2008 Allerton Security Investment Games of Interdependent Organizations
The paper develops a game‑theoretic framework for analyzing security investment decisions among autonomous organizations whose security outcomes are interdependent. Organizations are modeled as nodes in a weighted directed graph (a linear influence network) where each edge weight captures the magnitude and sign (positive or negative) of the externality that one organization’s investment has on another. Each organization chooses a non‑negative investment level; its utility is a concave benefit function of the effective investment (its own plus weighted contributions from neighbors) minus a linear cost. Under assumptions of continuous, strictly increasing, strictly concave benefit functions, the authors show that Nash equilibria of the resulting game correspond to solutions of a linear complementarity problem (LCP). They prove existence and uniqueness of the equilibrium when the influence matrix is strictly diagonally dominant, and that an asynchronous best‑response dynamics algorithm converges to this unique equilibrium from any starting point. The paper further introduces a constructive method for Pareto‑improving the equilibrium: by continuously increasing selected edge weights, two cooperating nodes can reduce their own investments while leaving all other nodes’ investments unchanged. This is formalized via a trajectory of weight matrices and equilibrium investments. The framework is illustrated with a web‑authentication scenario where firms share user credentials; high‑risk firms (e.g., banks) end up over‑investing relative to low‑risk firms, and the Pareto‑improvement method can lower the high‑risk firms’ costs without harming others. The authors conclude by highlighting the model’s applicability to various security policy questions and suggest future work on relaxing diagonal‑dominance and allowing investment in other entities.
Topics
Linear Influence Network Model for Interdependent Security
Defines the weighted directed graph representation of organizations, the influence matrix \(W\), and how each node’s effective investment is a linear combination of its own investment and weighted contributions from neighbors.
Quasilinear Utility Specification and Assumptions
Specifies each organization’s utility as a concave benefit function of effective investment minus linear cost, and lists the six technical assumptions (continuity, monotonicity, concavity, zero at zero, marginal benefit > cost at zero, marginal benefit < cost at infinity).
Equilibrium Characterization via Linear Complementarity Problem
Shows that Nash equilibria satisfy first‑order conditions that can be expressed as an LCP, linking equilibrium investments to the matrix \(W\) and the vector \(b\) (where \(b_i\) solves \(V'_i(b_i)=c_i\)).
Existence, Uniqueness, and Diagonal Dominance Theorem
Proves that if \(W\) is strictly diagonally dominant, the game has a unique Nash equilibrium, providing a clear sufficient condition for well‑posedness.
Asynchronous Best‑Response Dynamics (ABRD) Convergence
Presents Algorithm 1, where agents update their best‑responses at possibly different times, and proves convergence to the unique equilibrium under diagonal dominance.
Pareto‑Improving Weight Adjustments for Two Nodes
Constructs continuous trajectories of the influence matrix that increase selected edge weights, causing two cooperating nodes to strictly lower their investments while all other nodes’ investments remain unchanged.
Lemma 5.2 on Invariant Sub‑Equilibria
Establishes that if internal edge weights of a subgraph and the product of external weights with neighbor investments stay constant, the subgraph’s equilibrium investments are unchanged.
Web Authentication Case Study
Applies the model to a network of firms sharing passwords, distinguishing high‑risk and low‑risk firms, showing how equilibrium leads high‑risk firms to over‑invest, and demonstrating the Pareto‑improvement process on this concrete network.