This supplementary document contains a formal analysis of some of the extensions briefly discussed in Section 5 of the published version. Section A1 considers the game in which agents receive signals about the size of past attacks. Section A2 considers the game with observable shocks to the fundamentals. Section A3 considers the variant in which agents observe the shocks with a one-period lag. Section A4 considers the game with short-lived agents in which the fundamentals follow a random walk. Finally, Section A5 collects the proofs of the formal results contained in this document.