Security Games over Lexicographic Orders

Security is rarely single-dimensional and is in most practical instances a tradeoff between dependent, and occasionally conflicting goals. The simplest method of multi-criteria optimization and games with vector-valued payoffs, is transforming such games into ones with scalar payoffs, and looking for Pareto-optimal behavior. This usually requires an explicit weighting of security goals, whereas practice often only lets us rank security goals in terms of importance, but hardly admits a crisp numerical weight being assigned. This work picks up the issue of optimizing security goals in descending order of importance, coming to the computation of an optimal solution w.r.t. lexicographic orders. This is interesting in two ways, as it (i) is theoretically nontrivial since lexicographic orders do not generally admit representations by continuous utility functions, hence render Nash’s classical result inapplicable, and (ii) practically relevant since it avoids ambiguities by subjective (and perhaps unsupported) importance weight assignments. We corroborate our results by giving numerical examples, showing a method to design zero-sum games with a set of a-priori chosen Nash equilibria. This simple instance of mechanism design may be of independent interest.