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Across domains as varied as ecology, economics, and online platforms, individuals are incentivized to position themselves within resource-rich strategy spaces. In this work we introduce a new spatial game framework called the influencer's game. Within this new framework players' strategies are represented as positions within a trait space. Players then receive a share of a portion of a resource based on their relative influence at that resource's points traits. For the influencer's game, we show that there is a unique symmetric Nash equilibrium that exists under strict log-concavity of the influence kernel. Using the unique equilibrium we show that the stability of the system depends on the influence kernel's parameters, number of players, and the resource distribution shape. For any dimension we demonstrate a stability threshold that depends on both the scatter and curvature of the players. When this threshold is crossed, the symmetric equilibrium destabilizes and gives way to a variety of bifurcation types; including, specialist clusters, limit cycles, and sub-games that reproduce the larger game's structure. We ground these results using real-world data for resources as a demonstration of the game's application. Finally we extend the game dynamics to use multi-agent reinforcement learning and further show that the predicted bifurcation structure persists under partial observability and discretized strategy spaces. In all, the influencer's game provides a novel framework for predicting when competitive pressure drives convergence versus fragmentation with possible applications across domains.

We introduce Coarse Q-learning, a reinforcement learning framework in which a decision-maker partitions alternatives into coarse similarity classes, chooses among available classes using logit, and learns class valuations by averaging experienced payoffs. Even in stationary bandit problems with independently drawn menus, choices change the composition of pooled feedback. By deriving the continuous-time mean field ODE limit, we show this endogenous sampling can generate a unique globally attracting mixed equilibrium with limiting indifference, multiple stable strict rankings, or persistent cycling. Learning can reduce welfare even when the coarse representation admits efficient choice. An application demonstrates how digital platforms may exploit this feedback by promoting high-quality products to sustain purchases of inferior products under a common brand.

In practice, auction data are often endogenously censored and anonymous, revealing only limited outcome statistics rather than full bid profiles. We study robust auction design when the seller observes only the distribution of a single anonymous order statistic of the bidders' values and seeks to maximize worst-case expected revenue over all product distributions consistent with the observed statistic. We show that simple and widely used mechanisms are robustly optimal. Specifically, posted pricing is robustly optimal given the distribution of the highest value; the Myerson auction designed for the unique consistent i.i.d. distribution, with uniformly random tie-breaking, is robustly optimal given the lowest value distribution; and the second-price auction with an optimal reserve is robustly optimal when an intermediate order statistic is observed and the implied i.i.d. distribution is regular above its reserve. Without any regularity assumption, when the second-highest value is observed, the best worst-case revenue of a second-price auction with a reserve is still at least a $2-\sqrt{2}\approx 0.586$ fraction of the optimal worst-case revenue. More generally, for a broad class of monotone symmetric mechanisms depending only on the top $k$ order statistics, including multi-unit and position auctions, the worst-case revenue is attained under the i.i.d. distribution consistent with the observed $k$-th order statistic. Our results provide a tractable foundation for non-discriminatory auction design, where fairness and privacy are intrinsic consequences of the information structure rather than imposed constraints.

Colonel Blotto is a fundamental resource-allocation game in which two players divide their budgets across battlefields, and the higher allocation on each battlefield wins its value. The problem of computing an equilibrium efficiently has remained unresolved for its classical continuous version with arbitrary budgets and with values that may vary across battlefields but are the same for both players. We give a fully polynomial-time approximation scheme that computes an $\varepsilon$-Nash equilibrium in time polynomial in the input length and $1/\varepsilon$, and nearly linear in the input length for fixed $\varepsilon$. For sufficiently balanced budgets and battlefield values, we also compute exact equilibria in linear time.