Treffer: Game-Theoretic Optimization on School Safety: Resource Allocation Against Strategic Attacks.
Title:
Game-Theoretic Optimization on School Safety: Resource Allocation Against Strategic Attacks.
Authors:
Unal Eyi S; Department of Industrial and Systems Engineering, University at Buffalo, Buffalo, New York, USA., Zhuang J; Department of Industrial and Systems Engineering, University at Buffalo, Buffalo, New York, USA.
Source:
Risk analysis : an official publication of the Society for Risk Analysis [Risk Anal] 2025 Dec; Vol. 45 (12), pp. 5023-5042. Date of Electronic Publication: 2025 Dec 05.
Publication Type:
Journal Article
Language:
English
Journal Info:
Publisher: Blackwell Publishers Country of Publication: United States NLM ID: 8109978 Publication Model: Print-Electronic Cited Medium: Internet ISSN: 1539-6924 (Electronic) Linking ISSN: 02724332 NLM ISO Abbreviation: Risk Anal Subsets: MEDLINE
Imprint Name(s):
Publication: 2002- : Malden, MA : Blackwell Publishers
Original Publication: New York : Plenum Press, c1981-
Original Publication: New York : Plenum Press, c1981-
MeSH Terms:
References:
Aziz, R. A., M. He, and J. Zhuang. 2020. “An Attacker–Defender Resource Allocation Game With Substitution and Complementary Effects.” Risk Analysis 40, no. 7: 1481–1506. https://doi.org/10.1111/risa.13483.
Bushman, B. J., et al. 2016. “Youth Violence: What We Know And What We Need To Know.” American Psychologist 71, no. 1: 17–39. https://doi.org/10.1037/a0039687.
Center for Collegiate Mental Health. 2023. “Available Data.” Accessed April 12, 2025. https://ccmh.memberclicks.net/available‐data.
Chen, X., Y. Dong, and J. Zhuang. 2024. “Countering Terrorist Propaganda: Competitive Resource Allocation To Communities.” IISE Transactions 57, no. 1: 44–59. https://doi.org/10.1080/24725854.2023.2294457.
Cox, Jr, L. A. 2009. “Game Theory and Risk Analysis.” Risk Analysis: An International Journal 29, no. 8: 1062–1068. https://doi.org/10.1111/j.1539‐6924.2009.01247.x.
Ferguson, C. J., M. Coulson, and J. Barnett. 2011. “Psychological Profiles of School Shooters: Positive Directions and One Big Wrong Turn.” Journal of Police Crisis Negotiations 11, no. 2: 141–158. https://doi.org/10.1080/15332586.2011.581523.
Fisher, J. 2024. “Lessons Learned from the Robb Elementary School, Uvalde, Texas School Shooting.” Open Journal of Social Sciences 12, no. 4: 78–96.
Freilich, J. D., S. M. Chermak, N. M. Connell, B. R. Klein, and E. A. Greene‐Colozzi. 2021. Understanding the Causes of School Violence Using Open Source Data. Technical Report 301145, Office of Justice Programs, National Criminal Justice Reference Service. https://www.ncjrs.gov/pdffiles1/nij/grants/301145.pdf.
Hausken, K., and J. Zhuang. 2016. “The Strategic Interaction Between A Company and the Government Surrounding Disasters.” Annals of Operations Research 237, no. 1: 27–40. https://doi.org/10.1007/s10479‐014‐1684‐5.
He, F., J. Zhuang, and N. S. V. Rao. 2012. “Game‐Theoretic Analysis of Attack and Defense in Cyber‐Physical Network Infrastructures.” In Proceedings of the IISE Annual Conference, 3399–3406. Institute of Industrial and Systems Engineers.
Hunt, K., P. Agarwal, and J. Zhuang. 2021. “Technology Adoption for Airport Security: Modeling Public Disclosure and Secrecy in an Attacker‐Defender Game.” Reliability Engineering & System Safety 207: 107355. https://doi.org/10.1016/j.ress.2020.107355.
Hunt, K., and J. Zhuang. 2024. “A Review of Attacker‐Defender Games: Current State and Paths Forward.” European Journal of Operational Research 313, no. 2: 401–417. https://doi.org/10.1016/j.ejor.2023.04.009.
Jacobo, J. 2022. “These are the Security Measures the Uvalde School District had in Place.” ABC News, May 26, 2022. Accessed September 2, 2025. https://abcnews.go.com/US/security‐measures‐uvalde‐school‐district‐place/story?id=84966706.
Langman, P. 2009. Why Kids Kill: Inside the Minds of School Shooters Macmillan.
Leary, M. R., R. M. Kowalski, L. Smith, and S. Phillips. 2003. “Teasing, Rejection, and Violence: Case Studies of the School Shootings.” Aggressive Behavior: Official Journal of the International Society for Research on Aggression 29, no. 3: 202–214. https://doi.org/10.1002/ab.10061.
Mahbub, N., A. Le, and J. Zhuang. 2022. “Online Crowd‐Funding Strategy: A Game‐Theoretical Approach To A Kickstarter Case Study.” Annals of Operations Research 315, no. 2: 1019–1036. https://doi.org/10.1007/s10479‐020‐03857‐5.
Manshaei, M. H., Q. Zhu, T. Alpcan, T. Bacşar, and J.‐P. Hubaux. 2013. “Game Theory Meets Network Security and Privacy.” ACM Computing Surveys (CSUR) 45, no. 3: 1–39. https://doi.org/10.1145/2480741.2480742.
Moon, B., and L. F. Alarid. 2015. “School Bullying, Low Self‐Control, and Opportunity.” Journal of Interpersonal Violence 30, no. 5: 839–856. https://doi.org/10.1016/j.eswa.2024.123712.
New York State Assembly. 2022. 2022‐23 Assembly Budget Proposal. New York State Assembly. https://nyassembly.gov/2022budget/?sec=assembly.
Newman, K. S., C. Fox, D. Harding, J. Mehta, and W. Roth. 2008. Rampage: The Social Roots of School Shootings. Basic Books.
Nickerson, A. B., M. E. Manges, G. M. Bellavia, J. A. Livingston, L. N. Jenkins, and T. H. Feeley. 2023. “Bystander Intervention in Bullying and Sexual Harassment: Role of Personal and Perceived Peer Norms.” International Journal of Bullying Prevention 5, no. 2: 135–150. https://doi.org/10.1007/s42380‐022‐00125‐x.
Pillai, P. S., and S. Rao. 2014. “Resource Allocation in Cloud Computing Using the Uncertainty Principle of Game Theory.” IEEE Systems Journal 10, no. 2: 637–648.
Riedman, D. 2023. K‐12 School Shooting Database. Center for Homeland Defense and Security, Naval Postgraduate School. https://k12ssdb.org/.
Riedman, D. 2025. K‐12 School Shooting Database. Center for Homeland Defense and Security, Naval Postgraduate School. https://k12ssdb.org/all‐shootings accessed: 2025‐09‐02.
Safe and Sound Schools. 2022. State of School Safety Report. Safe and Sound Schools. https://safeandsoundschools.org/resources/state‐of‐school‐safety‐reports/.
Sallhammar, K., B. E. Helvik, and S. J. Knapskog. 2005. “Incorporating Attacker Behavior in Stochastic Models of Security.” In Proceedings of the International Conference on Security and Management, 79–85. CSREA Press.
Schildkraut, J., and K. Grogan. 2019. Are Metal Detectors Effective at Making Schools Safer? WestEd. https://wested.org/schoolsafetyreport.pdf.
Shan, X., and J. Zhuang. 2013a. “Cost of Equity in Homeland Security Resource Allocation in the Face of a Strategic Attacker.” Risk Analysis 33, no. 6: 1083–1099. https://doi.org/10.1111/j.1539‐6924.2012.01919.x.
Shan, X., and J. Zhuang. 2013b. “Hybrid Defensive Resource Allocations in the Face of Partially Strategic Attackers in a Sequential Defender–Attacker Game.” European Journal of Operational Research 228, no. 1: 262–272. https://doi.org/10.1016/j.ejor.2013.01.029.
Song, C., and J. Zhuang. 2017. “Two‐Stage Security Screening Strategies in the Face of Strategic Applicants, Congestions and Screening Errors.” Annals of Operations Research 258, no. 2: 237–262. https://doi.org/10.1007/s10479‐015‐2043‐x.
Suplina, N. 2022. The Economic Cost of Gun Violence. Everytown Research & Policy. Accessed September 2, 2025. https://everytownresearch.org/report/the‐economic‐cost‐of‐gun‐violence/.
Tambe, M. 2011. Security and Game Theory: Algorithms, Deployed Systems, Lessons Learned. Cambridge University Press.
Unal‐Eyi, S., and J. Zhuang. 2025. “A Scoping Review of the Literature on School Shooting From 1990 to 2023.” Trauma, Violence, & Abuse, ahead of print. https://doi.org/10.1177/15248380251350106.
Unson, I., and J. Zhuang. 2024. “Resource Allocation in Multi‐Layer, Continuous Defense, Security Games versus Strategic Attackers.” Risk Sciences, ahead of print. https://doi.org/10.1016/j.risk.2024.100010.
Vossekuil, B. 2002. The Final Report and Findings of the Safe School Initiative: Implications for the Prevention of School Attacks in the United States. U.S. Secret Service and U.S. Department of Education.
Wei, G., A. V. Vasilakos, Y. Zheng, and N. Xiong. 2010. “A Game‐Theoretic Method of Fair Resource Allocation for Cloud Computing Services.” Journal of Supercomputing 54: 252–269. https://doi.org/10.1007/s11227‐009‐0318‐1.
Wei, W., X. Fan, H. Song, X. Fan, and J. Yang. 2016. “Imperfect Information Dynamic Stackelberg Game Based Resource Allocation Using Hidden Markov for Cloud Computing.” IEEE Transactions on Services Computing 11, no. 1: 78–89. https://doi.org/10.1109/TSC.2016.2528246.
Wei, Z., and J. Zhuang. 2025. “Modeling Defensive Resource Allocation in Multilayered Systems Under Probabilistic and Strategic Risks.” Risk Analysis 45, no. 1: 177–193. https://doi.org/10.1111/risa.15070.
Xiao, W., K. Huang, X. Luo, and Y. Hong. 2013. “Study on Physical Layer Security with Game Theory.” In Proceedings of the 2013 IEEE 4th International Conference on Software Engineering and Service Science 697–700. IEEE. https://doi.org/10.1109/ICSESS.2013.6615402.
Yong, W. 2016. “Research on Network Security Situation Prediction Based on Markov Game Theory.” International Journal of Security and its Applications 10, no. 9: 301–308. https://doi.org/10.1007/s11277‐017‐5202‐3.
Zhang, A., K. Wang, J. Zhang, and B. A. Oudekerk. 2019. Indicators of School Crime and Safety: 2018. Technical Report NCES 2019‐047/NCJ 252571, National Center for Education Statistics.
Zhang, J., and J. Zhuang. 2019. “Modeling A Multi‐Target Attacker‐Defender Game With Multiple Attack Types.” Reliability Engineering & System Safety 185: 465–475. https://doi.org/10.1016/j.ress.2019.01.015.
Zhuang, J., and V. M. Bier. 2007. “Balancing Terrorism and Natural Disasters‐Defensive Strategy With Endogenous Attacker Effort.” Operations Research 55, no. 5: 976–991. https://doi.org/10.1287/opre.1070.0434.
Zhuang, J., and V. M. Bier. 2011. “Secrecy and Deception at Equilibrium, With Applications to Anti‐Terrorism Resource Allocation.” Defence and Peace Economics 22, no. 1: 43–61. https://doi.org/10.1080/10242694.2010.491668.
Zhuang, J., G. D. Saxton, and H. Wu. 2014. “Publicity vs. Impact in Nonprofit Disclosures and Donor Preferences: A Sequential Game With One Nonprofit Organization and N Donors.” Annals of Operations Research 221, no. 1: 469–491. https://doi.org/10.1007/s10479‐011‐0967‐3.
Bushman, B. J., et al. 2016. “Youth Violence: What We Know And What We Need To Know.” American Psychologist 71, no. 1: 17–39. https://doi.org/10.1037/a0039687.
Center for Collegiate Mental Health. 2023. “Available Data.” Accessed April 12, 2025. https://ccmh.memberclicks.net/available‐data.
Chen, X., Y. Dong, and J. Zhuang. 2024. “Countering Terrorist Propaganda: Competitive Resource Allocation To Communities.” IISE Transactions 57, no. 1: 44–59. https://doi.org/10.1080/24725854.2023.2294457.
Cox, Jr, L. A. 2009. “Game Theory and Risk Analysis.” Risk Analysis: An International Journal 29, no. 8: 1062–1068. https://doi.org/10.1111/j.1539‐6924.2009.01247.x.
Ferguson, C. J., M. Coulson, and J. Barnett. 2011. “Psychological Profiles of School Shooters: Positive Directions and One Big Wrong Turn.” Journal of Police Crisis Negotiations 11, no. 2: 141–158. https://doi.org/10.1080/15332586.2011.581523.
Fisher, J. 2024. “Lessons Learned from the Robb Elementary School, Uvalde, Texas School Shooting.” Open Journal of Social Sciences 12, no. 4: 78–96.
Freilich, J. D., S. M. Chermak, N. M. Connell, B. R. Klein, and E. A. Greene‐Colozzi. 2021. Understanding the Causes of School Violence Using Open Source Data. Technical Report 301145, Office of Justice Programs, National Criminal Justice Reference Service. https://www.ncjrs.gov/pdffiles1/nij/grants/301145.pdf.
Hausken, K., and J. Zhuang. 2016. “The Strategic Interaction Between A Company and the Government Surrounding Disasters.” Annals of Operations Research 237, no. 1: 27–40. https://doi.org/10.1007/s10479‐014‐1684‐5.
He, F., J. Zhuang, and N. S. V. Rao. 2012. “Game‐Theoretic Analysis of Attack and Defense in Cyber‐Physical Network Infrastructures.” In Proceedings of the IISE Annual Conference, 3399–3406. Institute of Industrial and Systems Engineers.
Hunt, K., P. Agarwal, and J. Zhuang. 2021. “Technology Adoption for Airport Security: Modeling Public Disclosure and Secrecy in an Attacker‐Defender Game.” Reliability Engineering & System Safety 207: 107355. https://doi.org/10.1016/j.ress.2020.107355.
Hunt, K., and J. Zhuang. 2024. “A Review of Attacker‐Defender Games: Current State and Paths Forward.” European Journal of Operational Research 313, no. 2: 401–417. https://doi.org/10.1016/j.ejor.2023.04.009.
Jacobo, J. 2022. “These are the Security Measures the Uvalde School District had in Place.” ABC News, May 26, 2022. Accessed September 2, 2025. https://abcnews.go.com/US/security‐measures‐uvalde‐school‐district‐place/story?id=84966706.
Langman, P. 2009. Why Kids Kill: Inside the Minds of School Shooters Macmillan.
Leary, M. R., R. M. Kowalski, L. Smith, and S. Phillips. 2003. “Teasing, Rejection, and Violence: Case Studies of the School Shootings.” Aggressive Behavior: Official Journal of the International Society for Research on Aggression 29, no. 3: 202–214. https://doi.org/10.1002/ab.10061.
Mahbub, N., A. Le, and J. Zhuang. 2022. “Online Crowd‐Funding Strategy: A Game‐Theoretical Approach To A Kickstarter Case Study.” Annals of Operations Research 315, no. 2: 1019–1036. https://doi.org/10.1007/s10479‐020‐03857‐5.
Manshaei, M. H., Q. Zhu, T. Alpcan, T. Bacşar, and J.‐P. Hubaux. 2013. “Game Theory Meets Network Security and Privacy.” ACM Computing Surveys (CSUR) 45, no. 3: 1–39. https://doi.org/10.1145/2480741.2480742.
Moon, B., and L. F. Alarid. 2015. “School Bullying, Low Self‐Control, and Opportunity.” Journal of Interpersonal Violence 30, no. 5: 839–856. https://doi.org/10.1016/j.eswa.2024.123712.
New York State Assembly. 2022. 2022‐23 Assembly Budget Proposal. New York State Assembly. https://nyassembly.gov/2022budget/?sec=assembly.
Newman, K. S., C. Fox, D. Harding, J. Mehta, and W. Roth. 2008. Rampage: The Social Roots of School Shootings. Basic Books.
Nickerson, A. B., M. E. Manges, G. M. Bellavia, J. A. Livingston, L. N. Jenkins, and T. H. Feeley. 2023. “Bystander Intervention in Bullying and Sexual Harassment: Role of Personal and Perceived Peer Norms.” International Journal of Bullying Prevention 5, no. 2: 135–150. https://doi.org/10.1007/s42380‐022‐00125‐x.
Pillai, P. S., and S. Rao. 2014. “Resource Allocation in Cloud Computing Using the Uncertainty Principle of Game Theory.” IEEE Systems Journal 10, no. 2: 637–648.
Riedman, D. 2023. K‐12 School Shooting Database. Center for Homeland Defense and Security, Naval Postgraduate School. https://k12ssdb.org/.
Riedman, D. 2025. K‐12 School Shooting Database. Center for Homeland Defense and Security, Naval Postgraduate School. https://k12ssdb.org/all‐shootings accessed: 2025‐09‐02.
Safe and Sound Schools. 2022. State of School Safety Report. Safe and Sound Schools. https://safeandsoundschools.org/resources/state‐of‐school‐safety‐reports/.
Sallhammar, K., B. E. Helvik, and S. J. Knapskog. 2005. “Incorporating Attacker Behavior in Stochastic Models of Security.” In Proceedings of the International Conference on Security and Management, 79–85. CSREA Press.
Schildkraut, J., and K. Grogan. 2019. Are Metal Detectors Effective at Making Schools Safer? WestEd. https://wested.org/schoolsafetyreport.pdf.
Shan, X., and J. Zhuang. 2013a. “Cost of Equity in Homeland Security Resource Allocation in the Face of a Strategic Attacker.” Risk Analysis 33, no. 6: 1083–1099. https://doi.org/10.1111/j.1539‐6924.2012.01919.x.
Shan, X., and J. Zhuang. 2013b. “Hybrid Defensive Resource Allocations in the Face of Partially Strategic Attackers in a Sequential Defender–Attacker Game.” European Journal of Operational Research 228, no. 1: 262–272. https://doi.org/10.1016/j.ejor.2013.01.029.
Song, C., and J. Zhuang. 2017. “Two‐Stage Security Screening Strategies in the Face of Strategic Applicants, Congestions and Screening Errors.” Annals of Operations Research 258, no. 2: 237–262. https://doi.org/10.1007/s10479‐015‐2043‐x.
Suplina, N. 2022. The Economic Cost of Gun Violence. Everytown Research & Policy. Accessed September 2, 2025. https://everytownresearch.org/report/the‐economic‐cost‐of‐gun‐violence/.
Tambe, M. 2011. Security and Game Theory: Algorithms, Deployed Systems, Lessons Learned. Cambridge University Press.
Unal‐Eyi, S., and J. Zhuang. 2025. “A Scoping Review of the Literature on School Shooting From 1990 to 2023.” Trauma, Violence, & Abuse, ahead of print. https://doi.org/10.1177/15248380251350106.
Unson, I., and J. Zhuang. 2024. “Resource Allocation in Multi‐Layer, Continuous Defense, Security Games versus Strategic Attackers.” Risk Sciences, ahead of print. https://doi.org/10.1016/j.risk.2024.100010.
Vossekuil, B. 2002. The Final Report and Findings of the Safe School Initiative: Implications for the Prevention of School Attacks in the United States. U.S. Secret Service and U.S. Department of Education.
Wei, G., A. V. Vasilakos, Y. Zheng, and N. Xiong. 2010. “A Game‐Theoretic Method of Fair Resource Allocation for Cloud Computing Services.” Journal of Supercomputing 54: 252–269. https://doi.org/10.1007/s11227‐009‐0318‐1.
Wei, W., X. Fan, H. Song, X. Fan, and J. Yang. 2016. “Imperfect Information Dynamic Stackelberg Game Based Resource Allocation Using Hidden Markov for Cloud Computing.” IEEE Transactions on Services Computing 11, no. 1: 78–89. https://doi.org/10.1109/TSC.2016.2528246.
Wei, Z., and J. Zhuang. 2025. “Modeling Defensive Resource Allocation in Multilayered Systems Under Probabilistic and Strategic Risks.” Risk Analysis 45, no. 1: 177–193. https://doi.org/10.1111/risa.15070.
Xiao, W., K. Huang, X. Luo, and Y. Hong. 2013. “Study on Physical Layer Security with Game Theory.” In Proceedings of the 2013 IEEE 4th International Conference on Software Engineering and Service Science 697–700. IEEE. https://doi.org/10.1109/ICSESS.2013.6615402.
Yong, W. 2016. “Research on Network Security Situation Prediction Based on Markov Game Theory.” International Journal of Security and its Applications 10, no. 9: 301–308. https://doi.org/10.1007/s11277‐017‐5202‐3.
Zhang, A., K. Wang, J. Zhang, and B. A. Oudekerk. 2019. Indicators of School Crime and Safety: 2018. Technical Report NCES 2019‐047/NCJ 252571, National Center for Education Statistics.
Zhang, J., and J. Zhuang. 2019. “Modeling A Multi‐Target Attacker‐Defender Game With Multiple Attack Types.” Reliability Engineering & System Safety 185: 465–475. https://doi.org/10.1016/j.ress.2019.01.015.
Zhuang, J., and V. M. Bier. 2007. “Balancing Terrorism and Natural Disasters‐Defensive Strategy With Endogenous Attacker Effort.” Operations Research 55, no. 5: 976–991. https://doi.org/10.1287/opre.1070.0434.
Zhuang, J., and V. M. Bier. 2011. “Secrecy and Deception at Equilibrium, With Applications to Anti‐Terrorism Resource Allocation.” Defence and Peace Economics 22, no. 1: 43–61. https://doi.org/10.1080/10242694.2010.491668.
Zhuang, J., G. D. Saxton, and H. Wu. 2014. “Publicity vs. Impact in Nonprofit Disclosures and Donor Preferences: A Sequential Game With One Nonprofit Organization and N Donors.” Annals of Operations Research 221, no. 1: 469–491. https://doi.org/10.1007/s10479‐011‐0967‐3.
Grant Information:
22STESE00001-04-00 U.S. Department of Homeland Security
Contributed Indexing:
Keywords: game theory; mental health interventions; physical security; resource allocation; school safety; strategic optimization
Entry Date(s):
Date Created: 20251205 Date Completed: 20251229 Latest Revision: 20251229
Update Code:
20251230
DOI:
10.1111/risa.70156
PMID:
41350116
Database:
MEDLINE
Weitere Informationen
School security remains a critical concern due to the increasing frequency of violent incidents, requiring a strategic balance between physical security measures and mental health programs. This study develops a game-theoretic framework to model the interaction between a school as the defender and a potential attacker, aiming to identify optimal investment decisions across two complementary layers of defense. Numerical illustrations calibrated with data from U.S. school shootings provide empirical support for the analysis. Through one-way and two-way sensitivity analyses, robustness tests, and scenario-based indifference curve analysis, we explore how attacker and defender valuations, intervention effectiveness, and defensive costs influence equilibrium strategies. The findings show that physical security measures have the strongest deterrent effect, but lasting protection depends on balanced investment in both security and mental health once their effectiveness exceeds critical thresholds. While physical security offers immediate deterrence, mental health interventions are essential for addressing underlying risk factors, emphasizing the complementary nature of the two approaches. The framework contributes to evidence-based decision-making for educational institutions and suggests future extensions to include external threats, incomplete information, and dynamic investment strategies.
(© 2025 Society for Risk Analysis.)