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.

翻译：布拉托博弈是博弈论中最古老的模型之一，最初由博雷尔于1921年提出，至今已逾百年。然而，该模型最初旨在模拟两个中央集权军队在零和"战线"上的对抗，这一特定场景在现代应用中较为有限。本研究提出并探讨了"私人布拉托博弈"——一种与众包及社交媒体相关的变体。私人布拉托博弈的关键区别在于：个体主体独立行动，无需中央"指挥官"协调。该模型自然衍生自多种场景，例如活动团体在多议题上的竞争、党派筹款人在多州选举中的角逐，或持有政治偏见的社交媒体用户将新闻文章标记为虚假信息等。我们完整刻画了私人布拉托博弈的纳什稳定性。具体而言，研究表明结果函数对博弈结局具有关键影响：我们分别考察了战线胜负由多数原则（中位数结果）决定，或由考虑所有主体的平滑机制（均值结果）决定的情形。我们进一步分析了这两种结果函数如何影响"错误配置努力"（即其选择无法影响最终结果的主体数量）。总体而言，当稳定配置存在时，均值结果能确保主体在各战线上接近均匀分布，从而最小化错误配置努力。然而，均值结果函数在稳定配置存在与否上呈现出混沌模式。对于中位数结果，我们精确刻画了稳定配置存在的条件，但证明该结果函数常导致主体在各战线上的极端不均衡分配。