The Inference Report

We identify a homological barrier to stable learning of local Nash equilibria, also known as first-order Nash equilibria (FONE), in constrained non-concave differentiable games. We consider continuous deterministic learning dynamics that fix every FONE point and aim to attract all initial conditions to the full FONE set. We show that this convergence guarantee is impossible when the target set has nontrivial positive-degree cohomology, such as that arising from a loop, and the state space is compact and contractible. We exhibit this barrier in a simple non-concave quadratic zero-sum game on a box whose exact FONE set consists of a diagonal six-edge loop and an isolated point. Consequently, no such dynamics can make the full FONE set a stable global attractor. We further show that degree two is the minimal polynomial degree at which this barrier can arise and extend the construction to obtain nontrivial homology in arbitrarily high degrees. Finally, we establish that the barrier persists for sufficiently small approximate-FONE targets under small perturbations, yielding an open, positive-volume family of quadratic zero-sum games with the same homological barrier.

Learning to profit from trade requires learning both sides of a market: what a seller is willing to accept and what a buyer is willing to pay. We study repeated bilateral trade, where an intermediary posts a purchase price to a seller and a sale price to a buyer whose valuations are drawn from an unknown joint distribution. Trade occurs only if both parties accept their respective prices, in which case the intermediary earns the difference between the two prices. The intermediary observes only whether trade occurs, without observing either valuation. We show that learning both sides of the market is no harder, in terms of regret, than learning just one. Specifically, we design an algorithm achieving regret $\widetilde{\mathcal{O}}(T^{2/3})$ against the best fixed pair of prices, matching, up to logarithmic factors, the classical lower bound for one-dimensional dynamic pricing. Our result shows that the classical one-dimensional regret rate survives the transition to two dimensions, raising the question of whether this absence of a curse of dimensionality extends to arbitrarily high dimensions.

We present a unified potential method for contextual search and bilateral trade with one-bit feedback and obtain order-tight algorithms for both problems. Our framework starts from a Bayesian perspective in which the unknown target (for contextual search) and the unknown pair of valuations (for bilateral trade) are sampled from a known prior distribution. In that model, the decision maker bases its prediction on the posterior distribution, obtained by conditioning on feedback received in previous rounds. For Bayesian contextual search, we present an algorithm with a total loss bounded by $O(\mathrm{Tr} \sqrt Σ)$, where $Σ$ is the covariance matrix of the target's prior; and for contextual bilateral trade, our algorithm's regret is bounded by $O(\mathrm{Tr} \sqrt{Σ_S}+\mathrm{Tr} \sqrt{Σ_B})$, where $Σ_S$ and $Σ_B$ are the covariance matrices of the seller's and buyer's valuations. We then use ideas from this Bayesian setting and adapt a ``dual'' of this trace-square-root potential to the classical worst-case setting, where the target and valuations are chosen adversarially from the unit Euclidean ball. For both contextual search and bilateral trade, we obtain tight $O(\sqrt d)$-regret algorithms. These results match the existing lower bounds up to a constant factor and improve on the previous guarantees of $O(d\log d)$ for contextual search and of $O(d6^d)$ for bilateral trade.

Leaderboards are a popular way to compare large language model (LLM) capabilities and can influence how model developers invest resources. We ask when competition on leaderboards leads developers to produce models with similar capability profiles, which we call monoculture, or similar performance levels. We study this by formulating leaderboard competition as a game in which model developers invest resources across capabilities to maximize their win rate. We show that similar production functions can induce similar capability profiles, while comparative advantage can sustain distinct capability profiles. We then study convergence in performance levels and show that diminishing returns can lead to an increasingly crowded competitive frontier. Using data from LM Arena and OpenRouter, we find that developers improve more in categories where they lag competitors, while still retaining category-specific advantages. At the same time, we find that the frontier of models becomes increasingly crowded, alongside increasingly diversified model usage patterns. Our results suggest that increasing competition does not need to produce interchangeable models, but can instead sustain competitive models with distinct strengths.

We give a deterministic polynomial-time algorithm for computing an exact extensive-form proper equilibrium in a finite two-player zero-sum extensive-form game with perfect recall and rational data. Properness is understood in the agent normal form: an action at an information set is penalized relative to a better action while all other local behavioral choices, including the player's own future actions, remain fixed. Our algorithm assigns each information set a geometric permutahedron of allowable action probabilities. A scaled doubly stochastic representation embeds these restrictions in a polynomial-size sequence-form polytope; a saddle point then satisfies the local $\varepsilon$-properness inequalities. To obtain an exact equilibrium rather than a solution at an arbitrarily small numerical perturbation, we establish sign stability of a parametric linear system, construct a persistent feasible basis, and extract the limits of behavioral ratios. The resulting probabilities are rational and have polynomial encoding length, including at information sets whose realization probabilities vanish.

In many modern multi-agent systems, a mediator seeks to influence strategic decision dynamics using only outcome-level feedback and without knowing agents' utilities. In this work, we study how a mediator can learn an unknown game and steer adaptive agents toward outcomes that maximize her utility using only trajectory feedback. The mediator knows the game form and her own objective, but not the agents' utilities. Moreover, she commits to nonnegative payments and provides recommendations and contexts, but receives only a single realized terminal outcome rather than full-information feedback. We develop halving-type algorithms that learn utility differences in normal-form games and conditional continuation-value differences in extensive-form games, achieving a sample complexity of $\widetilde O(\varepsilon^{-2})$ that matches worst-case lower bounds up to logarithmic factors. We then combine these learning procedures with a Lagrangian formulation in an online Learn-Compute-Steer algorithm that interleaves learning, equilibrium computation, and steering toward mediator-optimal extensive-form correlated equilibria (EFCE). With high probability, the Learn-Compute-Steer algorithm achieves a directness gap and an optimality gap of $\widetilde O(T^{-1/4})$, while time-average payments are of $\widetilde O(T^{-1/8})$.