在本文中，我們提出了一種涉及n 個代理的貝葉斯博弈來解決
資源分配問題。我們的主要目標是確定在此上下文中每個代理的
最佳資源分配。
為了實現這一目標，我們首先開發了一個回報函數，作為代理
評估其行為的可取性的參考點。條件預期收益是通過考慮獨立隨
機變量之和的概率來計算的，並優先考慮最小的請求。我們利用
馬爾可夫不等式和切爾諾夫界限等近似來估計此計算過程中的概
率。
此外，我們引入了最佳響應的概念，它將條件預期收益轉化為
微分方程。通過求解這個方程並確定均衡策略，我們可以有效地
確定每個智能體的資源分配。
另外，我們還提出了數值結果，說明了遊戲中均衡策略隨著代
理數量和可用容量變化的變化。這些結果與其他資源分配方案進
行了比較，清楚地證明了我們的方法相對於其他方法的優越性。

In this paper, we propose a Bayesian game involving n agents to tackle
the resource allocation problem. Our main goal is to determine the optimal
distribution of resources for each agent within this context.
To accomplish this, we initially develop a payoff function that acts
as a reference point for agents to evaluate the desirability of their actions.
The conditional expected payoff is computed by considering the
probability of the sum of independent random variables, prioritizing the
smallest requests. We utilize approximations such as Markov’s Inequality
and the Chernoff Bound to estimate the probability in this calculation
process.
Moreover, we introduce the concept of the best response, which transforms
the conditional expected payoff into a differential equation. By
solving this equation and identifying the equilibrium strategy, we can
effectively determine the allocation of resources for each agent.
Additionally, we present numerical findings that illustrate the variability
of the equilibrium strategy in our game as the number of agents
and the available capacity change. These results are compared against
other resource allocation schemes, clearly demonstrating the superiority
of our method over the alternatives.

[1] H. Zhang, C. Jiang, N. C. Beaulieu, X. Chu, X. Wen, and M. Tao,
“Resource allocation in spectrum-sharing ofdma femtocells with
heterogeneous services,” IEEE Transactions on Communications,
vol. 62, no. 7, pp. 2366–2377, 2014.
[2] J. Tang, D. K. C. So, E. Alsusa, K. A. Hamdi, A. Shojaeifard, and
K.-K. Wong, “Energy-efficient heterogeneous cellular networks
with spectrum underlay and overlay access,” IEEE Transactions on
Vehicular Technology, vol. 67, no. 3, pp. 2439–2453, 2018.
[3] D.-S. Lee, C.-S. Chang, R. Zhang, and M.-P. Lee, “Resource allocation
for urllc and embb traffic in uplink wireless networks,” 2022.
[4] N. Mokari, F. Alavi, S. Parsaeefard, and T. Le-Ngoc, “Limitedfeedback
resource allocation in heterogeneous cellular networks,”
IEEE Transactions on Vehicular Technology, vol. 65, no. 4, pp.
2509–2521, 2016.
[5] F. Fu and M. van der Schaar, “Learning to compete for resources in
wireless stochastic games,” IEEE Transactions on Vehicular Technology,
vol. 58, no. 4, pp. 1904–1919, 2009.
[6] S. Li, X. Hu, and Y. Du, “Deep reinforcement learning and game
theory for computation offloading in dynamic edge computing markets,”
IEEE Access, vol. 9, pp. 121 456–121 466, 2021.
[7] A. M. Seid, G. O. Boateng, B. Mareri, G. Sun, andW. Jiang, “Multiagent
drl for task offloading and resource allocation in multi-uav
enabled iot edge network,” IEEE Transactions on Network and Service
Management, vol. 18, no. 4, pp. 4531–4547, 2021.
[8] Y. Azimi, S. Yousefi, H. Kalbkhani, and T. Kunz, “Energy-efficient
deep reinforcement learning assisted resource allocation for 5g-ran
slicing,” IEEE Transactions on Vehicular Technology, vol. 71, no. 1,
pp. 856–871, 2022.
[9] Y. Chen and H. Wu, “Resource allocation for edge collaboration
with deep deterministic policy gradient in smart railway,” in
2021 IEEE 4th International Conference on Electronics Technology
(ICET), 2021, pp. 1163–1167.
[10] J. Zhao, X. Hu, and X. Du, “Algorithm of task offloading
and resource allocation based on reinforcement learning
in edge computing,” in 2021 IEEE 5th Information Technology,
Networking,Electronic and Automation Control Conference
(ITNEC), vol. 5, 2021, pp. 1266–1269.
[11] J. N. Tsitsiklis and Y. Xu, “Bayesian proportional resource allocation
games,” in 2010 48th Annual Allerton Conference on Communication,
Control, and Computing (Allerton), 2010, pp. 1556–1561.
[12] A. Garnaev and W. Trappe, “Fair resource allocation under an unknown
jamming attack: a bayesian game,” in 2014 IEEE International
Workshop on Information Forensics and Security (WIFS),
2014, pp. 227–232.
[13] A. Deligiannis and S. Lambotharan, “A bayesian game theoretic
framework for resource allocation in multistatic radar networks,” in
2017 IEEE Radar Conference (RadarConf), 2017, pp. 0546–0551.
[14] K. Akkarajitsakul, E. Hossain, and D. Niyato, “Distributed resource
allocation in wireless networks under uncertainty and application of
bayesian game,” IEEE Communications Magazine, vol. 49, no. 8,
pp. 120–127, 2011.
[15] M. J. Osborne, An introduction to game theory. New York: Oxford
University Press, 2004.
[16] R. Nelson, Probability, stochastic processes, and queueing theory:
The mathematics of computer performance modelling. New York
City: Springer-Verlag, 1995.