  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.
