Resource allocation in a MAC with and without security via game theoretic learning
 Shahid M. Shah^{1}Email authorView ORCID ID profile,
 Krishna Chaitanya A.^{1} and
 Vinod Sharma^{2}
https://doi.org/10.1186/s1363801709018
© The Author(s) 2017
Received: 6 June 2016
Accepted: 7 June 2017
Published: 26 June 2017
Abstract
In this paper, we study a Kuser fading multiple access channel (FMAC), with and without an eavesdropper (Eve). In the system without Eve, we assume that each user knows only its own channel gain and is completely ignorant about the other users’ channel state. The legitimate receiver sends a short acknowledgement message Acknowledge (ACK) if the message is correctly decoded and a No Acknowledge (NACK) if the message is not correctly decoded. Under these assumptions, we use game theoretic learning setup to make transmitters learn about the power allocation under each state. We use multiplicative weight noregret algorithm to achieve an εcoarse correlated equilibrium. We also consider the case where a user can receive other users’ ACK/NACK messages. Now, we can maximize a weighted sum utility and achieve Pareto optimal points. We also obtain Nash bargaining solutions, which are Pareto points that are fairer to the transmitting users. Fairness among users is quantified using Jain’s index.
With Eve, we first assume each user knows only its own channel gain to the receiver as well as to Eve. The receiver decides whether to send an ACK or a NACK to the transmitting user based on the secrecyrate condition. We use the above developed algorithms to get the equilibrium points. Next, we study the case where each user knows only the distribution of the channel state of Eve. Finally, we also consider the system where the users do not know even the distribution of the Eve’s channel.
Keywords
1 Introduction
A multiple access channel (MAC) is a basic building block in wireless networks [1]. Also, it models the uplink in a wireless cellular system. Therefore, it has been studied extensively over the years ([2–4]). More recent, it has also received attention from information theoretic security point of view. In this paper, we study a MAC with and without an eavesdropper using game theoretic techniques. This allows operating in the capacity region which is fair to the users and also provides distributed algorithms with local information at the users. First, we provide a literature survey on this problem.
A general Muser fading MAC is considered in [5] where the receiver has perfect channel state knowledge and broadcasts channel state information of all the users to all the transmitters. The authors prove that the capacity region of a Muser MAC has a polymatroid structure, and they exploit this structural property to find the optimal power and rate control policy. Timevarying additive white Gaussian noise (AWGN) MAC is studied in [6] where it is assumed that only the receiver can track the channel and not the transmitters. In that case, the transmitters allocate fixed powers (which satisfy the average power constraint) and transmit data over the channel.
In [7], the authors propose a distributed power allocation scheme using Game Theory. The authors assume that each user knows the channel gain of other users also, in addition to knowing his own channel gain. The authors prove that the sumrate point on the capacity region is a Nash equilibrium when the decoding strategy of the receiver is not known to the transmitters. The authors also prove the existence of a Stackelberg equilibrium in which the receiver acts as a leader and the transmitters play a lowlevel game. Using repeated games, the authors prove that each point on the capacity region of a fading MAC is achieved by some power control policy. In [8], the authors prove stronger results by assuming that each user knows only its own channel gain but knows the distribution of channel gains of the other users. Under these conditions, the authors prove the existence and uniqueness of a Bayesian equilibrium. In an orthogonal multiple access channel, the authors in [9] have used evolutionary game theory to obtain a power allocation scheme, while assuming that each user knows the channel gain of all users via feedback.
With security constraints, a multiple access wiretap channel (MACWT) has been well studied in literature. One of the early works is reported in [10] where only one user has confidential messages to be transmitted. The authors have obtained upper bounds on the secrecyrate regions. In [11], the authors consider a more general setup wherein they consider a discrete memoryless multiple access channel where the transmitting users receive a noisy version of each others’ conversation, and they do not trust each other. In this scenario, the authors have obtained an achievable secrecyrate region and some outer bounds. In some special cases, this provides secrecy capacity region. A multiple access wiretap channel with feedback has been studied in [12]. An achievable region of a Gaussian multiple access wiretap channel (GMACWT) was obtained in [13] (the secrecy capacity region is still an open problem).
In the above work, weak secrecy criterion is used. A strong secrecybased achievable rate region for a MACWT is reported in [14]. In [15], the authors find secure degrees of freedom for a MACWT. More recently in [16], the authors have studied a compound MACWT and have characterized inner and outer bounds on the secrecy capacity region. In [17], the authors have studied a fading MACWT with full Channel State Information (CSI) of Eve and also when each user knows the channel state of all the users to the receiver but is ignorant of the instantaneous value of channel state to the eavesdropper (only its distribution is known). But knowing other users’ channel gains to the legitimate receiver may also not be practical: it needs a lot of signalling overhead and feedback information. Hence, in this paper, we present a gametheoretic solution to the resource allocation scheme under the hypothesis that each user only knows its own channel gain and is completely ignorant of other users’ channels (not even the distributions).
In interference channel model [18], the authors use learning algorithms to study a stochastic game and learn optimal power allocation policies. The authors use noregret algorithm to prove the existence of a correlated equilibrium. It is assumed that each user knows power allocation policy of other users, which is not always realistic. The same authors extend this work to the case where each user knows only his own channel gain and does not know the power levels used by other users. The authors prove the existence of a coarse correlated equilibrium using multiplicative weights noregret algorithm [19].
In this paper, we first consider a fading MAC (FMAC) without security constraint. We assume each user knows only its individual channel gain (unlike [8], we do not assume that it knows the distributions of channel gains of others). Since the receiver is receiving data from all the users, it is quite practical to assume that the receiver has channel state information of all the transmitting users. Once a user sends a codeword corresponding to a particular message, the receiver sends an ACK if it decodes it successfully, else it sends a NACK. Each user computes a utility based on the ACK/NACK. We use multiplicative weight noregret algorithm to obtain a coarse correlated equilibrium (CCE). We also assume in the later part of the paper that each user can decode ACK/NACK of other users and hence knows their utility. Then we aim to maximize the sum utility and propose an algorithm to obtain a Pareto point (PP). We also find a Nash bargaining solution (NBS) which provides a Pareto point and ensures fairness among users. We also study the case where users can transmit at multiple rates rather than fixed rates.
Next, we consider a fading MACWT where we first assume that each user knows its channel gains to the receiver and Eve. In this case, we repeat all the algorithms which we used for a FMAC (without security), i.e. multiplicative weight (MW), PP, and NBS, and also consider the multiple rates case. Since it is not practical to assume instantaneous channel gain of the eavesdropper to be known at the transmitter and the receiver, we next consider the case where the receiver only knows the distribution of the Eve’s channel gains. The receiver calculates secrecy outage and sends an ACK/NACK based on that. We again obtain a CCE, PP and a NBS. To the best of our knowledge, this is the first paper which is using game theory on MACWT.
Finally, we compare the sum rates obtained via all these algorithms to the global CSI case and also with the sum rate obtained in [17].
The rest of the paper is organized as follows. In Section 2, we describe the channel model for FMAC without the security constraint and formulate the problem. In Section 2.1, we use multiplicative weight algorithm to obtain a CCE. In Sections 2.2 and 2.3, we obtain Pareto optimal points and NBS. In Section 3, we consider a fading MACWT. First, we consider the case when CSI of Eve is available at the transmitters. In Section 3.1, we consider the case when CSI of Eve is not available at the transmitters (only its distribution is known) and obtain a CCE, a NBS and a PP for both the scenarios. Section 4 generalizes these results to multirate scenario. In Section 5, we compare the various schemes on an example. Finally, in Section 6, we conclude the paper.
2 Fading MAC: without security constraint

\(X(t)=\frac {1}{\sigma _{b}}\widetilde {X}(t)\)

\(\widehat {H}(t)=\frac {1}{\sigma _{b}}\widetilde {H}(t)\)

\(\eta _{b}=\frac {1}{\sigma _{b}}\widetilde {\eta }(t)\), where now \(\eta (t) \sim \mathcal {N}(0,1)\)
Let \(H_{i}(t)\triangleq \left \widehat {H}_{i}(t)\right ^{2}\). The channel gains are assumed discrete valued, in the sets \(\mathcal {H}_{i} \triangleq \{h_{i}^{(1)}, \ldots, h_{i}^{(M)}\}\), where M is the number of possible states. Also, {H _{ i }(t),t≥0} are independent and identically distributed (iid) sequences with distributions \(\left \{\alpha _{i}^{(1)}, \ldots, \alpha _{i}^{(M)}\right \}\). To transmit any codeword, user i can choose any power level from the set \(\mathcal {P}_{i} \triangleq \left \{P_{i}^{(1)},\ldots, P_{i}^{(M)}\right \}\). Also, user i has average power constraint \(\overline {P}_{i}\).
Assumption of discrete channel states Before we describe our game model, we first clarify our assumptions of discrete channel states. The assumption that the channel states take values from discrete sets is based on two observations: (1) In practice, the channel gains are estimated by transmitting known pilots from the transmitter to receiver (a base station in case of uplink scenario), and then the receiver feeds back the estimated value of channel gains up to some precision level. Hence, if n bits are used to represent the value of channel gains (as in digital communication, the information is represented by a finite number of bits), the total possible number of channel states is 2^{ n } [8]. (2) The continuous state of a channel (e.g. Rayleigh distribution) can be quantized to obtain any level of accuracy by a finite set. Also, algorithms in game theory require a finite set. This finite CSI is a commonly made assumption in literature [8, 20, 21].
User i transmits at a fixed rate r _{ i } (to be generalized later) via a usual point to point channel encoder. If the receiver successfully decodes a message, it sends an ACK to that particular user. Otherwise, it sends a NACK. We assume that the NACK and ACK are transmitted at low rates so that these can be received with negligible error at the intended transmitter. The goal of each user is to maximize its probability of successful transmission.
Each user i is assumed to know its own channel gain H _{ i }(t) at time t. Since the receiver can estimate the channel gain of all the users (either by receiving known pilots or by using initial data received), the receiver can use successive cancellation decoding strategy to decode all the users.
Hence, this stochastic game can be modelled as a oneshot game in which player i maximizes its utility (9). In the rest of the paper, we develop algorithms to compute equilibrium points for this game.
2.1 Multiplicative weight algorithm for learning CCE
In this section, we use MW algorithm [22] to compute coarse correlated equilibrium (CCE). The cost of each user can be defined as \(\mathfrak {C}_{i}((a_{i},a_{i})\triangleq \nu _{i}(a_{i},a_{i})\). A coarse correlated equilibrium is defined as follows:
Definition 2.1
for all \(\hat {a}_{i}\) and for all users i.
A coarse correlated equilibrium exists for any finite game unlike a Nash equilibrium. A CCE is a generalization of a mixed strategy. A Nash equilibrium and every mixed strategy Nash equilibrium is also a CCE. We now define external regret that plays a key role in the MW algorithm.
Definition 2.2
The MW algorithm is a noregret algorithm, meaning that the users update their strategies based on the received cost such that the external regret converges to zero ([22, 23]). The MW algorithm is presented in Algorithm 1.
2.1.1 Explanation of Algorithm 1
We provide a brief explanation of Algorithm 1. In each iteration t, a _{ i } is the action chosen by user i with probability \(\Phi _{i}^{t}(a_{i})=\frac {w_{i}^{(t)}(a_{i})}{\sum _{a_{i}}w_{i}^{(t)}(a_{i})}\), and then it pays an average cost \(c_{i}^{(t)}(a_{i})\). Based on the cost incurred, the users update their weight as \(w_{i}^{(t+1)}(a_{i})=w_{i}^{(t)}(a_{i})[1\epsilon ^{\prime }]^{c_{i}^{(t)}(a_{i})}\). Number of iterations needed to get an εapproximate point is \(T>2^{(\ln (n)/\epsilon ^{2})1}\), as given by Theorem 2.1 below.
Theorem 2.1
(Remark 2.3 of chapter 17 from [24]) After T iterations, where \(T>2^{\frac {\ln n}{\epsilon ^{2}}1}\), of noregret dynamics (MW algorithm), every player of a cost minimization game has regret (12) at most ε for each of its strategies. Let \(\psi ^{(t)}=\prod _{i=1}^{K} \Phi _{i}^{(t)}\) denote the outcome distribution at time t and \(\psi =\frac {1}{T}\sum _{t=1}^{T} \psi ^{(t)}\), the time averaged history of these distributions. Then ψ is an εcoarse correlated equilibrium.
2.2 Pareto optimal points
In a wireless environment, it is realistic to assume that the ACK/NACK bits sent to a particular user can be successfully decoded by all the other users also (because these are sent at a low rate using robust codes). In that case, all users can learn about the utility of each other at time t. We show in this section that this information can be used to get a socially optimal Pareto point which generally provides a better performance than a CCE.
Definition 2.3
An action profile \(a\in \mathcal {P}\) is a Pareto point if there does not exist another profile \(\tilde {a}\) such that \(\nu _{i}(\tilde {a})\geq \nu _{i}(a), \forall ~i\in \mathcal {K}\) and \(\nu _{j}(\tilde {a})>\nu _{i}(a)\) for some j≠i.
is a Pareto point [25].
In Algorithm 2 below, we provide a distributed algorithm in which the users update their strategies in a sequential fashion so as to improve Ω(a). This distributed algorithm is the variation of a heuristic stochastic local search algorithm. In this algorithm, each user chooses a random action and uses it for a fixed number of time slots (say T). Then each user finds weighted sum of the utilities (since each user receives ACK/NACK of other users, it can calculate this quantity). After T slots, a user experiments randomly with probability say, ρ, and then with some probability updates the action profile according to its channel state. Now, one user uses this action for next T slots and the other users use the previous action. Based on the weighted sum of utilities, the particular user defines a benchmark. The details of algorithm in the scenario of interference channel can be found in [18].
2.3 Nash bargaining solution
The Pareto points obtained in Section 4 are socially optimal but may not be fair to all users: some users may get much more rates than others. To obtain fair Pareto points, we use the concept of Nash Bargaining Solution (NBS) [26].
This bargaining problem is denoted by \((\mathcal {V},\delta)\).
The aim of the bargaining problem is to find a bargaining solution which is Pareto optimal and satisfies the axioms of symmetry, invariance and independence of irrelevant alternatives [27].
Theorem 2.2

Each user chooses an action that gives higher power level to the channel state that has higher probability of occurrence. In other words, among the set of feasible actions, choose a subset of pure strategies that gives the highest power level to the channel state with highest probability of occurrence. We shrink the subset by considering the actions that give higher power level to the second frequently occurring channel state, and we repeat this process until we get a single strategy.

If all the channel states occur with equal probability, we follow the above procedure by considering the value of the channel gain instead of the probabilities of occurrence of the channel gains.
Hence, the algorithm for NBS is the following Algorithm 3.
From [26], if the set of utilities \(\mathcal {V}\) is convex, then a Nash bargaining solution is also proportionally fair. In our problem, \(\mathcal {V}\) is convex, and hence, the solution is proportionally fair also.
2.3.1 Fairness comparison via Jain’s index
It can be easily shown that \(1/K \leq \mathcal {J}(\mathbf {r})\leq 1\). An allocation policy is fair if \(\mathcal {J}(\mathbf {r})\) is close to 1, and policy is called unfair if it is close to 1/K.
3 Fading MAC: with security constraints
Nomenclature
Symbol  Definition 

K  Number of transmitting users 
\(\widetilde {H}_{i}(t)\)  Channel gain to Bob 
\(\widetilde {G}_{i}(t)\)  Channel gain to Eve 
M  Possible values of channel gain 
\(\mathcal {P}_{i}\)  Action space 
\(\overline {P}_{i}\)  Power constraint for user i 
π(i)  ith element of permutation 
of index set  
\(\alpha _{i}^{(j)}\)  pmf of H _{ i }(t) 
n  Maximum no. of action of a user 
\(\omega _{i}^{(t)}\left (a_{i}^{(t)},H_{i}(t)\right)\)  Instantaneous reward for user i 
for given action \(a_{i}^{(t)}\)  
r _{ i }  Rate of user i 
\(\beta _{i}^{(j)}\)  pmf of Eve’s channel state, G _{ i }(t) 
Φ _{ i }(t)  Empirical distribution 
over action space for user i  
a _{ i }  action choosen by user i 
δ _{ i }  Disagreement value for user i 
η _{ b }(t)  AWGN at Bob 
η _{ e }(t)  AWGN at Eve 
\(\mathcal {J}(\mathbf {r})\)  Jain’s index 
\(\mathfrak {C}(a_{i},a_{i})\)  Cost of each user 
c(a _{ i })^{(t)}  Average cost of user i up to time t 
ε  Regret for cost minimization game 
ε ^{′}  Weight update factor 
\(\mathbbm {1}_{\{A\}}\)  Indicator function 
\(\mathcal {V}\)  Utility set for Nash bargaining 
Δ  Disagreement strategy for 
Nash bargaining solution  
δ _{ i }  Disagreement value for user i 
where \(\widetilde {\eta }_{e}(t)\) is white Gaussian noise, with distribution \(\mathcal {N}(0,\sigma _{e}^{2})\) and independent of \(\{\widetilde {\eta }_{b}(t)\}\) and the channel gain processes and \(\{\widetilde {X}_{i}(t)\}\).
where \(X(t)=\widetilde {X}(t)/\sigma _{e}\), \(\widehat {G}(t)=\widetilde {G}(t)/\sigma _{e}\) and \(\eta _{e}(t)=\widetilde {\eta }_{e}(t)/\sigma _{e}\).
We define \(G_{i}(t)\triangleq \left \widehat {G}_{i}(t)\right ^{2}\). The channel gains of Eves’ channels are assumed discrete valued, in the set \(\mathcal {G}_{i} \triangleq \left \{g_{i}^{(1)},\ldots,g_{i}^{(M)}\right \}\). Also, {G _{ i }(t),t≥0} are iid independent of each other and also of the sequences {H _{ i }(t)}, with distribution \(\left \{\beta _{i}^{(1)},\ldots,\beta _{i}^{(M)}\right \}\) respectively. User i transmits at a fixed rate r _{ i } via wiretap coding. If the receiver successfully decodes (see details below in this subsection), it sends an ACK to that particular user. Otherwise, it sends a NACK. We assume that the NACK and ACK are transmitted at low rates so that these can be received with negligible error at the intended transmitter. The goal of each user is to maximize the probability of successful transmission.
Each user i is assumed to know its own channel gains H _{ i }(t) and G _{ i }(t) at time t. Since the receiver can estimate the channel gain of all the users (either by receiving known pilots or by using initial data received), the receiver can use successive decoding strategy to decode all the users.
otherwise a NACK, where (a)^{+}= max(0,a). The above constraint follows from the achievable secrecyrate region of a Gaussian MACWT as discussed in [13]. Each user i takes action (allocating power) \(P_{i}^{(j)}\) when its channel gains are \(H_{i}^{(j)}\) and \(G_{i}^{(j)}\) to transmit at its rate.
Now, we can use all the algorithms of Sections 2.1, 2.2 and 2.3 to obtain a CCE, PP and NBS.
3.1 Fading MACWT with individual main channel CSI only
where probability is over the channel gains G _{ π(i)}. The receiver sends an ACK if \(P_{o}^{S}<\epsilon \), else the receiver sends a NACK. Hence, we define utility of user i as
where \(\mathbbm {1}_{\{C\}}\) is an indicator function. With these utility functions, we can use the algorithms provided in Sections 2.1, 2.2 and 2.3.
3.2 Avoiding security breach
In the previous sections, we assumed that when the legitimate receiver cannot securely decode the message, it sends a NACK. This is useful for the transmitters to learn the equilibrium point. But the messages transmitted during those slots may be decoded by Eve (with probability >ε in Section 3.1). Now, we modify the system a little so as to use the above coding scheme but mitigate this secrecy loss also.
We assume that each slot is comprised of two subslots. The channel gain does not change during the whole slot. In the first part of the slot, we transmit a dummy (random) message. If Bob sends an ACK to user i, then the actual confidential message can be transmitted by user i in the second subslot at the same power. If Bob sends a NACK, then user i should not use the second subslot. We can make the second subslot much larger than the first subslot so that the rate loss due to the dummy messages is minimal.
4 Transmission at multiple rates
We can now use all the existing algorithms to compute CCE, PP and NBS.
5 Numerical results
In this section, we provide several examples using the algorithms developed in this paper. We divide our examples into two parts: (1) FMAC (without security constraint) and (2) FMACWT.
We first discuss the quantization of channel gains from various continuous fading distributions.
5.1 General distribution
with mean e ^{ μ } and variance \(e^{\mu +\sigma ^{2}/2}\).
5.1.1 FMAC (without security constraint)
We also observe the following behaviours of the three schemes (Figs. 2 and 4). At any given SNR, the three schemes allocate resources (power, rate) to each user. In CCE, each user optimizes its own objective function (here probability of successful transmission), without worrying about the other users. Hence, Jain’s index will be sometimes fair and sometimes unfair. Thus, there will be fluctuations. In Pareto policy, the sum throughput of all the users is optimized. But the allocation of resources to individual users can often be unfair. This is a known weakness of Pareto points. Hence, Jain’s index is above 0.5. Nash Bargaining is designed to be a fairer scheme. Hence, Jain’s index for it is close to 1. At any given SNR, we expect it to provide a fairer solution than the CCE and a Pareto point.
5.1.2 FMACWT (with security constraint)
6 Conclusions
In this paper, a Kuser fading multiple access channel with and without security constraints is studied. First, we consider a FMAC without the security constraints. Under the assumption of individual CSI of users, we propose the problem of power allocation as a stochastic game when the receiver sends an ACK or a NACK depending on whether it was able to decode the message or not. We have used multiplicative weight noregret algorithm to obtain a coarse correlated equilibrium (CCE). Then we consider the case when the users can decode ACK/NACK of each other. In this scenario, we provide an algorithm to maximize the weighted sum utility of all the users and obtain a Pareto optimal point. PP is socially optimal but may be unfair to individual users. Next, we consider the case where the users can cooperate with each other so as to disagree with the policy which will be unfair to individual user. We then obtain a Nash bargaining solution, which in addition to being Pareto optimal is also fair to each user. We use Jain’s index to quantify the fairness between different algorithms.
Next, we study a Kuser fading multiple access wiretap channel with CSI of Eve available to the users. We use the previous algorithms to obtain a CCE, PP and NBS. Next, we consider the case where each user does not know the CSI of Eve but only its distribution. In that case, we use secrecy outage as the criterion for the receiver to send an ACK or a NACK. Here also, we use the previous algorithms to obtain a CCE, PP or NBS. Finally, we show that our algorithms can be extended to the case where a user can transmit at different rates. At the end, we provide a few examples to compute different solutions and compare them under different CSI scenarios.
Declarations
Acknowledgements
The first author would like to thank Deekshit PK for the useful discussions.
Authors’ contributions
The author names are in order of contribution, i.e. first author has the major contribution, second author helped in writing the algorithms and third author is a research advisor. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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