A fair and efficient pricing strategy for slotted Aloha in MPR models Dandan Wang, Cristina Comaniciu, Uf Tureli Stevens Institute of Technology Hoboken, NJ 07094, USA Email: {dwang3, ccomanic, utureli}@stevens.edu Abstract—In this paper, we propose a fair pricing strategy to improve the performance of a slotted Aloha system in which users act selfishly to improve their own utility. Based on a game theoretic framework, we show that both throughput and revenue can be optimized by appropriately selecting a pricing strategy for the selfish users. Our proposed solution considers multipacket reception capabilities for the physical layer. The network enforces fairness among different users by employing a pricing policy that favors equal access probabilities. I. I NTRODUCTION Medium Access Control (MAC) is central to the successful deployment of modern wireless networks, where users are ex- pected to manage resources in a decentralized fashion [1], [2]. In this context, a recent focus in MAC protocol design has been on the behavior of access control protocols in the presence of selfish users that seek to maximize their own performance [3], [4], [5]. At a first glance, it may seem that selfish users running their own MAC strategies could lead to protocol failures by constantly colliding in an attempt to maximize their individual throughput. The results in [6] have contradicted this hypothesis and have showed that a distributed Aloha based MAC protocol for selfish users is viable and stable. Their analysis uses a game theoretic formulation. However, the obtained throughput for the system analyzed in [6] is lower than that of a centrally controlled Aloha, and depends on the cost associated with the users’ transmissions. The result in [6] motivates our current research, which seeks to improve the selfish users’ Aloha throughput by introducing a differentiated fair pricing mechanism. In this paper, we show that the throughput of the centralized slotted Aloha can be achieved in a network in which selfish users access the network attempting to maximize their own utility, by creating differentiated incentives to transmit, which are correlated with the transmission cost for each user. Game theoretic formulations for analyzing MAC protocols [7],[8], and in particular slotted Aloha, were recently proposed in the literature, including the work in [6], its extension for multipacket reception (MPR) models in [9], and a pricing strategy in [10] for an Aloha network of heterogeneous users with inelastic bandwidth requirements. The game model in [10] constructs a concave utility function of the price the users are willing to pay and the throughput they want to get associated with the charge of the network. Therefore, the network can achieve a required throughput using this pricing strategy. However, [10] does not model the transmission cost of each transmission, such as the energy consumption asso- ciated with packet transmission. Another shortcoming of the model considered in [10] is that all the users are assumed to be willing to pay the same price to the network and the network charges are the same for all the users, irrespective of their channel quality or energy consumption. In this paper, we extend the work in [6] and [9] to allow for a more realistic model in which users have differentiated trans- mission costs for packet transmissions (e.g., energy consump- tion based on channel quality), and their utilities are affected by this transmission cost, as well as the delay experienced due to the access control. To regulate the transmission attempts for all users, we introduce a pricing strategy which enforces fairness and maximizes the system performance. We consider two different optimization criteria: throughput and revenue maximization. We use our model to consider improvements at the physical layer that allow for multi-packet reception. The paper is organized as follows. In section II, a game theoretic formulation for slotted Aloha with energy constraints is proposed. The analysis of the Nash equilibrium for slotted Aloha with multipacket reception is presented in section III. Conclusions follow in section IV. II. GAME MODEL FOR SLOTTED ALOHA WITH PRICING In this section, we present a game-theoretic model for slotted Aloha. Each slot of the system is a one-stage game. At the beginning of each slot, the players learn the current state of the game, which is the number of users (N) who currently have packets to send. Each of these players has two possible actions: transmit (T) or wait (W). When a user transmits, its transmission can either succeed (S), or fail (F). The gain associated with a successful transmission is a normalized throughput of 1, while the cost of transmitting for user i is c i (e.g., energy cost), and the network’s current charge for this user is μ i . In our game, different nodes can have different transmission costs. If player i transmits and succeeds in a given slot, then that player will receive a payoff of 1 -c i -μ i for that slot (throughput - energy cost - price paid). If the user refrains from transmission in a particular slot (waits), this will result in one slot delay for that user. This delay is associated with the loss in the throughput the player could have achieved if it would have transmitted successfully. Thus, the payoff for this waiting user i can be determined as -(1-c i -μ i ) (the negative