Approximating Power Indices to Assess Cybersecurity Criticality

This paper describes our work in developing approximation algorithms to calculate the Banzhaf Power Index (BPI) in a bicooperative game (that is, games with two coalitions) with large n for the number of players. Our motivation for this work is applying a cooperative game-theoretic framework to cybersecurity scenarios: our past experience with network defense made us receptive to the principle that differences in the criticality of players or network resources in a coalition setting is not always proportional to their differences in weighting or numbers of votes. Hence, calculating a game-theoretic power measure makes sense as a basis for both assessments and allocation decisions. The challenge is that for most real-world scenarios, the value of n is too high for an exact algorithm to solve in time to be actionable in a network defense scenario. We describe our approximation algorithm, and show empirical results that demonstrate that it produces solid estimates of the BPIs that would result from an exact calculation. Therefore, this approximation approach has utility in scenarios where it is imperative to deliver timely results and network membership can be dynamic.