Game-theoretic rate region for multi-user channels

Ali.khanafer 11:57, 30 May 2011 (PDT)

Understanding the fundamental limits of communications in networks of selfish users has been receiving growing attention. The competitive nature of users impacts the very definition of information-theoretic achievable rate regions. Because the communication schemes employed by certain users impede the achievable rates of other users through interference, the capacity region by definition requires cooperative optimization of the encoding schemes. In practice, however, users might refrain from embarking in such cooperation to satisfy their own needs. Recent research focused on studying communication networks from a conflict standpoint using cooperative and noncooperative game theory. Clearly, not all rate tuples in a given capacity region are achievable from a game-theoretic point of view. A noncooperative framework can be utilized to find the rate tuples within the information-theoretic rate region that are Nash equilibrium (NE) solutions. The collection of all such rate tuples is often termed the NE rate region. In a cooperative setting, game-theoretic tools such as the Nash bargaining solution (NBS) can be utilized to find a unique operation point.

Interference Channel
The interference channel (IC) is the most basic model in multi-user information theory. . User $i$ sends a message $x_i$ to receiver $i$ over the channel $h_{ii}$ which experiences interference from user $j$, through the channel $h_{ij}$. Additive white Gaussian noise (AWGN) $z_i$ is added at receiver $i$. In the two-user case, we can classify Gaussian ICs in three categories depending on the channel conditions: The complete description of the capacity region of the IC remains an open problem in information theory literature. The region is only known for specific channel classes: the very weak interference and the very strong interference. However, recently proposed a scheme that achieves the capacity region to within one bit for all ranges of channel parameters. Noncooperative and cooperative game-theoretical approaches were taken to identify the achievable rate tuples in a competition.
 * Strong Interference: $|h_{12}|^2 \geq 1$ and $|h_{21}|^2 \geq 1$.
 * Mixed Interference: either $0 < |h_{12}|^2 <1$ and $|h_{21}|^2 \geq 1$ or $0 < |h_{21}|^2 <1$ and $|h_{12}|^2 \geq 1$.
 * Weak Interference: $0 < |h_{12}|^2 <1$ and $0 < |h_{21}|^2 <1$.

The NE rate region
In, Berry and Tse focus on the two-user IC. They view the communication process as a nonzero-sum game. They manage to obtain the full NE region of the deterministic IC, and approximate the region to within one bit in the Gaussian case, which parallels the information-theoretic results in. For a given $\eta > 0$, the authors define an $\eta$-IC game as follows. Each user $i \in \{1,2\}$ adopts a strategy $s_i$ which describes his encoding scheme and includes:


 * The number of information bits $B_i$ and the block length $N_i$ of the codewords.
 * The set of codewords $\mathcal{C}_i$ and the encoder $f_i: \{1,...,2^{B_i}\} \times \Omega_i \rightarrow \mathcal{C}_i$, where $\Omega_i$ represents the randomness in the mapping.
 * The rate of the code $R_i(s_i) = B_i/N_i$.

The outcome of the game is, naturally, the rate that both users achieve \begin{eqnarray} \mathcal{U}_i(s_1,s_2) = \left\{ \begin{array}{ll} R_i(s_i), & \mbox{if $p_i^{(k)}(s_1,s_2) \leq \eta$ $\forall k$}\\ 0, & \mbox{otherwise,}\end{array} \right. \end{eqnarray} where $p_i^{(k)}(s_1,s_2)$ is the average bit error probability of the $k$-th block which depends on the detection scheme. Because both users aim to maximize their utility, they compete in a nonzero-sum game and the equilibrium solution they seek is the NE. They further define a $(1-\eta)$-reliable strategy to be a choice $(s_1,s_2)$ which achieves rates $(R_1(s_1),R_2(s_2))$ and guarantees $p_i^{(k)}(s_1,s_2) \leq \eta$ $\forall i,k$. Given the above notions, the authors provide the following two definitions.

Definition: The capacity region $\mathcal{C}$ of the IC is the closure of the set of all rate tuples $(R_1,R_2)$ such that for every $\bar{\eta} >0$ and $\eta \in (0,\bar{\eta})$, there exists a $(1-\eta)$-reliable strategy pair $(s_1,s_2)$ which achieves the rate tuple $(R_1,R_2)$.

Definition: The NE region $\mathcal{C}^{\epsilon}_{NE}$ of the IC is the closure of the set of all rate tuples $(R_1,R_2)$ such that for every $\epsilon>0$, $\bar{\eta}(\epsilon)>0$ and $\eta \in (0,\bar{\eta}(\epsilon))$, there exists a $(1-\eta)$-reliable strategy pair $(s_1,s_2)$ which achieves the rate tuple $(R_1,R_2)$ and is an $\epsilon$-NE.

An alternative definition to $\mathcal{C}^{\epsilon}_{NE}$ would be to require $(s_1,s_2)$ to be a NE of the $\eta$-game. However, this would require finding an encoding scheme which achieves the optimal rate tuple with a non-zero error probability $\eta$. This is a rather difficult problem, and it is unknown if a solution exists in some cases. Thus, the authors define $\mathcal{C}^{\epsilon}_{NE}$ in terms of $\epsilon$-NE to avoid such difficulties. The parameter $\bar{\eta}(\epsilon)$ is essential to the definition of $\mathcal{C}^{\epsilon}_{NE}$. If we did not have such a constraint, one can set $\eta = 1$ making the target bit error probability $100\%$. Clearly, the resulting $\eta$-game does not have a $\epsilon$-NE as any user can choose any strategy and improve his utility arbitrarily. Hence, an $\epsilon$-NE for a given $\eta$-game is not necessarily an $\epsilon$-NE for a $\tilde{\eta}$-game, $\tilde{\eta}>\eta$. On the other hand, $\mathcal{C}$ is well defined without the need for the parameter $\bar{\eta}$ as a $(1-\eta)$-reliable strategy is also $(1-\tilde{\eta})$-reliable when $\tilde{\eta} > \eta$.

The Linear Deterministic Model
The NE region was completely characterized for the deterministic IC. The deterministic model of the IC was first proposed in order to gain insights about the capacity region of the Gaussian IC. The model approximates the more complicated Gaussian case and captures the effects of noise and interference.

The number of bits above the noise margin in the link between each transmitter and its receiver is denoted by $n_{ii}$ which corresponds to $\log(\mbox{SNR}_i)$, where SNR$_i$ is the signal-to-noise ratio of the corresponding point-to-point Gaussian channel. Similarly, $n_{ij}$ corresponds to $\log(\mbox{INR}_{ij})$, where INR$_{ij}$ is the interference-to-noise ratio of the corresponding Gaussian channel, and determines the number of least significant bits that see interference from the signal of the other user. In Fig., the message of user $1$ consists of $4$ levels (or bits). At the receiver, the least significant bit is truncated; hence, $n_{11} = 3$. Also, the two most significant bits of the message of user $2$ interfere with two least significant bits of the signal received at receiver $1$; hence, $n_{12} = 2$.

To find the NE solutions, the authors define the following two-dimensional box \begin{equation} \label{box} \mathcal{B} = \{(R_1,R_2): L_i \leq R_i \leq U_i, \forall i \in \{1,2\}\}, \end{equation} where \begin{align} & L_i =& (n_{ii} - n_{ij})^+ \label{DetBounds1} \\ & U_i =& \left\{ \begin{array}{ll} n_{ii} - \min(L_j,n_{ij}), \mbox{if $n_{ij}\leq n_{ii}$}\\ \min((n_{ij}-L_j)^+,n_{ii}), \mbox{otherwise.}\end{array} \right. \label{DetBounds2} \end{align} The box $\mathcal{B}$ has an intuitive interpretation. The lower bound $L_i$ represents the number of levels user $i$ can transmit above the interference margin caused by the signal transmitted of user $j$. The upper bound $U_i$ is the number of levels at receiver $i$ that receive signals from transmitter $i$ and are free of interference from the top $L_j$ levels of the signal of transmitter $j$. In Fig., $L_1=1$, $L_2=3$, $U_1=1$, and $U_2=3$.

Without any cooperation between the two users, and regardless of the strategy employed by the other user, user $i$ can always achieve a rate $L_i$ since those are the interference-free levels over which the user can transmit with maximum rate. Hence, we conclude that if a pair of strategies $(s_1,s_2)$ is an $\epsilon$-NE, they must indeed achieve a rate tuple $(R_1,R_2) \geq (L_1,L_2)$. Otherwise, if a strategy $s_i' \neq s_i$ achieves a rate $R_i < L_i$, then user $i$ will definitely deviate to improve his payoff by at least $\epsilon$. However, transmitting over the $L_i$ levels will cause interference on a subset of levels available at receiver $j$. User $j$ can only transmit with maximum rate over the remaining levels, which is exactly the definition of $U_j$. With the above discussion in mind, we have the following lemma.

Lemma: $\mathcal{C}^{\epsilon}_{NE} \subseteq \mathcal{C} \cap \mathcal{B}$.

A rigorous proof is presented in where Fano's inequality is utilized to obtain bounds on the NE rates in terms of the average error probability. The lemma identifies the non-equilibrium rates. It remains to find the rates in $\mathcal{C} \cap \mathcal{B}$ that are NE solutions. To this end, the authors introduce a modified version of the Han-Kobayashi schemes presented in along with two definitions:

Definition: A randomized Han-Kobayashi (RHK) scheme for a block-length $N$ is one where each user divides his message set $\{1,...,2^{B_i}\}$ into the direct product of a private message set containing $2^{NR_{ip}}$ messages and a common message set containing $2^{NR_{ic}}$ messages, where $NR_{ip}+NR_{ic} = B_i$. Moreover, each user is allowed to have a random common message set containing $2^{NR_{ir}}$ equally likely messages.

The private messages of each user is to be decoded by its respective receiver only, whereas the common messages can be decoded by both receivers. Note that $R_i = R_{ip}+R_{ic}$.

Definition A rate tuple $(R_{1c},R_{1p},R_{2c},R_{2p})$ \textbf{fully utilizes the interference free levels} for user $i$ if $R_{ip}\geq a_i$ and $R_{ic}\geq b_i$, where $a_i = (n_{ii}-n_{ji}-n_{ij})^+$ and $b_i = (n_{ii}-\max(n_{ii}-n_{ji},n_{ij}))^+$.

Here, $a_i$ and $b_i$ represent the private and common interference free levels at user $i$, respectively. Note that if a rate tuple fully utilizes the interference free levels, then $R_i \geq L_i$, which is a required condition for any NE as discussed above.

Definition A rate tuple $R = (R_{1c},R_{1p},R_{1r},R_{2c},R_{2p},R_{2r})$ is self-saturated at receiver $i$ if $R$ lies on the boundary of $\mathcal{R}^m_i$ at receiver $i$, where $\mathcal{R}^m_i$ is the set of all rate tuples satisfying: \begin{eqnarray} R_{ic}+R_{ip}+R_{jc}+R_{jr} &\leq& \max\{n_{ii},n_{ij}\} \\ R_{ip}+R_{jc}+R_{jr} &\leq& \max\{n_{ij},(n_{ii}-n_{ji})^+\} \\ R_{ip} &\leq& (n_{ii}-n_{ji})^+ \\ R_{ic}+R_{ip} &\leq& n_{ii} \end{eqnarray}

Self-saturation when $R_i \geq L_i$ implies that a user cannot deviate without worsening his pay-off, because if this was possible one of the rate constraints of the capacity region obtained by an RHK would be violated, which leads to a high probability of error.

The following lemma uses the above definitions to identify the NE rate tuples.

Lemma: For any pair $(R_1,R_2) \in \mathcal{C} \cap \mathcal{B}$, there exists a non-randomized Han-Kobayashi ($R_{1r}=R_{2r}=0$) rate split $R=(R_{1c},R_{1p},R_{2c},R_{2p})$ that fully utilizes the interference free levels at each transmitter. For any such $R$, there exists random common rates $R_{1r},R_{2r} \geq 0$ such that $(R_{1c},R_{1p},R_{1r},R_{2c},R_{2p},R_{2r})$ is self-saturated at both receivers. It then follows that $(R_{1c}+R_{1p},R_{2c}+R_{2p}) \in \mathcal{C}^{\epsilon}_{NE}$.

The proof for this lemma involves using Fano's inequality and standard random coding arguments. Combining the above two lemmas, we know exactly which rates are NE.

Theorem $\mathcal{C}^{\epsilon}_{NE} = \mathcal{B}\cap \mathcal{C}$.