I recreated the centralized model based on Pareto optimality presented in Inter-Operator Spectrum Sharing for Cellular Networks using Game Theory (Kamal et al.) and applied it to a fictional problem. I formulated the problem based on real-world data, to test and review the model. The model is based on a two player non-zero sum game. A link to the code is provided at the end of this post.
In the Chicago area, T-Mobile has an LTE channel size of 15 MHz on the AWS – Band 66 spectrum and covers about 68 Million people nationwide. In the same area AT&T has an LTE channel size of 10 MHz on the AWS – Band 66 spectrum, but covers 138 Million people nationwide. Consider a scenario in which T-Mobile notices that its user arrival rates are 80% of that of AT&T in the Chicago area for a certain time of day. They propose that since AT&T has many more customers and T-Mobile has unused spectrum, they pool their AWS – Band 66 spectrum together to make a pool of 25 MHz. The two “players” agree to a pricing structure of $500/unit of satisfaction (reward), a price of $200/MHz, a blocking penalty of $40/MHz. The data rate is capped at 7.93 Mbps (upper bound of AT&T). The percentage of the pool for each player was allocated based on the percentage of customers nationwide, with 1 MHz kept in reserve by the meta operator. The rest of the parameters are as presented in Inter-Operator Spectrum Sharing for Cellular Networks using Game Theory. For which user arrival rate will the reward be minimized for both players (i.e. which user arrival rates should be avoided by both players)?
The model was able to capture the user arrival rate with minimum reward for both players. T-Mobile has a higher reward across the board and wins this deal as seen in Figure 1, because it uses much less of the pool and is less subject to penalization. However, given the pricing structure, there is no loser. It is here that the shortcomings of the centralized model based on PO are exposed. The reward defined as dollars per unit of satisfaction is a very loose definition that will have to determined on a case to case basis. Defining an appropriate reward and penalty was the main difficulty when constructing this model, because they must be derived from acquired data, such as customer surveys, which is not available to this review. This makes it difficult to test different cases of the models presented in the paper and verify robustness.
The Pareto Front shown in Figure 2 reveals the trade-offs between the T-Mobile and AT&T rewards. The boundaries were extracted from the optimization region shown in Figure 1. The point of mutually minimum rewards from the Pareto front, (7926, 3430), corresponds to user arrival rates of 1.6 s-1. This means that user arrival rates of about 1.6 s-1 should be avoided when using this model to dodge mutually sub-optimal rewards for both companies, irrespective of pricing structure.
Link to code
- Kamal, H., Coupechoux, M., and Godlewski, P., “Inter-Operator Spectrum Sharing for Cellular Networks using Game Theory”, IEEE, 2016, pp. 425-429