor is identical across subchannels. We further assume that channel gains are drawn from a fixed distribution that is common knowledge to the users. We make the simplifying assumption of flat fading, i.e., constant gains across subchannels, to develop the model. A user’s strategic decision consists of an allocation of power across the available subchannels to maximize the available data rate (measured via Shannon capacity).
In Section 2 we consider a simultaneous-move game between the devices under this model. We study two scenarios: first, a game where all channel gains are unknown to both users; and second, a game where a user knows the gain between its own TX-RX pair as well as the interference power gain from the other transmitter at its own receiver (also called incident channel gains), but it does not know the channel gain between the TX-RX pair of the other user or the interference it causes to the other receiver. In these two scenarios, we show that there exists a unique symmetric Bayes-Nash equilibrium 2 , where both users equally spread their power over the band (regardless of the channel gains observed). In this equilibrium, the actions played after channel gains are realized are also a Nash equilibrium of the complete information game.
While simultaneous-move games are a good model for competition between devices with equal priority to shared resources, they are not appropriate for a setting where one device is a natural incumbent, such as primary/secondary device competition. In such two-tiered models for spectrum sharing, some radio bands may be allocated to both primary and secondary users. The primary users have priority over the secondary users and we use game theory to analyze competition in such scenarios. In Section 3, we consider a two-stage sequential Bayesian game where one device (the primary) moves before the other (the secondary); we find that asymmetric equilibria can be sustained where the devices sometimes operate in disjoint subchannels (called “sharing” the bandwidth), provided interference between them is sufficiently large. We also add an entry stage to the game, where the secondary device decides whether or not it wants to operate in the primary’s band in the first place; we also characterize Nash equilibria
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