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 Open Access
Equilibria of channel selection games in parallel multiple access channels
 Samir M Perlaza^{1}Email author,
 Samson Lasaulce^{2} and
 Mérouane Debbah^{3}
https://doi.org/10.1186/16871499201315
© Perlaza et al.; licensee Springer. 2013
 Received: 8 December 2011
 Accepted: 15 November 2012
 Published: 25 January 2013
Abstract
In this article, the parallel multiple access channel is studied under the assumption that transmitters maximize their individual spectral efficiency by selfishly tuning their power allocation policy. Two particular scenarios are studied: (a) transmitters are allowed to use all the available channels; and (b) transmitters are constrained to use a single channel. Both scenarios are modeled by oneshot games and the corresponding sets of Nash equilibria (NE) are fully characterized under the assumption that the receiver treads the multiple access interference as noise. In both cases, the set of NE is nonempty. In the case in which transmitters use a single channel, an upper bound of the cardinality of the NE set is provided in terms of the number of transmitters and number of channels. In particular, it is shown that in fully loaded networks, the sum spectral efficiency at the NE in scenario (a) is at most equal to the sum spectral efficiency at the NE in scenario (b). A formal proof of this observation, known in general as a Braess paradox, is provided in the case of two transmitters and two channels. In general scenarios, we conjecture that the same effect holds as long as the network is kept fully loaded, as shown by numerical examples. Moreover, the price of anarchy and the price of stability in both games are also studied. Interestingly, under certain conditions on the channel gains, Pareto optimality can be achieved at some NE if and only if the number of channels equals or exceeds the number of transmitters. Finally, simulations are presented to verify the theoretical results.
1 Introduction
Multiple access channels (MAC) correspond to a communication scenario where several transmitters communicate with a single receiver through a common channel [1]. In parallel MAC, each transmitter can exploit a common set of orthogonal channels to communicate with the receiver. Often, channel orthogonality is assumed in the frequency domain, and thus, channels can be understood as different nonoverlapping frequency bands. This model allows one to study communication scenarios such as 802.11based wireless local area networks [2, 3], distributed soft or hard handovers in cellular systems [4], or throughputmaximizing power control in multicarrier code division multiple access (MCCDMA) systems [5].
In this article, we analyze the parallel MAC assuming that transmitters selfishly maximize their individual spectral efficiency (ISE) by autonomously selecting a single channel to perform their transmission. Here, the channel selection (CS) policy is not imposed by the receiver to the transmitters, which justifies the terms decentralized parallel MAC. The motivation for studying this scenario and, in particular, the limitation of using a single channel for transmitting stems from the fact that this is often a practical constraint in some wireless networks, for instance, WiFi networks. Moreover, the choice of enforcing radio devices to use only a subset of all the available channels has been proved to be beneficial in the case of rateefficient centralized parallel MAC when using successive interference cancellation [4]. In the case of energyefficient decentralized parallel MAC, it has been shown that using a single channel is a dominant strategy [5]. Within this framework, we study this scenario, to which we refer to as CS problem, using a oneshot game model. The players (the transmitters) have discrete action sets (the channels) and their utility function (performance metric) corresponds to their ISE. Our interest focuses on the analysis of the set of Nash equilibria (NE) [6] of this game. The relevance of the NE relies, in part, on the fact that it describes a network state where the channel used by each transmitter is individually optimal with respect to the channels adopted by all the other transmitters in the network. Another reason is that an NE can be reached in a fully decentralized fashion when radio devices interact during certain time following particular behavioral rules [7, 8].
We distinguish the CS game described above from the power allocation (PA) game. In the PA game, transmitters can simultaneously use all the available channels, and thus, the set of actions is a convex and closed set [9]. Indeed, the PA problem in decentralized parallel MAC has been well investigated in the wireless literature [9–13]. In these works, the main contribution consists in conducting a complete analysis of the set of NE of the corresponding oneshot games. Nonetheless, very few is known about the set of equilibria in decentralized parallel MAC, with CS policies. As we shall see, by dropping the convexity of the set of actions, fundamental differences arise. For instance, the uniqueness of the NE no longer holds.
Although the parallel MAC is, in terms of signal model, a special case of the multipleinput multipleoutput (MIMO) interference channel (IC), the NE analysis of the MIMOIC [14–18] does not directly apply to the case of parallel MAC. First, [10, 11] address the case of fast fading links and shows substantial differences between both models especially in terms of the uniqueness of the NE. Second, even if identical channel variation models are considered in the MIMOIC and parallel MAC, the sufficient condition for the uniqueness of the NE derived in [14, 19] for static or block fading IC can be shown to hold with probability zero in parallel MAC [9]. In particular, the sufficient condition for the uniqueness of the NE provided in [14] is not necessary in general [20]. The (more explicit) sufficient conditions for uniqueness given by Scutari et al. [19] are generally not verified in parallel MAC. More precisely, from [19] it is implied that there exists a unique pure NE with high probability when for each pointtopoint communication the signal dominates the interference. This condition is clearly not verified in parallel MAC. For instance, in a 2user parallel MAC, if one user’s signal is dominated by the interference, the converse holds for the other user. Finally, we highlight that in the parallel MAC, when singleuser decoding (SUD) is assumed at the receiver, both the compact PA and CS games have a special structure, namely, they are potential games [21] or more specifically, bestresponse potential games [22]. This structure is not observed in the case of the MIMOIC model. In Section 3, we discuss the implications and advantages derived from this fact.
Within this context, the main contributions of this article are described hereunder.

The set of NE of the decentralized parallel MAC is fully described in the case of CS policies and SUD at the receiver. This set is shown to be nonempty and an upper bound of its cardinality is provided as a function of the number of transmitters and available channels.

In the 2transmitter 2channel case, it is formally proved that for any realization of the channel gains, there always exists at least one NE in the CS game that produces a higher or equal network spectral efficiency (NSE) than the unique NE in the PA game. In wireless communications, this kind of observations is often associated with a Braesstype paradox [23] as in [20, 24, 25], where similar observations have been made in other scenarios. For an arbitrary number of transmitters and channels, we only provide numerical results that support the aforementioned claim.

The set of NE of the decentralized parallel MAC is also studied in the asymptotic regime, that is for a large number of transmitters in the case of CS policies. In this context, we provide closedform expressions of the fraction of players which transmit over each channel as a function of the ratio between the channel bandwidth and the total available bandwidth (sum of all channel bandwidths).

Finally, we study the efficiency of the set of equilibria in terms of the price of anarchy (PoA) and the price of stability (PoS). We provide closedform conditions over which no loss of performance is observed at the equilibrium in both the PA and the CS games.
The content of this article can briefly be summarized as follows. In Section 3, the system and game models are described. In Section 3, we revisit the existing results regarding the PA problem and we provide new results on the CS problem in terms of existence and uniqueness of the NE. In Section 3, the contribution aforementioned are fully detailed. In Section 3, we present simulation results in order to verify our theoretical results. Finally, the article is concluded by Section 3.
2 Models
2.1 System model
where p_{k,max} is the maximum transmit power of transmitter k. A PA vector for transmitter k is any vector p_{ k } = (p_{k,1}, …, p_{k,S}) with nonnegative entries satisfying (2). The noise vector w is an Sdimensional zero mean Gaussian random variable with independent, equal variance real and imaginary parts. Here, $\ud53c\left(\mathit{w}{\mathit{w}}^{H}\right)=\text{diag}\left({\sigma}_{1}^{2},\dots ,{\sigma}_{S}^{2}\right)$, where ${\sigma}_{s}^{2}$ represents the noise power over channel s. We, respectively, denote the noise spectral density and the bandwidth of channel s by N_{0} and B_{ s }, thus, ${\sigma}_{s}^{2}={N}_{0}{B}_{s}$. The total bandwidth is denoted by $B=\sum _{s=1}^{S}{B}_{s}$.
2.2 Game models
In the sequel, we respectively refer to the games ${\mathcal{G}}_{\left(a\right)}$ and ${\mathcal{G}}_{\left(b\right)}$ as the PA game and CS game. Moreover, we drop the superindices ^{(a)} or ^{(b)} when interchangeably referring to the game ${\mathcal{G}}_{\left(a\right)}$ or ${\mathcal{G}}_{\left(b\right)}$.
and g_{k,s} ≜ h_{k,s}^{2}. Note that from (6), it is implied that SUD is used at the receiver. The choice of SUD is basically due to scalability (in terms of signaling cost) and fairness for the decoding scheme. Clearly, optimality is not sought here, nonetheless, these constraints are inherit to the decentralized nature of the network.
As a solution concept for both ${\mathcal{G}}_{\left(a\right)}$ and ${\mathcal{G}}_{\left(b\right)}$, we focus on the notion of NE [6], which we define, using our notation, as follows
Definition 1 (Pure NE)
The relevance of the NE is that at such state, the PA or CS policy chosen by any transmitter is optimal with respect to the choice of all the other transmitters. Thus, in a decentralized network, the NE is a stable network state, since no player has a particular interest in deviating from the actual state.
In the following, we provide some fundamental results which we use in the further analysis of the games ${\mathcal{G}}_{\left(a\right)}$ and ${\mathcal{G}}_{\left(b\right)}$.
3 Auxiliary results
In this section, we introduce some existing results on the existence and uniqueness of the NE in the games ${\mathcal{G}}_{\left(a\right)}$ and ${\mathcal{G}}_{\left(b\right)}$. For doing so, we use the fact that both games ${\mathcal{G}}_{\left(a\right)}$ and ${\mathcal{G}}_{\left(b\right)}$ have been shown to be potential games (PG) [21, 28] in [4, 29], respectively. We conclude this section by introducing a new result which allows us to establish an upper bound on the number of NE that the game ${\mathcal{G}}_{\left(b\right)}$ might possess.
3.1 General results
The analysis presented in the following strongly relies on the fact that both games ${\mathcal{G}}_{\left(a\right)}$ and ${\mathcal{G}}_{\left(b\right)}$ are potential games. Thus, for the sake of completeness, we define exact PGs using our notation.
Definition 2 (Exact PG)
Interestingly, the first term ϕ(p) does not depend on the index of k and the term υ_{ k }(p_{−k}) does not depend on the actions of player k. Thus, the results initially mentioned by the authors of [4, 29] follow immediately from (8) and Definition 2.
Lemma 1
The strategic form games${\mathcal{G}}_{\left(i\right)}$, with i ∈{a, b}, are exact potential games with potential function$\varphi \left(\mathit{p}\right):{\mathcal{P}}^{\left(i\right)}\to {\mathbb{R}}_{+}$defined in (9).
The relevance of PG relies on the fact that it is a class of games for which the existence of at least one pure NE is always guaranteed [21]. In addition, many known learning procedures, such as, best response dynamics, fictitious play, and some reinforcement learning dynamics converge in PG. As a consequence, any of these dynamics can be used to implement algorithms to achieve an equilibrium in a fully decentralized fashion. Nonetheless, we leave the design of decentralized techniques for achieving NE out of the scope of this article and we focus on the analysis of the equilibria. We refer the interested readers to [7, 8, 13, 14, 30] for more details. In the following, we use Lemma 1 to analyze the set of NE of both our PA game ${\mathcal{G}}_{\left(a\right)}$ and our CS game ${\mathcal{G}}_{\left(b\right)}$.
3.2 Known results concerning the PA game ${\mathcal{G}}_{\mathbf{\left(}\mathbf{a}\mathbf{\right)}}$
In the following, we comment on the existence and uniqueness of the NE in ${\mathcal{G}}_{\left(a\right)}$.
3.2.1 Existence of an NE
Regarding the existence of pure NE, the following lemma is an immediate consequence of our Lemma 1 and Lemma 4.3 in [21].
Lemma 2 (Existence of a pure NE)
The game ${\mathcal{G}}_{\left(a\right)}$ has always at least one NE in pure strategies.
Regarding the existence of a mixed NE (i.e., a probability distribution on the possible actions which verifies Definition 1), it follows from [31], that the existence of at least one NE in mixed strategies always exists. This is basically because the action spaces ${\mathcal{P}}_{k}^{\left(a\right)}$ are compact spaces and the utility functions are continuous with respect to the action profile. However, in compact strategy spaces, mixed strategies are generally less attractive due to the difficulty of its implementation in wireless communications systems [8].
3.2.2 Uniqueness of the pure NE
In the game ${\mathcal{G}}_{\left(a\right)}$, the uniqueness of the NE has been shown to hold with probability one [9].
Theorem 1 (NE uniqueness in parallel MAC)
The game ${\mathcal{G}}_{\left(a\right)}$ has almost surely a unique pure NE.
A formal proof of Theorem 1 is provided in [9]. This proof is based on the concept of degeneracy which allows one to characterize the directions along which the potential remains constant. A simpler proof for the special case of 2players and 2channels is given in [32].
3.2.3 Determination of the NE
where β_{ k } is a Lagrangian multiplier chosen to saturate the power constraints (2). Note that this result shows the connections between the notion of NE and the wellknown waterfilling PA policy [13]. For a further discussion on this connection, the reader is referred to [7, 8].
3.3 New results concerning the CS game ${\mathcal{G}}_{\mathbf{\left(}\mathbf{b}\mathbf{\right)}}$
In the game ${\mathcal{G}}_{\left(b\right)}$, it can be checked that given a vector ${\mathit{p}}_{k}\in {\mathcal{P}}_{k}^{\left(b\right)}$, it follows that $\forall k\in \mathcal{K}$ and ∀p_{ k } ∈ [0, p_{k,max}], the potential function satisfies that ϕ(p_{ k }e_{ s },p_{−k}) ⩽ ϕ(p_{k,max}e_{ s },p_{−k}), where ${\mathit{e}}_{s}=\left({e}_{s,1},\dots ,{e}_{s,S}\right)\in {\mathbb{R}}^{S}$, $\forall r\in \mathcal{S}\setminus s$, e_{s,r} = 0, and e_{s,s} = 1. Thus, the problem of transmit power control disappears and there is no lost of generality by choosing the action sets as ${\mathcal{P}}_{k}^{\left(b\right)}$. Technically, the main difference between ${\mathcal{G}}_{\left(a\right)}$ and ${\mathcal{G}}_{\left(b\right)}$ is that the latter is a finite game ($\mathcal{K}\times \mathcal{S}<+\infty $). In the following, we investigate the consequences of this fact on the existence and multiplicity of the NE.
3.3.1 Existence of a pure NE
Regarding the existence of at least one NE in pure strategies, we have that from our Lemma 1 and Corollary 2.2 of [21], the following lemma holds.
Lemma 3 (Existence of a pure NE)
The game ${\mathcal{G}}_{\left(b\right)}$ has always at least one NE in pure strategies.
Regarding the existence of an equilibrium in mixed strategies, we have that given that the actions sets are discrete and finite, then the existence of at least one NE in mixed strategies is ensured [6].
Multiplicity of the pure NE
In the following, we show that multiple NE might exists in the game ${\mathcal{G}}_{\left(b\right)}$.
Proposition 1
The proof of Proposition 1 is given in Appendix. This proof relies on the fact that whenever a player deviates from a given action profile p, it achieves a different utility. That is, given a set of channels ${\left\{{g}_{\mathit{\text{ij}}}\right\}}_{\forall (i,j)\in \mathcal{K}\times \mathcal{S}}$, it holds that u_{ k }(p) ≠ u_{ k }(p^{′}), where p^{′} is obtained by letting only one transmitter to deviate from p. This holds from the fact that the channel realizations are drawn from a continuous distribution and thus, Pr(u_{ k }(p) = u_{ k }(p^{′})p ≠ p^{′}) = 0. Thus, under this condition, if p is an NE, then p^{′} is not and vice versa. Following this procedure, a set of action profiles can be eliminated and only some actions profiles which are NE candidates are left over. In this way, a bound on the maximum number of equilibria can be provided independently of the exact channel realizations and only relying on the parameters K and S. Indeed, this is one of the main properties of the bound provided by Proposition 1 since, a calculation of the exact number of NE becomes cumbersome when the number of transmitters K and channels S grows to infinity. As we shall see, the bound in Proposition 1 holds with strict equality in the case of K = 2 transmitters and S = 2 channels, that is, 1 ⩽ L ⩽ 2. For an arbitrary number of transmitters K > 2 and channels S > 2, the bound is not tight. This is basically because some of the action profiles obtained using the elimination processes described before might mutually be exclusive of the set of NE. For instance, in the game ${\mathcal{G}}_{\left(b\right)}$ with K = 3 and S = 2, the set of PA vectors p_{max}(e_{1}, e_{1}, e_{1}), p_{max}(e_{1}, e_{2}, e_{2}), p_{max}(e_{2}, e_{2}, e_{1}), and p_{max}(e_{2}, e_{1}, e_{2}) might all be NE candidates. Nonetheless, if p_{max}(e_{1}, e_{1}, e_{1}) is an equilibrium for a given vector of channels ${\left\{{g}_{\mathit{\text{ij}}}\right\}}_{\forall (i,j)\in \mathcal{K}\times \mathcal{S}}$, then the other three action sets are not NE for the same set of channels and vice versa. Thus, only three out of the four candidates can be NE simultaneously. The exact number of NE can be determined following the method described in the following section, but it requires the complete knowledge of the channel gains. Proposition 1 aims at providing an estimation based only on the parameters K and S.
3.3.3 Determination of the NE
From the definition of the matrix $\widehat{\mathit{A}}$, we have that a necessary and sufficient condition for a vertex v_{ i } to represent an NE action profile is to have a null outdegree in the oriented graph $\u011c$, i.e., there are no outgoing edges from the node v_{ i } (sink vertex). Finally, one can conclude that determining the set of NE in the game ${\mathcal{G}}_{\left(b\right)}$ boils down to identifying all the sink vertices in the oriented graph $\u011c$. Note that this method can be used only to determine the whole set of NE. It does not pretend to be an algorithm which can be directly implemented in decentralized wireless networks, since it requires complete information at each transmitter. Methods for achieving equilibria in wireless networks are described in [7, 8, 13, 14, 30].
4 Equilibrium performance analysis and special cases
From this analysis, we conclude that limiting the transmitters to use a single channel brings a better result in terms of NSE (15). Second, we consider the case of a large number of transmitters. This study leads to two important conclusions: (i) the fraction of players using a given channel depends mainly on the bandwidth of each channel and not on the exact channel realization nor the number of players and channels; (ii) in the asymptotic regime (K → ∞) both games exhibit the same performance. Before we start, let us introduce the notion of best response correspondence, since it plays a central role in the following analysis.
Definition 3 (Bestresponse correspondence)
is defined as the bestresponse correspondence of player$k\in \mathcal{K}$, given the actions p_{−k}adopted by all the other players.
4.1 The 2transmitter 2channel case
Consider the games ${\mathcal{G}}_{\left(a\right)}$ and ${\mathcal{G}}_{\left(b\right)}$ with K = 2 and S = 2. Assume also that $\forall k\in \mathcal{K}$, p_{k,max} = p_{max} and $\forall s\in \mathcal{S}$, ${\sigma}_{s}^{2}={\sigma}^{2}$ and ${B}_{s}=\frac{B}{S}$. Denote by $\text{SNR}=\frac{{p}_{\text{max}}}{{\sigma}^{2}}$ the average signaltonoise ratio (SNR) of each active communication.
4.1.1 The PA game
Note that, for all $k\in \mathcal{K}$ and for all ${\mathit{p}}_{k}\in {\mathcal{P}}^{\left(a\right)}$, the set BR_{ k }(p_{−k}) is a singleton (Definition 3) and thus, (17) represents a system of equations. By solving the resulting system of equations (17) for a given realization of the channels ${\left\{{g}_{\mathit{\text{ij}}}\right\}}_{\forall (i,j)\in \mathcal{K}\times \mathcal{S}}$, one can determine the NE of the game ${\mathcal{G}}_{\left(a\right)}$. We present such a solution in the following proposition.
Proposition 2 (NE in${\mathcal{G}}_{\mathbf{\left(}\mathit{a}\mathbf{\right)}}$)
Let the action profile${\mathit{p}}^{\u2020}=\left({\mathit{p}}_{1}^{\u2020},{\mathit{p}}_{2}^{\u2020}\right)\in {\mathcal{P}}^{\left(a\right)}$, with${\mathit{p}}_{1}^{\u2020}=\left({p}_{11}^{\u2020},{p}_{\text{max}}{p}_{11}^{\u2020}\right)$and${\mathit{p}}_{2}^{\u2020}=\left({p}_{\text{max}}{p}_{22}^{\u2020},{p}_{22}^{\u2020}\right)$be an NE action profile of the game${\mathcal{G}}_{\left(a\right)}$with channel realizations g = (g_{11}, g_{12}, g_{21}, g_{22}). Then, with probability one, p^{ † }is the unique NE and it can be written as follows:

Equilibrium 1: if$\mathit{g}\in {\mathcal{A}}_{1}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\left\{\mathit{g}\in {\mathbb{R}}_{+}^{4}:\frac{{g}_{11}}{{g}_{12}}\phantom{\rule{0.3em}{0ex}}\u2a7e\phantom{\rule{0.3em}{0ex}}\frac{1+\text{SNR}{g}_{11}}{1+\text{SNR}{g}_{22}},\phantom{\rule{0.3em}{0ex}}\frac{{g}_{21}}{{g}_{22}}\phantom{\rule{0.3em}{0ex}}\u2a7d\phantom{\rule{0.3em}{0ex}}\frac{1+\text{SNR}{g}_{11}}{1+\text{SNR}{g}_{22}}\right\},$then,${p}_{11}^{\u2020}={p}_{\text{max}}$and${p}_{22}^{\u2020}={p}_{\text{max}}$.

Equilibrium 2: if$\mathit{g}\in {\mathcal{A}}_{2}=\left\{\mathit{g}\in {\mathbb{R}}_{+}^{4}:\frac{{g}_{11}}{{g}_{12}}\u2a7e1+\text{SNR}\left({g}_{11}+{g}_{21}\right),\phantom{\rule{0.3em}{0ex}}\frac{{g}_{21}}{{g}_{22}}\u2a7e1+\text{SNR}\left({g}_{11}+{g}_{21}\right)\right\}$, then,${p}_{11}^{\u2020}={p}_{\text{max}}$and${p}_{22}^{\u2020}=0$.

Equilibrium 3: if$\mathit{g}\in {\mathcal{A}}_{3}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\left\{\mathit{g}\in {\mathbb{R}}_{+}^{4}:\frac{{g}_{11}}{{g}_{12}}\phantom{\rule{0.3em}{0ex}}\u2a7d\phantom{\rule{0.3em}{0ex}}\frac{1}{1+\text{SNR}\left({g}_{12}+{g}_{22}\right)},\phantom{\rule{0.3em}{0ex}}\frac{{g}_{21}}{{g}_{22}}\phantom{\rule{0.3em}{0ex}}\u2a7d\phantom{\rule{0.3em}{0ex}}\frac{1}{1+\text{SNR}\left({g}_{12}+{g}_{22}\right)}\right\}$then,${p}_{11}^{\u2020}=0$and${p}_{22}^{\u2020}={p}_{\text{max}}$.

Equilibrium 4: if$\mathit{g}\in {\mathcal{A}}_{4}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\left\{\mathit{g}\in {\mathbb{R}}_{+}^{4}:\frac{{g}_{11}}{{g}_{12}}\phantom{\rule{0.3em}{0ex}}\u2a7d\phantom{\rule{0.3em}{0ex}}\frac{1+\text{SNR}{g}_{21}}{1+\text{SNR}{g}_{12}},\phantom{\rule{0.3em}{0ex}}\frac{{g}_{21}}{{g}_{22}}\u2a7e\frac{1+\text{SNR}{g}_{21}}{1+\text{SNR}{g}_{12}}\right\}$, then,${p}_{11}^{\u2020}=0$and${p}_{22}^{\u2020}=0$.

Equilibrium 5: if$\mathit{g}\in {\mathcal{A}}_{5}=\left\{\mathit{g}\in {\mathbb{R}}_{+}^{4}:\frac{{g}_{11}}{{g}_{12}}\u2a7e\frac{{g}_{21}}{{g}_{22}},\phantom{\rule{0.3em}{0ex}}\frac{1+\text{SNR}{g}_{11}}{1+\text{SNR}{g}_{22}}<\frac{{g}_{21}}{{g}_{22}}<1+\text{SNR}\left({g}_{11}+{g}_{21}\right)\right\}$, then,${p}_{11}^{\u2020}={p}_{\text{max}}$and${p}_{22}^{\u2020}=\frac{1}{2}\left({p}_{\text{max}}\frac{{\sigma}^{2}}{{g}_{22}}+\frac{{\sigma}^{2}+{g}_{11}{p}_{\text{max}}}{{g}_{21}}\right)$.

Equilibrium 6: if$\mathit{g}\in {\mathcal{A}}_{6}=\left\{\mathit{g}\in {\mathbb{R}}_{+}^{4}:\frac{{g}_{11}}{{g}_{12}}\u2a7e\frac{{g}_{21}}{{g}_{22}},\phantom{\rule{0.3em}{0ex}}\frac{1}{1+\text{SNR}\left({g}_{12}+{g}_{22}\right)}<\frac{{g}_{11}}{{g}_{12}}<\frac{1+\text{SNR}{g}_{11}}{1+\text{SNR}{g}_{22}}\right\}$, then,${p}_{11}^{\u2020}=\frac{1}{2}\left({p}_{\text{max}}\frac{{\sigma}^{2}}{{g}_{11}}+\frac{{\sigma}^{2}+{p}_{\text{max}}{g}_{22}}{{g}_{12}}\right)$and${p}_{22}^{\u2020}={p}_{\text{max}}$.

Equilibrium 7: if$\mathit{g}\in {\mathcal{A}}_{7}=\left\{\mathit{g}\in {\mathbb{R}}_{+}^{4}:\frac{{g}_{11}}{{g}_{12}}\u2a7d\frac{{g}_{21}}{{g}_{22}},\phantom{\rule{0.3em}{0ex}}\frac{1+\text{SNR}{g}_{21}}{1+\text{SNR}{g}_{12}}<\frac{{g}_{11}}{{g}_{12}}<1+\text{SNR}\left({g}_{11}+{g}_{21}\right)\right\}$, then,${p}_{11}^{\u2020}=\frac{1}{2}\left({p}_{\text{max}}\frac{{\sigma}^{2}+{p}_{\text{max}}{g}_{21}}{{g}_{11}}+\frac{{\sigma}^{2}}{{g}_{12}}\right)$and${p}_{22}^{\u2020}=0$.

Equilibrium 8: if$\mathit{g}\in {\mathcal{A}}_{8}=\left\{\mathit{g}\in {\mathbb{R}}_{+}^{4}:\frac{{g}_{11}}{{g}_{12}}\u2a7d\frac{{g}_{21}}{{g}_{22}},\phantom{\rule{0.3em}{0ex}}\frac{1}{1+\text{SNR}\left({g}_{12}+{g}_{22}\right)}<\frac{{g}_{21}}{{g}_{22}}<\frac{1+\text{SNR}{g}_{21}}{1+\text{SNR}{g}_{12}}\right\}$, then,${p}_{11}^{\u2020}=0$and${p}_{22}^{\u2020}=\frac{1}{2}\left({p}_{\text{max}}\frac{{\sigma}^{2}+{g}_{12}{p}_{\text{max}}}{{g}_{22}}+\frac{{\sigma}^{2}}{{g}_{21}}\right)$.
The proof of Proposition 2 is given in Appendix.
Lemma 4
is an NE action profile of the game${\mathcal{G}}_{\left(a\right)}$.
The proof of Lemma 4 is the first part of the proof of Proposition 2. In the following section, we perform the same analysis presented above for the game ${\mathcal{G}}_{\left(b\right)}$.
4.1.2 The CS game
Potential function ϕ of the game ${\mathcal{G}}_{\mathbf{\left(}\mathit{a}\mathbf{\right)}}$ , with K = 2 and S = 2
${T}_{{x}_{1}}\setminus {T}_{{x}_{2}}$  p_{2} = (p_{max}, 0)  p_{2} = (0, p_{max}) 

p_{1} = (p_{max}, 0)  $\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\begin{array}{c}\frac{1}{2}{log}_{2}\left({\sigma}^{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{p}_{\text{max}}({g}_{11}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{g}_{21})\right)\\ +\phantom{\rule{0.3em}{0ex}}\frac{1}{2}{log}_{2}\left({\sigma}^{2}\right)\end{array}$  $\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\begin{array}{c}\frac{1}{2}{log}_{2}\left({\sigma}^{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{p}_{\text{max}}{g}_{11}\right)\\ +\phantom{\rule{0.3em}{0ex}}\frac{1}{2}{log}_{2}\left({\sigma}^{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{p}_{\text{max}}{g}_{22}\right)\end{array}$ 
p_{1} = (0, p_{max})  $\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\begin{array}{c}\frac{1}{2}{log}_{2}\left({\sigma}^{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{p}_{\text{max}}{g}_{12}\right)\\ +\phantom{\rule{0.3em}{0ex}}\frac{1}{2}{log}_{2}\left({\sigma}^{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{p}_{\text{max}}{g}_{21}\right)\end{array}$  $\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\begin{array}{c}\frac{1}{2}\phantom{\rule{0.3em}{0ex}}{log}_{2}\phantom{\rule{0.3em}{0ex}}\left({\sigma}^{2}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{p}_{\text{max}}({g}_{12}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{g}_{22})\phantom{\rule{0.3em}{0ex}}\right)\\ +\phantom{\rule{0.3em}{0ex}}\frac{1}{2}{log}_{2}\left({\sigma}^{2}\right)\end{array}$ 
Proposition 3 (NE in${\mathcal{G}}_{\mathbf{\left(}\mathit{b}\mathbf{\right)}}$)
Let the PA vector${\mathit{p}}^{\ast}=\left({\mathit{p}}_{1}^{\ast},{\mathit{p}}_{2}^{\ast}\right)\in {\mathcal{P}}^{\left(b\right)}$be one NE in the game${\mathcal{G}}_{\left(b\right)}$. Then, depending on the channel gains g = (g_{11}, g_{12}, g_{21}, g_{22}), the NE p^{∗}can be written as follows:

Equilibrium 1: when$\mathit{g}\in {\mathcal{B}}_{1}=\left\{\mathit{g}\in {\mathbb{R}}_{+}^{4}:\frac{{g}_{11}}{{g}_{12}}\u2a7e\frac{1}{1+\text{SNR}{g}_{22}}\mathit{\text{and}}\frac{{g}_{21}}{{g}_{22}}\u2a7d1+\mathit{\text{SNR}}{g}_{11}\right\}$, then,${\mathit{p}}_{1}^{\ast}=\left({p}_{\text{max}},0\right)$and${\mathit{p}}_{2}^{\ast}=\left(0,{p}_{\text{max}}\right)$.

Equilibrium 2: When$\mathit{g}\in {\mathcal{B}}_{2}=\left\{\mathit{g}\in {\mathbb{R}}_{+}^{4}:\frac{{g}_{11}}{{g}_{12}}\u2a7e1+\mathit{\text{SNR}}{g}_{21}\mathit{\text{and}}\frac{{g}_{21}}{{g}_{22}}\u2a7e1+\mathit{\text{SNR}}{g}_{11}\phantom{\rule{2.77695pt}{0ex}}\right\}$, then,${\mathit{p}}_{1}^{\ast}=\left({p}_{\text{max}},0\right)$and${\mathit{p}}_{2}^{\ast}=\left({p}_{\text{max}},0\right)$.

Equilibrium 3: when$\mathit{g}\in {\mathcal{B}}_{3}=\left\{\mathit{g}\in {\mathbb{R}}_{+}^{4}:\frac{{g}_{11}}{{g}_{12}}\u2a7d\frac{1}{1+\text{SNR}{g}_{22}}\mathit{\text{and}}\frac{{g}_{21}}{{g}_{22}}\u2a7d\frac{1}{1+\mathit{\text{SNR}}{g}_{12}}\right\}$, then,${\mathit{p}}_{1}^{\ast}=\left(0,{p}_{\text{max}}\right)$and${\mathit{p}}_{2}^{\ast}=\left(0,{p}_{\text{max}}\right)$.

Equilibrium 4: when$\mathit{g}\in {\mathcal{B}}_{4}=\left\{\mathit{g}\in {\mathbb{R}}_{+}^{4}:\frac{{g}_{11}}{{g}_{12}}\u2a7d1+\mathit{\text{SNR}}{g}_{12}\mathit{\text{and}}\frac{{g}_{21}}{{g}_{22}}\u2a7e\frac{1}{1+\mathit{\text{SNR}}{g}_{12}}\phantom{\rule{2.77695pt}{0ex}}\right\}$, then,${\mathit{p}}_{1}^{\ast}=\left(0,{p}_{\text{max}}\right)$and${\mathit{p}}_{2}^{\ast}=\left({p}_{\text{max}},0\right)$.
This confirms the fact that several NE might exist in the game ${\mathcal{G}}_{\left(b\right)}$ depending on the exact channel realization, as stated in Proposition 1. Moreover, one can also observe that there might exist an NE action profile which is not a global maximizer of the potential function (9) [33] (e.g., $\varphi \left({\mathit{p}}^{\ast}\right)<\varphi \left({\mathit{p}}_{2}^{+}\right)$).
In the sequel, the performance achieved by the transmitters at the equilibrium in both games are compared.
4.2 A Braesstype paradox
As suggested in [23], a Braesstype paradox refers to a counterintuitive observation consisting in a reduction of the individual utility at the equilibria, when the players are granted with a larger set of actions. That is, by letting the players to choose from a larger set of options, their individual benefit reduces. Recently, the Braesstype paradox has been also associated with the reduction of the sumutility instead of the individual utilities, as in [20, 25, 34] in the wireless communications arena.
In our particular case, the set of actions for player k in the game ${\mathcal{G}}_{\left(b\right)}$ is a subset of its set of actions in the game ${\mathcal{G}}_{\left(a\right)}$, i.e., $\forall k\in \mathcal{K}$, ${\mathcal{P}}_{k}^{\left(b\right)}\subseteq {\mathcal{P}}_{k}^{\left(a\right)}$. Interestingly, as observed in [23], reducing the set of actions of each player leads, in this particular game, to a better global performance. This effect has been reported in the parallel interference channel in [24, 25] under the consideration of particular channel conditions and later, more generally in [20]. However, a formal proof of the existence of this paradox is not provided in the aforementioned references. This observation has also been reported in the parallel MAC for the case of successive interference cancellation (SIC) in [35]. Nonetheless, the channel in [35] was not fully decentralized, as it required a central controller to dictate the channel policy to all the transmitters. In the following, we study this observation in more detail.
Let us denote by ${\mathit{p}}_{k}^{(\u2020,n)}$ the unique NE action profile of game ${\mathcal{G}}_{\left(a\right)}$, when the vector $\mathit{g}=\left({g}_{11},{g}_{12},{g}_{21},{g}_{22}\right)\in {\mathcal{A}}_{n}$, for all n ∈ {1 …, 8}. Let us also denote by p^{(∗,n)} one of the NE action profiles of game ${\mathcal{G}}_{\left(b\right)}$ when $\left({g}_{11},{g}_{12},{g}_{21},{g}_{22}\right)\in {\mathcal{B}}_{n}$, for all n ∈cv{1, … 4}. The sets ${\mathcal{A}}_{n}$ and ${\mathcal{B}}_{n}$ are defined in Propositions 2 and 3. Then, for a finite SNR level, one can observe that $\forall n\in \{1,\dots ,4\},{\mathcal{A}}_{n}\cap {\mathcal{B}}_{n}={\mathcal{A}}_{n}$ and $\forall \mathit{g}=\left({g}_{11},{g}_{12},{g}_{21},{g}_{22}\right)\in {\mathcal{A}}_{n}$, the following equality always holds ${\mathit{p}}_{k}^{(\u2020,n)}={\mathit{p}}_{k}^{(\ast ,n)}$, which implies the same network performance. However, when the NE of both games are different, one cannot easily compare the utilities achieved by each player since they depend on the exact channel realizations. Fortunately, the analysis largely simplifies by considering either a low SNR regime or a high SNR regime and more general conclusions can be stated. The performance comparison between games ${\mathcal{G}}_{\left(a\right)}$ and ${\mathcal{G}}_{\left(b\right)}$ for the low SNR regime is presented in the following proposition.
Proposition 4
The proof of Proposition 4 is given in Appendix. From, Proposition 4, it can be stated that at the low SINR regime, players achieve the NE by simply choosing the channel with the highest channel gain independently of the other player’s action. The performance comparison between games ${\mathcal{G}}_{\left(a\right)}$ and ${\mathcal{G}}_{\left(b\right)}$ for the high SNR regime is presented in the following proposition.
Proposition 5
and$\delta \u2a7e0$.
The proof of Proposition 5 is given in Appendix. From the proof of Proposition 5, it can be stated that in none of the games, players transmit simultaneously on the same channels. Now, from Propositions 4 and 5, it can be concluded that at low SNR both games ${\mathcal{G}}_{\left(a\right)}$ and ${\mathcal{G}}_{\left(b\right)}$ induce the same NSE. On the contrary, the game ${\mathcal{G}}_{\left(b\right)}$ always induces a higher or equal NSE than the game ${\mathcal{G}}_{\left(a\right)}$ in the high SNR regime. This counterintuitive result implies a Braesstype paradox, as suggested in the beginning of this section.
The case of large systems
In this section, we deal with the games ${\mathcal{G}}_{\left(b\right)}$ for the case of large networks, i.e., networks with a large number of transmitters. Within this scenario, the dominant parameter to analyze these games is the fraction of transmitters using a particular channel. As we shall see, contrary to the case of small number of transmitters and channels analyzed in the previous section, in the case of large networks, each player becomes indifferent to the action adopted by each of the other players. Here, each player is rather concerned with the fractions of players simultaneously playing the same action, i.e., using the same channel. Hence, one of the interesting issues to be solved is the determination of the repartition of the users between the different channels at the NE.
The above provides a very simple relation between the repartition of the users among the available channels in the asymptotic regime. Indeed, it can be implied that the number of transmitters using a given channel s is proportional to the bandwidth allocated to the corresponding channel. In particular, note that this result generalizes the work in [36].
Although the number of pure NE in the CS game might be conjectured to be large, it is in fact relatively small in the asymptotic regime. Indeed, (28) indicates that when the number of users is large, the fraction of pure NE goes to zero whenever the number of channels is greater or equal to 3. This result shows the difficulty of using methodologies such as the one proposed in Section 3 to study the set of NE or the difficulty of achieving equilibria using decentralized learning algorithms as proposed in [7, 8].
PoA and PoS
where the equality in (35) holds since the maximizers of the function ϕ are the set of NE as shown in Section 3.
respectively.
Here, strict equality holds for all $k\in \mathcal{K}$, only when transmitters do not share the same channels with other transmitters. In the following, we use this reasoning to study the PoA and the PoS of both games ${\mathcal{G}}_{\left(a\right)}$ and ${\mathcal{G}}_{\left(b\right)}$.
PoA and PoS in ${\mathcal{G}}_{\left(a\right)}$
Following Proposition 2, it becomes clear that strict equality in (42) is observed at the equilibrium of the game ${\mathcal{G}}_{\left(a\right)}$, if and only if, transmitters use different channels to communicate with the receiver. In the case in which K = 2 and S = 2, according to Proposition 2, such condition is fulfilled when g = (g_{11}, g_{12}, g_{21}, g_{22}) satisfies either $\mathit{g}\in {\mathcal{A}}_{1}$ or $\mathit{g}\in {\mathcal{A}}_{4}$. This observation leads us to the following lemma.
Lemma 5 (PoA and PoS of${\mathcal{G}}_{\mathbf{\left(}\mathit{a}\mathbf{\right)}}$)
with strict inequality.
From Lemma 5, the following corollaries follow by taking the limits SNR→0 and SNR→∞, respectively.
Corollary 1 (PoA in the low SNR regime)
with ϵ > 0 arbitrarily small.
Corollary 2 (PoA in the high SNR regime)
In a more general case, for instance K > 2 and S > 2, we state the following result:
Proposition 6 (PoA of the${\mathcal{G}}_{\mathbf{\left(}\mathit{a}\mathbf{\right)}}$)
Note that from Proposition 6, it can be concluded that in the case in which K > S, the PoA is strictly bigger than one, since the condition (57) does not hold. Thus, as long as K > S there always exists a loss in performance due to the decentralization of the network. Nonetheless, depending on the channel realizations, in the case in which K ⩽ S, global optimal performance can be achieved at the equilibrium.
PoA and PoS in ${\mathcal{G}}_{\left(b\right)}$
with strict inequality for any channel realization in (61). This is basically because two different NE action profiles generate the same sum utility with probability zero, when the channel gains are generated following a continuous distribution. Thus, the numerator is monotonically increasing while the denominator is monotonically decreasing in (40). On the contrary, in the case of the PoS (62), equality can be observed whenever p^{(‡, b)} = p^{(⊤, b)} and (33) holds with equality, which is the case when there exists an equilibrium where every transmitter uses a different channel from all the other transmitters.
Following the same arguments as in the case of the game ${\mathcal{G}}_{\left(a\right)}$, we state that a unitary PoA (PoA=1) is observed only when the set of NE is unitary and transmitters do not share the same channels to transmit. This condition according with Proposition 3 is observed in the game ${\mathcal{G}}_{\left(b\right)}$, with K = 2 and S = 2, when g = (g_{11}, g_{12}, g_{21}, g_{22}) satisfies that $\mathit{g}\in \{{\mathcal{B}}_{1}\cup {\mathcal{B}}_{4}\}\setminus {\mathcal{\mathcal{B}}}_{5}$. This observation leads us to the following lemma.
Lemma 6 (PoA of the game${\mathcal{G}}_{\left(b\right)}$)
with strict inequality.
In the case of the PoS, we highlight that strict equality holds in (42) when transmitters use different channels to communicate with the receiver at the equilibrium. This condition is satisfied in the game ${\mathcal{G}}_{\left(b\right)}$, with K = 2 and S = 2, when g = (g_{11}, g_{12}, g_{21}, g_{22}) satisfies either $\mathit{g}\in {\mathcal{B}}_{1}$ or $\mathit{g}\in {\mathcal{B}}_{4}$. Thus, we write the following lemma.
Lemma 7 (PoS of the game${\mathcal{G}}_{\mathbf{\left(}\mathit{b}\mathbf{\right)}}$)
with strict inequality.
From Lemmas 6 and 7, the following corollaries follow by taking the limits SNR→0 and SNR→∞, respectively.
Corollary 3 (PoA and PoS in the low SNR regime)
with ϵ > 0 arbitrarily small.
Corollary 4 (PoA and PoS in the high SNR regime)
In Corollary 4, (77) follows from the fact that at high SNR, the game ${\mathcal{G}}_{\left(b\right)}$, with K = 2 and S = 2, always possesses two equilibria in pure strategies (Lemma 8). Thus, at the high SNR regime, the numerator of (29) arbitrarily increases while its denominator arbitrarily decreases. Similarly, at the high SNR regime both equilibria consist in action profiles where transmitters use different channels to communicate with the receiver. Thus, (42) holds with strict equality for the case of the NE with the highest NSE p^{⊤}, which justifies (78). In a more general case, for instance K > 2 and S > 2, we state the following result:
Proposition 7 (PoA of the ${\mathcal{G}}_{\left(b\right)}$)
Proposition 8 (PoS of the${\mathcal{G}}_{\mathbf{\left(}\mathit{b}\mathbf{\right)}}$)
Note that from Proposition 7, it can be concluded that in the case in which K > S, the PoA and the PoS are both strictly bigger than one, since the condition (84) does not hold. Thus, as in the previous case, as long as K > S there always exists a loss in performance due to the decentralization of the network. Nonetheless, depending on the channel realizations, in the case in which K ⩽ S, global optimal performance can be achieved at the equilibrium.
Numerical examples
In the previous sections, a mathematical argument has been provided to show that at the low and high SNR regimes, using a CS policy yields a higher or equal NSE than using a waterfilling PA policy. A formal proof has been provided for the case of K = 2 transmitters and S = 2 channels. Moreover, we highlight that a CS policy is evidently simpler than a PA policy in terms of implementation. Unfortunately, providing a formal proof for an arbitrary number of transmitters K and channels S at a finite SNR becomes a hard task since it will require to calculate all the types of NE depending on the exact channel realizations. Hence, for the case of arbitrary parameters K, S, and SNR, we provide only numerical examples to give an insight of the general behavior. First, we evaluate the impact of the SNR for a network with a fixed number of transmitters and channels. Second, we evaluate the impact of the network load, i.e., the number of transmitters per channel for a given fixed SNR.
Impact of the SNR $\frac{{\mathit{p}}_{\mathbf{\text{max}}}}{{\mathit{\sigma}}_{\mathbf{2}}}$
Impact of the number of transmitters (K)
Conclusions
In this article, it is shown to what extent the equilibrium analysis of the decentralized parallel MAC differs from those conducted for other channels like Gaussian MIMO interference channels and fast fading MIMO MAC. In particular, the special structure of parallel MAC and the assumption of SUD at the receiver leads to the potential game property. The CS game was merely introduced in the literature but not investigated in details as it is in this article. In particular, a graphtheoretic interpretation is used to characterize the number of NE. In the case where the number of transmitters is large, the fraction between pure NE and the total number of action profiles is relatively small, which makes both the analysis and the achievability of the NE a challenging task. Now, from a design point of view, we provide theoretical results and numerical examples to show that a fully loaded network, when transmitters use only one channel, can be more efficient than its counterpart when all the channels can be exploited by the transmitters. Although all of these results are encouraging about the relevance of gametheoretic analyses of PA problems, important practical issues have deliberately been ignored. For example, the impact of channel estimation is not assessed at all. Also, it is important to conduct a detailed analysis on the signaling cost involved by all the PA algorithms arising from this game formulations to learn NE.
Appendix
Proof of Proposition 1
and a_{n,n} = 0 for all $n\in \mathcal{I}$, where the set ${\mathcal{V}}_{n}$ is the set of indices of the adjacent vertices of vertex v_{ n }. In the following, we use the concept of distance between two vertices of the graph G. We define this concept using our notation:
Definition 4 (Shortest Path)
and thus, with probability one, none on the action profiles in ${\mathcal{V}}_{\left(n\right)}$ are NE. By iteratively following this procedure over the remaining set of action profiles $\mathcal{I}\setminus \{n\cup {\mathcal{V}}_{\left(n\right)}\}$, some action profiles can be eliminated and only those that are potentially NE are left over.
NE candidates. This establishes an upper bound for L and completes the proof.
Proof of Proposition 2
In the following, we use the notation −c to denote the element other than c in the binary set $\mathcal{C}$.