本研究提出了一個貝葉斯博弈模型來分配資源，在這個遊戲中，有 c 單位的資源需要分配給 n 個玩家，每個玩家 i 有一個需求量 v_i 單位的資源。收益的設置方式使得玩家 i 在獲得不超過 v_i 單位的資源時感到滿足。我們假設資源是根據最小的需求優先原則分配給玩家，針對這個遊戲，我們分析了平衡策略函數。

在研究中，我們考慮了玩家在資源分配過程中的策略選擇。通過貝葉斯博弈模型，玩家根據自己對其他玩家需求的估計來制定策略。通過對平衡策略的研究，分析結果顯示，平衡策略函數有兩種模式：誠實模式和保留模式。誠實模式中，玩家根據自己的真實需求進行請求，資源將根據真實需求分配；保留模式中，玩家根據預期的競爭情況適當降低請求，以增加獲得資源的機會，這種模式使得玩家具備自主性決策。

我們的研究在資源分配問題中通過求解及分析微分積分方程確定策略轉折點，有效地計算不同參數情況下的平衡策略，並以數值迭代結果加以驗證，這為實際應用中的資源分配問題提供了重要參考。

In this paper we propose a Bayesian game to allocate resources. In this game, there are c units of resources to be allocated to n players. Agent i has a demand of v_i units of resources. Payoffs are setup such that player i is contented with no more than v_i units of resources. We assume that resources are granted to the players from the smallest request first. For this game, we analyze the equilibrium strategy functions. In the examples we study, we show that the equilibrium strategy functions are in a truthful mode or in a hold-back mode.

[1] Adnan Abid, Muhammad Faraz Manzoor, Muhammad Shoaib Farooq, Uzma Farooq, and Muzammil Hussain. Challenges and issues of resource allocation
techniques in cloud computing. KSII Transactions on Internet & Information Systems, 14(7), 2020.
[2] Xuemei Li and Li Da Xu. A review of internet of things — resource allocation. IEEE Internet of Things Journal, 8(11):8657–8666, 2021.
[3] Yongjun Xu, Guan Gui, Haris Gacanin, and Fumiyuki Adachi. A survey on resource allocation for 5G heterogeneous networks: Current research, future trends, and challenges. IEEE Communications Surveys & Tutorials, 23(2):668–695, 2021.
[4] Francesca Fossati, Sahar Hoteit, Stefano Moretti, and Stefano Secci. Fair resource allocation in systems with complete information sharing. IEEE/ACM Transactions on Networking, 26(6):2801–2814, 2018.
[5] F. P. Kelly, A. K. Maulloo, and D. K. H. Tan. Rate control for communication networks: Shadow prices, proportional fairness and stability. J. Oper. Res. Soc., 49(3):237–252, March 1998.
[6] W. Ogryczak et al. Fair optimization and networks: A survey. J. Appl. Math., 2014, Sep. 2014.
[7] J. Mo and J. Walrand. Fair end-to-end window-based congestion control. IEEE/ACM Trans. on Networking, 8(5):556–567, October 2000.
[8] R. Jain, D. M. Chiu, , and W. R. Hawe. A quantitative measure of fairness and discrimination for resource allocation in shared computer system. Eastern research laboratory, digital equipment corporation, Hudson, MA, USA, 1984.
[9] Duan-Shin Lee, Cheng-Shang Chang, Ruhui Zhang, and Mao-Pin Lee. Resource allocation for URLLC and eMBB traffic in uplink wireless networks. Performance Evaluation, 161, September 2023.
[10] Zhenhua Deng. Distributed algorithm design for resource allocation problems of high-order multiagent systems. IEEE Transactions on Control of Network Systems, 8(1):177–186, 2021.
[11] B. Shao, M. Li, and X. Shi. Distributed resource allocation algorithm for general linear multiagent systems. IEEE Access, 10:74691–74701, 2022.
[12] Judit Bar-Ilan and David Peleg. Distributed resource allocation algorithms. In International Workshop on Distributed Algorithms, pages 277–291. Springer, 1992.
[13] Yingmo Jie, Cheng Guo, Kim-Kwang Raymond Choo, Charles Zhechao Liu, and Mingchu Li. Game-theoretic resource allocation for fog-based industrial internet of things environment. IEEE Internet of Things Journal, 7(4):3041– 3052, 2020.
[14] I.J. Curiel, M. Maschler, and S.H. Tijs. Bankruptcy games. Zeitschrift f¨ur Operations Research, 31:A143–A159, 1987.
[15] Jun Huang, Ying Yin, Yi Sun, Yanxiao Zhao, Cong-cong Xing, and Qiang Duan. Game theoretic resource allocation for multicell D2D communica-tions with incomplete information. In 2015 IEEE International Conference on Communications (ICC), pages 3039–3044. IEEE, 2015.
[16] Fei Teng and Fr´ed´eric Magoul`es. A new game theoretical resource allocation algorithm for cloud computing. In Advances in Grid and Pervasive Computing: 5th International Conference, GPC 2010, Hualien, Taiwan, May 10-13, 2010. Proceedings 5, pages 321–330. Springer, 2010.
[17] A. S. Tanenbaum. Modern Operating Systems. Pearson Education, Inc., 3 edition, 2008.
[18] F. Guillemin and A. Simonian. Stationary analysis of the shortest queue first service policy. Queueing System, 77:393–426, 2014.
[19] Linus E. Schrage and Louis W. Miller. The queue M/G/1 with the shortest remaining processing time discipline. Operations Research, 14(4):670 – 684, 1966.
[20] Madelon A. de Kemp, Michel Mandjes, and Neil Olver. Performance of the smallest-variance-first rule in appointment sequencing. Operations Research, 69(6):1909 – 1935, November-December 2021.
[21] Moty Amar, Dan Ariely, Shahar Ayal, Cynthia E. Cryder, and Scott I. Rick. Winning the battle but losing the war: The psychology of debt management. Journal of Marketing Research, XLVIII (Special Issue 2011):S38 – S50, 2011.
[22] Martin J. Osborne. An introduction to game theory. Oxford University Press, New York, 2004.
[23] S. Ghahramani. Fundamentals of Probability with Stochastic Processes. CRC Press, Boca Raton, Florida, 4 edition, 2019.
[24] T. Heinrich, Y. Jang, L. Mungo, M. Pangallo, A. Scott, B. Tarbush, and S. Wiese. Best-response dynamics, playing sequences, and convergence to equilibrium in random games. Int J Game Theory, 52:703 – 735, 2023.