Dr. David Blackwell was a mathematician and statistician of the first rank, whose contributions to statistical theory, game theory, and decision theory predated many of the algorithmic breakthroughs that define modern artificial intelligence. This survey examines three of his most consequential theoretical results the Rao Blackwell theorem, the Blackwell Approachability theorem, and the Blackwell Informativeness theorem (comparison of experiments) and traces their direct influence on contemporary AI and machine learning. We show that these results, developed primarily in the 1940s and 1950s, remain technically live across modern subfields including Markov Chain Monte Carlo inference, autonomous mobile robot navigation (SLAM), generative model training, no-regret online learning, reinforcement learning from human feedback (RLHF), large language model alignment, and information design. NVIDIAs 2024 decision to name their flagship GPU architecture (Blackwell) provides vivid testament to his enduring relevance. We also document an emerging frontier: explicit Rao Blackwellized variance reduction in LLM RLHF pipelines, recently proposed but not yet standard practice. Together, Blackwell theorems form a unified framework addressing information compression, sequential decision making under uncertainty, and the comparison of information sources precisely the problems at the core of modern AI.

In this paper, we focus on solving Extensive-F orm Games (EFGs).

The Counterfactual Regret Minimization (CFR) algorithm Zinkevich et al. [2007] is crucial for solving extensive-form games with incomplete information.

This tutorial provides a survey of algorithms for Defensive Forecasting, where predictions are derived not by prognostication but by correcting past mistakes. Pioneered by Vovk, Defensive Forecasting frames the goal of prediction as a sequential game, and derives predictions to minimize metrics no matter what outcomes occur. We present an elementary introduction to this general theory and derive simple, near-optimal algorithms for online learning, calibration, prediction with expert advice, and online conformal prediction.

Recent advancements in artificial intelligence (AI) have leveraged large-scale games as benchmarks to gauge progress, with AI now frequently outperforming human capabilities. Traditionally, this success has largely relied on solving Nash equilibrium (NE) using variations of the counterfactual regret minimization (CFR) method in games with incomplete information. However, the variety of Nash equilibria has been largely overlooked in previous research, limiting the adaptability of AI to meet diverse human preferences. To address this challenge, where AI is powerful but struggles to meet customization needs, we introduce a novel approach: Preference-CFR, which incorporates two new parameters: preference degree and vulnerability degree. These parameters allow for greater flexibility in AI strategy development without compromising convergence. Our method significantly alters the distribution of final strategies, enabling the creation of customized AI models that better align with individual user needs. Using Texas Hold'em as a case study, our experiments demonstrate how Preference CFR can be adjusted to either emphasize customization, prioritizing user preferences, or to enhance performance, striking a balance between the depth of customization and strategic optimality.

We consider Blackwell approachability, a very powerful and geometric tool in game theory, used for example to design strategies of the uninformed player in repeated games with incomplete information. We extend this theory to "generalized quitting games" , a class of repeated stochastic games in which each player may have quitting actions, such as the Big-Match. We provide three simple geometric and strongly related conditions for the weak approachability of a convex target set. The first is sufficient: it guarantees that, for any fixed horizon, a player has a strategy ensuring that the expected time-average payoff vector converges to the target set as horizon goes to infinity. The third is necessary: if it is not satisfied, the opponent can weakly exclude the target set. In the special case where only the approaching player can quit the game (Big-Match of type I), the three conditions are equivalent and coincide with Blackwell's condition. Consequently, we obtain a full characterization and prove that the game is weakly determined-every convex set is either weakly approachable or weakly excludable. In games where only the opponent can quit (Big-Match of type II), none of our conditions is both sufficient and necessary for weak approachability. We provide a continuous time sufficient condition using techniques coming from differential games, and show its usefulness in practice, in the spirit of Vieille's seminal work for weak approachability.Finally, we study uniform approachability where the strategy should not depend on the horizon and demonstrate that, in contrast with classical Blackwell approacha-bility for convex sets, weak approachability does not imply uniform approachability.