In game theory, a Blotto game [12] is a twoperson zerosum game where the players are tasked to simultaneously distribute their limited resources over several objects, and the player allocating the most resources to an object wins the object. Intuitively, the goal of the players is to win the highest number of objects and their resource allocation strategies are crucial in determining the outcome. The payoff of the game is then equal to the total number of objects won. In fact, this game has widely been used to characterize some competitive real environments: allocating campaign budgets for elections, distributing soldiers for battlefields, etc. In wireless communications, Blotto game has been first used to model the PA problem under malicious jamming attacks for cognitive radio networks [15]. In the last few decades, intensive research effort has been devoted to study the equilibrium properties of Blotto games. In [16], a purestrategy symmetric monotonic Bayesian equilibrium is found for a Blotto game with incomplete information. The idea of deriving the equilibrium point is very constructive and some of the equilibrium properties are useful in solving the fairness, efficiency, and complexity problems in the SABG to be proposed in this article.
3.1 Blotto game formulation
In this study, we extend the twoperson Blotto game into a stochastic Kplayer Blotto game [16] which is more appropriate to model the SA problem in the OFDMA system described in Section 2. In this game, each user is allocated a total budget of which is to be spent on N subcarriers subject to the constraint where is the BA strategy of the k th user on the n th subcarrier. The budget could be in the form of fictitious credit [10] issued by the BS for bidding purposes.
Let denote the Blotto game where is the index set for the bidders (users), and are the PA and BA strategy sets, respectively, while u_{
k
} (•) is the utility function for the k th user. In this game, every user tries to allocate their power and budget to maximize their utility functions (or win as many subcarriers as possible). To preserve high efficiency, the users should be encouraged to win the "good" subcarriers. (Note: "good" subcarriers refer to the subcarriers with good channel conditions for a particular user.) In a fair auction, the user who allocates the highest budget on a subcarrier has the highest chance to win that subcarrier. Therefore, a rational user will always allocate more budget to the good subcarriers. To achieve competitive fairness and high efficiency, the utility function can be formulated as
where is the probability of the k th user winning the n th subcarrier. From Equation 10, and are the power and budget vectors for the k th user, respectively, where and . The utility function in Equation 10 combines PA and BA where p_{
k
} and b_{
k
}are both coupled on each subcarrier and under the budget and power constraints of each user. Unlike other game theoretic approaches, the utility function in Equation 10 does not represent the actual throughput that a user can achieve at the end of the game. Instead, it provides a mechanism for users to allocate budget to different subcarriers based on their expected throughput on the subcarriers. The payoff of a user is equal to the number of subcarriers won, which is then quantified in the (actual) throughput.
In the SABG, users compete with each others in an auction market. Unlike the conventional auctions, the proposed SABG allows users to simultaneously bid for all subcarriers. Owing to limited budget, the users need to spend it wisely to win as many good subcarriers as possible to maximize their own throughput. To this end, optimal PA and BA strategies are required for the SABG such that Equation 10 can be maximized subject to the power and budget constraints, i.e.,
The approach used in [16] to derive the equilibrium state in Blotto games is adopted in this article to analyze the proposed PA and BA strategies for the SABG as given in the following proposition.
Proposition 1: If all the users maximize their utility function according to Equation 11, then the optimal PA strategy for the k th user on all subcarriers is
where Δ _{
k
} is the waterfilling level of the k th user and x^{+} = max{x,0}. The optimal BA strategy for the k th user on all subcarriers is
Proof: See Appendix.
It is worth mentioning that the PA strategy given in Equation 12 is found to correspond to the traditional waterfilling PA techniques which can provide high spectral efficiency [3]. Since the PA strategy is independent of the BA strategy, the transmit power can be waterfilled onto different subcarriers before implementing BA.
After BA, the BS which acts as an auctioneer will assign the available subcarriers to the users based on their biddings. In general, SA is a combinatorial problem which requires complex algorithms to obtain the optimal solution [5, 9]. In this study, we modify the binary constraint in Equation 5 into a probabilistic SA decision to facilitate design of a lowcomplexity SA algorithm. Thus, Equation 5 becomes
Next, we define the set to include all the subcarriers that have been allocated to the k th user, i.e.,
In this section, a fair, efficient, and lowcomplexity SA scheme has been proposed. Note, however, that the BA strategy proposed in Proposition 1 is not symmetrical as what has been derived in [16]. Therefore, the symmetric monotonic Bayesian equilibrium in [16] is not applicable in the SABG. In the following section, we will study the equilibrium of the SABG.
3.2 Existence and uniqueness of NE in the SABG
In the SABG, the individual user maximizes its own utility function in a distributed fashion. Since the proposed PA strategy is independent of the proposed BA strategy and does not require knowledge of other users' action for implementation, the PA is an iterative waterfilling algorithm without divergence problem. However, the BA strategy proposed in Equation 13 demonstrates some strategic interdependence among the users. The utility function of each user is governed by its own strategy as well as those of other users. Hence, the BA which optimizes individual utility also depends on the BA of other users in the system. Thus, it is necessary to characterize a set of BA strategies whereby all the users are satisfied with the utility attained given the BA strategies of other users. Such an equilibrium operating point is called a NE in game theory [17]. The NE concept offers a predictable and stable outcome for a game where multiple agents with conflicting interests compete through selfoptimization and reach a state from which no player wishes to deviate [17]. In other words, at a NE, given the BA of other users, no user can improve its subcarrier utility level by making individual changes in the BA. Nevertheless, such a state may not necessarily exist. Thus, we first need to investigate the existence of NE in the SABG.
Theorem 1: NE exists in the SABG.
Proof: According to the implicit function theorem [18], a Jacobian matrix must be nonsingular at the point of existence. By using Equation 13, we define
where F_{
k
} , are differentiable functions. Taking the partial derivative , results in 1 on the main diagonal of the Jacobian matrix while the terms outside the main diagonal can be expressed as
Since , , (normally in Mbps), , i ≠ k is always of the order of 10^{3} or smaller. Therefore, the values of the terms , , i ≠ k are extremely small and will only have a negligible impact on the nonsingularity of the corresponding Jacobian matrix. Furthermore, since the Jacobian matrix is a continuous function with respect to , solutions exist for the entire range of large values of . In conclusion, by assuming that , , are large enough, the solution for Equation 16 exists and hence the existence of NE can be ensured.
We next prove the uniqueness of this NE, which ensures convergence of the algorithm to be proposed in Section 4. For this purpose, we use a discretetime model where time is divided into iterations and we assume that all the users only act once in one iteration and remain static during that iteration. Let and be the BA strategies of the k th user on the n th subcarrier at the next and current iterations, respectively, where t is the iteration number. We can rewrite Equation 16 as
Using Equaiton 18, the uniqueness of the NE which exists in the proposed games can be shown next.
Theorem 2: The SABG have the unique NE.
Proof: In [18], it is shown that if a fixed point exists and if the function f satisfies three properties: positivity , monotonicity , and scalability , convergence to a fixed and unique point is guaranteed. For brevity, we drop the iteration index t in the following proof. Since all the elements on the righthand side (RHS) of Equation 18 are positive, and hence , which ensure positivity of . Next, to prove the monotonicity property, we modify Equation 18 and obtain
It is observed from Equation 19 that a decrease in to results in a decrease in to , , i ≠ k as is a continuous function of , , i ≠ k as shown in Equation 13. However, this decrease results in an increase in , , m ≠ n because more budget is available to be spent on other subcarriers and users tend to spend all additional budget to increase the chance of winning more subcarriers. Therefore, in the second term on the RHS of Equation 19, the denominator remains the same but the numerator decreases. Hence, on the RHS of Equation 19, the second term is always smaller than the first term. This ensures that and thus the monotonicity condition is satisfied. Similarly, to prove the scalability property, Equation 18 can be rewritten as
Since ε is a scalar multiplication constant for the summation in the second term on the RHS of Equation 20, the term remains the same. Hence, the first term on the RHS of Equation 20 which is multiplied with ε > 1 is always larger than the second term. This indicates that and thus the scalability condition is fulfilled. Since all the three properties are satisfied, the solution which exists in the SABG, and hence the corresponding NE, is unique.
3.3 Properties of the SABG
Some of the useful equilibrium properties in [16] can be adopted in the SABG and are summarized in the following theorem.
Theorem 3: The SABG has the following properties:

1.
All users in the SABG will adhere to the BA strategy in Equation 13 if the knowledge of the BA strategies of other users is not available, but it is believed that the available budget will be allocated according to Equation 13.

2.
The NE which exists in the SABG has a monotonic property where the user with the highest budget has the highest chance of winning more subcarriers.

3.
All the users in the SABG compete for subcarriers. Even a small amount of additional budget improves the chance of winning more subcarriers.

4.
All the users in the SABG tend to fully spend their budget to increase the chance of winning more subcarriers.
Proof: Since a unique monotonic equilibrium exists in the SABG, the latter inherits all the properties of a Blotto game listed in [16].
From the second property in Theorem 3, the BS can manipulate the amount of budget available to each user to achieve different objectives. In this study, we aim to enforce competitive fairness into the SABG while incurring a minimal loss in total throughput particularly in correlated fading channels. The following proposition gives the condition to achieve fairness in the SABG.
Proposition 2: The SABG can achieve competitive fairness if each user is allocated an equal amount of budget.
Proof: Consider two users (users 1 and 2) whose achievable throughputs on two subcarriers are shown in Figure 1 where . According to [11], competitive fairness can be achieved if their BA strategies are . Using Equation 13, we have
The following inequalities can be formulated using the competitive fairness conditions
The inequalities in Equation 22 can be simplified as
Given that , it is noted from the inequalities (23) that and such that ε ≈ 0 as (Mbps) are large enough. Therefore, the only solution that satisfies the above inequalities is . This condition is both necessary and sufficient to achieve competitive fairness in the SABG. According to [11], the proof for the above twouser case can be extended to the Kuser general case (K > 2) by introducing multivariable inequalities. Using the same approach above, the solution can be obtained as , as proved in [11].
In general, the SABG scheme is adaptive and can be adjusted to reach two extreme states (i.e., maximalrate and maxmin fairness) by manipulating the budget allocated to each user. The adaptive SABG can be achieved using
where B is the amount of budget allocated by the BS and 0 ≤ β ≤ 1 is an adjustable coefficient controlled by the BS. When β = 0.5, it is shown in Equation 24 that every user is allocated an equal amount of budget and competitive fairness is attained.
Proposition 3: The throughput achieved by the SABG approximates maximalrate throughput if β = 1 while maxmin fairness is achieved by the SABG if β = 0.
Proof: Let β = 1 and Equation 24 indicate that the amount of budget allocated to every user is proportional to the achievable throughput where users with better channel conditions obtain more budget. As shown in Equation 14, having more budget will increase the probability of acquiring subcarriers. In other words, subcarriers are always allocated to users with the best channel gains. Instead of assigning subcarriers based on the budget allocated, the SA decision can be modified as
The binary decision in Equation 25 corresponds to the maximalrate scheme. On the other hand, let β = 0 and Equation 24 indicate that the amount of budget allocated to every user is inversely proportional to the achievable throughput where users with worse channel conditions obtain more budget. In this scenario, users with poor channel conditions are assigned more subcarriers to improve their throughput. Therefore, Equation 14 can be modified as
where Equation 26 corresponds to the maxmin fairness scheme.