Private Blotto: Viewpoint Competition with Polarized Agents

Colonel Blotto games are one of the oldest settings in game theory, originally proposed over a century ago in Borel 1921. However, they were originally designed to model two centrally-controlled armies competing over zero-sum "fronts", a specific scenario with limited modern-day application. In this work, we propose and study Private Blotto games, a variant connected to crowdsourcing and social media. One key difference in Private Blotto is that individual agents act independently, without being coordinated by a central "Colonel". This model naturally arises from scenarios such as activist groups competing over multiple issues, partisan fund-raisers competing over elections in multiple states, or politically-biased social media users labeling news articles as misinformation. In this work, we completely characterize the Nash Stability of the Private Blotto game. Specifically, we show that the outcome function has a critical impact on the outcome of the game: we study whether a front is won by majority rule (median outcome) or a smoother outcome taking into account all agents (mean outcome). We study how this impacts the amount of "misallocated effort", or agents whose choices doesn't influence the final outcome. In general, mean outcome ensures that, if a stable arrangement exists, agents are close to evenly spaced across fronts, minimizing misallocated effort. However, mean outcome functions also have chaotic patterns as to when stable arrangements do and do not exist. For median outcome, we exactly characterize when a stable arrangement exists, but show that this outcome function frequently results in extremely unbalanced allocation of agents across fronts.