Allocation of Resources for Protecting Public Goods against Uncertain Threats Generated by Agents

This paper analyses a framework for designing robust decisions against uncertain threats to public goods generated by multiple agents. The agents can be intentional attackers such as terrorists, agents accumulating values in flood or earthquake prone locations, or agents generating extreme events such as electricity outage and recent BP oil spill, etc. Instead of using a leader-follower game theoretic framework, this paper proposes a decision theoretic model based on two-stage stochastic optimization (STO) models for advising optimal resource allocations (or regulations) in situations characterized by uncertain perceptions of agent behaviors. In particular, the stochastic mini-max model and multi- shortfalls) is advanced in the context of quantile optimization for dealing with potential extreme events. Proposed framework can deal with both direct and indirect judgments on the decision makers perception about uncertain agent behaviors, either directly by probability density estimation, or indirectly by probabilistic inversion. The quantified distributions are treated as input to the stochastic optimization models in order to address inherent uncertainties. Robust decisions can then be obtained against all possible threats, especially with extreme consequences. This paper also introduces and compares three different computational algorithms which can be used to solve arising two-stage STO problems, including bilateral descent method, linear programming approximation and stochastic quasi-gradient method. A numerical example of high dimensionlity is presented for illustration of their performance under large number of scenarios typically required for dealing with low probability extreme events. Case studies include deensive resource allocations among cities and security of electricity networks.