Self-Play

Regularized self-play -- the family behind DeepNash's Stratego play -- drives a two-player zero-sum policy to a Nash equilibrium by best-responding to a slowly moving, entropy-regularized reference policy $ρ$. When the game has a polytope of value-equivalent equilibria, the regularizer silently breaks the tie: with a uniform reference it selects the maximum-entropy member, the I-projection of $ρ$ onto the Nash set. Can the reference be used to choose the equilibrium on purpose? On five exactly solvable games plus a 2-D polytope, with exact best responses and equivalence tests over independent seeds, anchoring the reference at a target member and refining steers self-play to that member with mean coordinate error 0.007 at median exploitability $5\times10^{-5}$, TOST-equivalent to the request within $\pm0.05$; the anchoring persists through refinement and follows the reference, not the initialization. Selection follows the reach-weighted I-projection (slope 0.969 [0.950, 0.987]). We report with equal emphasis where the story breaks: fixed off-manifold references cost 0.08-0.25 exploitability; stiff or flat families require a smaller mirror step, set by a pre-registered rule; boundary targets undershoot; curvature predicts where boundary saturation bites (rank correlation 0.90, p=0.037) while interior precision is curvature-independent. Table and MLP steering maps are equivalent within $\pm0.03$ at every target (30 seeds); matched control arms show attention's robust signature is excess seed variance, any systematic shift bounded at 0.018 and not significant. Against a best response the selection-robustness trade-off is degenerate: steering matters only against fixed, non-equilibrium opponents. The recipe -- anchor the reference at the desired member and refine -- reinterprets the KL anchor of RLHF-style RL as a selection knob, not only a stability leash.

We give deterministic and uncoupled learning dynamics for finite multiplayer general-sum games under full-information feedback that achieve constant individual swap regret, independent of the horizon $T$. With $n$ players and at most $m$ actions each, the individual swap regret of every player is $O(\sqrt{n} m \log m \log^{5/2}(nm))$ at every finite horizon. Each player predicts the deviation gains, then uses these predictions to update a row-stochastic transition matrix, and plays its stationary distribution. The proof combines a potential argument exploiting stationarity with a two-scale higher-order prediction analysis, using rooted-tree representations to handle the nonlinear dependence of deviation gains on the stationary distributions. An adversarially robust variant, obtained through a generic common-prefix switching wrapper, preserves the self-play bound up to a universal constant and guarantees individual swap regret at most $7\sqrt{m T \log m}$ in the adversarial setting.

Recovering unmanned aerial vehicles (UAVs) in maritime environments is challenging due to wind turbulence and ship-deck motion, making it a valuable test case for alternative control and learning approaches as conventional landing approaches often become unreliable. We study simulated mid-air capture of quadrotor UAVs by a ship-mounted robotic arm, learning robust cooperative control policies with Heterogeneous-Agent Proximal Policy Optimization (HAPPO) Reinforcement Learning. We train with HAPPO using a curriculum and an adversarial wind agent (HARL-AC) in NVIDIA Isaac Lab, and compare the obtained control policies against those generated through curriculum-based domain randomization and a benchmark trained on a single sea state. In-distribution evaluation on sea states $0/4/5$ shows comparable success for HARL-AC and domain randomization of up to $97.5\%$. On out-of-distribution sea states $7/8/10$, HARL-AC generalizes better, achieving up to $16\%$ higher median success rate at sea state 10, and substantially lower crash rates of up to $14\%$ compared to the domain randomization policy. Furthermore, we show that the adversarially trained policy shows more cautious behavior, slightly increasing timeouts by $<3\%$, but yields safer recovery behavior in severe, unseen conditions.

Combining search with function approximation has driven major advances in game-playing programs, making self-play algorithms more competitive than ever. Still, the computational overhead of the most popular methods, based on Monte Carlo Tree Search (MCTS), can be substantial. In this work, we investigate whether simpler methods remain competitive in non-trivial, moderately sized games such as Connect Four, Hex(7x7) and synthetic games. We train a minimal self-play implementation of Approximate Value Iteration (AVI) and use ground-truth oracles for exact evaluation. Contrary to expectations, our results demonstrate the surprising effectiveness of AVI: it learns more accurate value functions than those learned by AlphaZero, while its one-step-lookahead greedy policies remain competitive with MCTS-based policies at substantially lower training and inference costs. Preliminary experiments on Othello and Go(9x9) show that AVI trains stably on larger games and learns effective value functions. These findings suggest that the success of MCTS-based methods may have eclipsed simpler approaches that have become increasingly practical with modern deep-learning tools.

Training a competitive Text-to-SQL agent usually depends on human-annotated natural-language/SQL pairs, which are expensive, domain-specific, and a bottleneck for scaling to new databases. We show it is possible to train a competitive solver with zero annotated pairs. We introduce SQL-Zero, a proposer-solver self-play in which a challenger and a solver start from the same base LLM and the only ground truth is execution against the database itself. The challenger generates SQL pairs calibrated to the solver's current difficulty (targeting "hard but solvable"), and both roles are updated with GRPO in alternating turns, with a template-level repetition penalty on the challenger to prevent diversity collapse. Training on BIRD databases with no labels, self-play improves over the zero-shot base on BIRD dev by 6.6 points at 3B and 7.3 points at 7B. It also scores higher than a matched control trained under the same recipe on human BIRD gold over the same databases, although an exact paired test does not resolve that margin. Transfer depends on scale: at 3B every iteration outperforms the base on unseen Spider databases and under lexical perturbation (Spider-Syn), where it also degrades less than the matched BIRD-gold control, whereas at 7B only the first iteration preserves transfer.

The dominance of Neural Networks (NNs) in RL is partially due to their incremental learning capability, which naturally suits the online, non-stationary nature of self-play training. However, gradient-boosted trees like LightGBM are widely recognised as the state of the art for tabular data in supervised learning, often outperforming NNs in accuracy and efficiency. Game states are inherently tabular---discrete actions, categorical card identities, structured board positions---which makes them an ideal candidate for tree-based methods. We introduce LUGL (Local Updates, Global Learning), a framework that decouples data collection from model fitting, enabling non-incremental learners such as GBTs to operate in RL settings where they would otherwise fail due to distributional shift. LUGL alternates between a local updates phase, where the agent plays self-play games and accumulates tabular updates (Q-values, V-values, policies, or regret values) in a finite table, and a global learning phase, where the table is used to train a function approximator that generalises to unseen states before the table is reset. We test our approach in four standard perfect-information games (Tic-tac-toe, Connect-4, Othello, and Hex) and five imperfect-information games (Kuhn's poker, Leduc Hold'em, Liar's Dice, Goofspiel, and Flop5 Hold'em), and show that our results are competitive with or superior to DQN and DeepCFR. Our experiments demonstrate that the community's strong bias towards NNs in game-playing may be unwarranted, since LightGBM-based agents achieve competitive or superior performance across all tested benchmarks.

Training autonomous driving policies through pure self-play has recently shown promising results. Following Gigaflow and Puffer- Drive, we train driving policies in a similar self-play fashion, but extend the models from MLPs to Transformers and train on the high-definition map of a real city, where we ultimately aim to deploy them. On the CARLA and Waymax benchmarks, our policies fall short of Gigaflow, and we trace the gap to specific failure modes, including reward hacking at traffic lights and a missing incentive to stop at stop signs. We further analyze which traffic rules emerge from self-play and how closely they match human driving, and we confirm that reward conditioning yields the intended diversity of driving behaviors. A demonstration of a trained policy is available at https://laursisask-ut.github.io/eccvdemo.

Continuous self-improvement requires an ever-expanding pool of self-generated, diverse, adaptive goals. For language agents, existing training environment pools (hand-curated, statically synthesized, or frozen-verifier) keep the goal distribution fixed as the learner scales. We introduce SPADE (Self-Play in Adaptive Synthetic Executable Environments), a self-play RL framework in which a single LLM plays two roles: an Environment Designer that writes complete, long-horizon training environments as executable code with an OpenAI Gym-style reset()/step() interface, and a Reasoning Agent that learns to act in them. Each is a stateful, multi-turn environment (state transitions, reward functions, and verification code), so one interface spans reasoning problems and multi-step agentic tool use. The Reasoning Agent's regret is estimated using the gap between its reward with and without privileged hints; in optimizing this regret signal the Environment Designer learns to target environments at the edge of the agent's capabilities while keeping them feasible. Through extensive experimentation, we find several components critical to success: grounding the Environment Designer on documents sampled from a large pretraining corpus, and giving it an accumulated environment memory. Scaling to 30B-parameter models, SPADE improves over the strongest fixed-environment baseline by +5.3 on average across eight held-out math, science, code, and reasoning benchmarks, and lifts the tool-use setting by +5.7 on BFCL-v4 multi-turn and +13.9 on ACEBench-Agent; on the games setting, the margin over the strongest baseline grows with model scale. By making environment design itself a learnable component, SPADE takes a concrete step toward open-ended self-improvement.

Large language model agents deployed without a central controller are often assumed to require communication to coordinate their actions. We ask what remains possible without it: when independent instances of the same model cannot communicate, can they still reason about their counterparts well enough to exceed the standard game-theoretic baseline for uncoordinated play? We introduce a benchmark of one-shot, no-communication games in which each of thirteen language models is told only that its counterparts are running the same model and is evaluated against the Nash equilibrium of the underlying game. In two-player matrix games spanning seven archetypes and two to ten actions per player, two frontier-hosted models consistently exceed their Nash benchmark, approaching the optimal joint outcome in several archetypes, while most open-weight models achieve only partial gains that vary sharply by game structure. Performance degrades substantially in team-based games with four or more interchangeable agents, particularly as the action space grows, suggesting that whatever capability drives self-play gains in dyadic games does not transfer to larger multi-agent teams.

LLM agents are increasingly deployed in multi-agent social settings where they must cooperate, negotiate, and adapt to other agents. Measuring and improving these social skills is hard because, unlike math or logic, social interaction offers no objective ground truth: evaluations fall back on LLM judges, which are costly, subjective, and noisy, and models get no reliable signal to learn from. To address both, we first introduce Social Gym, an environment of 21 multi-agent social games (e.g., Werewolves, Resistance, Spyfall) whose rule-decided outcomes make agent performance verifiable and objective, with an Elo tournament that produces a cross-game leaderboard. Benchmarking experiments show that while GPT-5-mini tops the leaderboard, no model excels at all games uniformly or in all game roles, pointing to limitations of social reasoning. Motivated by this, we additionally propose SPaRTan (Self-Play and Reflect-Transfer), a training-free self-improvement loop: a model plays a game, reflects on its trajectories and their outcomes to produce a transferable playbook, and applies that playbook in subsequent games. Our results show that SPaRTan playbooks help GPT-5-mini agents level their performance on weaker roles, but largely do not improve Qwen3-32B's performance. Together, Social Gym and SPaRTan offer a reproducible, verifiable foundation for measuring and improving LLM social reasoning without weight updates.

Mixed strategy equilibrium predicts i.i.d play: past actions should not help predict future decisions. Human players, however, systematically depart from this benchmark, and in O'Neill's zero sum card game, these departures can be predicted by black box sequence models such as LSTMs. This paper asks whether that predictive power can be achieved by transparent alternatives that also reveal the behavioural structure behind it. Using 84,060 decisions from 2,802 pairs, the analysis first benchmarks naive and behavioral models against interpretable machine learning and deep learning models, then evaluates the modified EWA specifications of prior work against these benchmarks and uses the LASSO diagnostics to motivate a further nested frequency tracking extension. The results show that repeat or avoid behavior, especially players' management of their own recent action histories, accounts for most of the interpretable and strategically exploitable signal, while frequency tracking adds little out of sample.

Reasoning in large language models (LLMs) is often grounded in human text, human demonstrations, and human-generated rationales. For equilibrium reasoning in complex games, however, relying on human data can be suboptimal. In fact, human play is often guided by intuition and heuristics and can deviate substantially from game equilibrium. This discrepancy is amplified in games with mixed-strategy equilibria, where human data is heavily biased toward pure strategies. Consequently, conditioning LLMs on this data yields weak game strategies. To grant LLMs the reasoning capacity in games, in this work, we study how to elicit equilibrium play using solver output. We propose Mixed-Strategy Decision Tree (MDT), which articulates the silent optimality of the equilibrium into sparse strategic rules that both humans and LLMs could understand. Using solver output rather than human annotation allows us to extend the input to arbitrarily new states and continuations. We instantiate this study on No-Limit Texas Hold'em by querying a solver oracle for over \textbf{250 million mixed-strategy decisions}; MDT together with other techniques \textbf{reduces the $\ell_1$ distance to the equilibrium by $52.6\%$} across $8$ different LLM configurations. A Route-only ablation tests the incremental contribution of the shadow-based contrast, while complete River-endgame and Liar's Dice experiments evaluate strategic fidelity and portability beyond the original NLH communication setting.

Training LLM-based search agents requires high-quality search data: tasks that demand genuine multi-hop retrieval and trajectories that use search tools effectively. Existing pipelines often depend on human-written tasks, expert demonstrations, or stronger teacher models. We present SearchMaster, a self-play framework that trains a single LLM from search tasks it generates, solves, and verifies in a local search environment. The key challenge is that self-generated tasks and rollouts can yield misleading signals: pseudo multi-hop questions, success-rate difficulty estimates that ignore search depth, and rollouts with excessive opening but little targeted evidence acquisition. SearchMaster addresses these failure modes with three controls. An Evidence-Chain Generator (ECG) grounds task generation in explicit cross-document evidence chains to reduce pseudo multi-hop questions. A Search-Depth Reward (SDR) scores task difficulty by the search depth of successful rollouts rather than success rate alone, keeping retained tasks search-intensive. An Over-Opening Penalty (OOP) regulates tool use by discouraging excessive document opening, avoiding long but shallow browsing. Verified Proposer and Solver rollouts are then jointly optimized with GRPO. Across six deep-search benchmarks, SearchMaster improves a Qwen3.5-9B backbone from 38.19% to 51.52% average accuracy, with a 30.1-point gain on BrowseComp-Plus. These results show that grounded and regulated self-play can provide effective search-agent training data without human-labeled QA pairs or expert demonstrations. The code is available at https://github.com/WentaoTan/SearchMaster.

Adversarial self-play is an appealing recipe for legal reasoning: have a student model draft an argument, have an adversary attack it, and reward the student when its argument survives the attack. We designed exactly such a training signal -- a verifiable "survival" reward in which both the student's cited authorities and the adversary's counter-authorities are checked by a citation verifier, so that survival is decided on verified grounds rather than rhetoric, and fabricated citations are automatically neutralized. We then asked a narrow but important question: does the competitive component itself -- the adversary and the survival reward -- add anything on top of an otherwise identical non-competitive training run? Across four independent tests -- a bootstrap comparison, a two-seed replication, a paired per-case adversarial-robustness comparison, and a blinded head-to-head judgment of generated arguments, plus a follow-up pilot with a deliberately strengthened self-play adversary -- the competitive component produced no reliable benefit. The blinded judgment gave a 49% win rate (binomial p approx. 1.000); the strengthened-adversary pilot gave a 50% win rate (32:32, p approx. 1.000). An early apparent +29% advantage reversed and proved to be a small-sample artifact. We report this as an honest negative result. The value of the paper is reproducibility and the sharing of concrete pitfalls: an initially promising metric that inverted on more data, and an adversarial-robustness metric that silently collapsed to plain recall once the adversary stopped citing the same authorities as the gold answer. This null is consistent with, and reconfirms in the legal domain, the conclusion of the companion coding-domain study (Kim, 2026, arXiv:2607.08255) that the value of multi-teacher curricula arises from constructing a verifiable environment rather than from competition itself.

Self-play agents can generate training problems without questions from target benchmarks, but their curricula lack persistent state: failures affect gradients yet do not explicitly shape future practice. External skill memories preserve procedural experience but are typically learned from fixed task distributions. We introduce \textbf{SESA} (Self-Evolving Skill-Augmented Agent), which makes procedural memory an evolving state of tool-augmented search self-play. A challenger poses problems, while a separately parameterized solver alone retrieves skills. Informative failures are distilled into reusable skills and written back to memory. The updated memory changes solver behavior and success, which changes the challenger's reward and the distribution of future problems; the resulting frontier produces new failures that rewrite memory. This bidirectional loop makes task generation and skill memory co-evolve. Because retrieved skills shape on-policy training trajectories, their benefits can enter the model parameters as well as remain in the external bank, enabling memory-free deployment and optional inference-time retrieval. Across seven open-domain and multi-hop question-answering benchmarks, SESA improves average accuracy over SSP by 1.2--3.2 points across multiple backbones and surpasses the skill-augmented SkillRL baseline by 0.9 points under a unified evaluation protocol. On Qwen3 models, SESA-Off retains 1.8--2.2 points of improvement over SSP, while the final skill bank adds a further 0.5--1.0 points. These results show that evolving skill memory is not merely an inference-time plug-in: it changes policy learning and the future training distribution while retaining value as optional external memory. Our code is available at https://github.com/Zenghuang-Fu/SESA-Self-Evolving-Search-Agents.

Self-play in simulation produces robust driving policies at scale. Demonstrations of such behavior have been made using privileged vectorized observations such as exact poses and velocities, even for occluded agents. This assumes that perception is solved and introduces a representation gap with the partial observation of a deployed agent driving from the perspective view of egocentric cameras. A common fix, distilling the privileged policy into a camera-input student, leaves the student imitating decisions its own view cannot justify. Instead, we establish perspective-view self-play as a practical training regime. We introduce Pictura, a GPU-accelerated multi-agent driving simulator that renders each agent's egocentric view at every step, mitigating the representation gap at its source. Pictura sustains up to 500K agent-steps/s (2M images/s) on a single H100. Using Pictura, we train Alberti by self-play with plain PPO. It is the first large-scale driving self-play policy trained directly from perspective images, without privileged observations. Training spans 50B agent steps for ~35M km of driving. It approaches the driving performance of its privileged vectorized counterpart, and transfers zero-shot to Waymo Open Motion Dataset layouts re-rendered in Pictura, where it outperforms privileged vectorized agents. Project page: https://valeoai.github.io/Pictura/

We introduce \textbf{GPT-Red}, an automated red-teaming agent that is trained to discover novel prompt injection attacks against frontier LLMs. The goal of this model is to evaluate and improve the robustness of our production systems. To this end, we use it to adversarially train GPT-5.6, our most robust model to prompt injections to date. To create GPT-Red, we design a scalable self-play algorithm where the model is tasked with attacking a diverse population of simultaneously-trained defender agents. We train the model on realistic red-teaming environments using compute on the same scale as some of our largest RL post-training runs, making it the single-largest LLM safety training run ever documented. GPT-Red excels at red-teaming: it reliably breaks our past models up to GPT-5.5, it finds more successful attacks than human red-teamers, and it generalizes to held-out environments, defender models, and harnesses. In the future, we expect that as we improve the robustness of each new GPT model, it will in turn will provide better learning signal for \textit{even stronger} red-teamer agents, thus unlocking a self-improvement flywheel.

We revisit the regret loss framework introduced in Park et al. (2025), which uses decision-theoretic regret as a direct loss function for training models to make better decisions, through the lens of probability-simplex policies. Our first result shows that a single-layer self-attention model trained with regret loss admits a stationary point whose forward-pass exactly matches smoothed fictitious play with the appropriate stepsize that ensures no-regret behavior-i.e., for any given policy input, the model outputs the same update that smoothed fictitious play would produce. In parallel, we also newly introduce a swap-regret loss function, which extends the regret-loss framework beyond external regret and enables models to directly optimize for swap-deviation robustness. We further show that this swap-regret loss admits a stationary point whose forward pass implements the corresponding swap-regret update induced by classical Blum-Mansour no-pass implementation algorithm, with each head implementing an external-regret update via smoothed fictitious play. Together, these results show that regret-trained attention can realize differentiable mechanisms whose deployment induces equilibrium behavior in games: external-regret dynamics lead to coarse correlated equilibrium, while swap-regret dynamics lead to correlated equilibrium. Thus, regret-based objectives steer minimal attention architectures toward online-learning dynamics with game-theoretic guarantees, without supervised traces of those algorithms.

LLM training is shifting from manual design and annotation to interaction-driven self-evolution. However, existing self-evolutionary methods face a fundamental dilemma between task diversity and verification reliability: environment-bound methods obtain precise feedback but confine learning to narrow domains, while open-ended self-generation broadens the task space but lacks reliable verification, allowing misleading rewards to pollute the training loop. We identify agent skills as a powerful middle ground to reconcile this tension: each skill ensures deep, verifiable execution in a specific scenario, while dynamic routing across skills maintains open-ended task variety. Leveraging this insight, we introduce Skill Self-Play (Skill-SP), a co-evolutionary framework comprising a proposer, a solver, and a dynamic skill controller. Orchestrated via a reinforcement learning loop, these components co-evolve in a continuous self-play loop: the proposer generates challenging tasks conditioned on dynamically sampled skills; the solver explores candidate solutions to push its capability boundaries; and the skill controller collects execution feedback to update and expand the skill library. This interactive co-evolution effectively bridges the gap between structured verification and open-ended exploration. Empirical evaluations on tool-use and reasoning benchmarks demonstrate that Skill-SP, serving as a robust evolution engine, consistently pushes the performance ceiling of competent backbones while catalyzing striking turnarounds for initially misaligned models. Our code is available at https://github.com/Qwen-Applications/skill-self-play.

Large Audio Language models (LALMs) have made rapid progress on acoustic understanding, yet they still struggle with fine-grained audio reasoning (e.g., recognizing event order, repetitions and duration). Existing post-training methods heavily rely on expensive external labels or provide only coarse semantic signals. To bridge this gap, we introduce Audio-Zero, the first label-free self-evolution framework in the field of LALMs that improves fine-grained auditory perception and reasoning. Audio-Zero constructs an auditory self-play game from unlabeled audio contrast pairs: most players hear a reference audio, while one odd listener hears a subtle variant. The model first generates clues describing what it hears and then identifies the odd listener by reasoning over inconsistencies among clues. Since the odd listener is known by construction, the game provides verifiable rewards without any annotated answers. Experiments with Qwen2-Audio-7B-Instruct and Qwen2.5-Omni-7B on TREA, MMAU Test-mini and MMAR show that Audio-Zero improves fine-grained audio reasoning while preserving broad audio understanding. Evolutionary and diagnostic analyses further reveal that increasingly fine-grained auditory descriptions emerge naturally from game pressure.

Training robust autonomous driving agents requires a simulator that is fast enough for reinforcement learning at scale, realistic enough to ground behavior in real-world map structure, and diverse enough to cover the safety-critical long tail that logged data rarely contains. We present TerraZero, a procedural driving simulator and self-play training stack. A configurable C engine runs simulation on the CPU and policy inference on the GPU over a zero-copy path, sustaining 1.3M agent-steps per second on a single server-grade GPU, far faster than existing object-level simulators, while keeping fidelity lighter single-agent systems omit: heterogeneous agents, multiple dynamics models, and full traffic-rule enforcement. TerraZero treats logged data only as a source of real-world map geometry, populating each map with randomized rule-based road users and signal controllers and randomizing agent dynamics, rewards, and sizes per episode, so a map yields an unbounded set of scenarios. Every reported policy trains from scratch by reinforcement learning alone on a compute-efficient self-play recipe across GPUs, with zero human demonstrations and no fallback planner at inference. Policies generalize zero-shot across cities and datasets, including emergent left-hand-traffic driving without explicit supervision. As an ego policy, TerraZero is the first fully learned policy to top the InterPlan long-tail benchmark, ahead of larger learned planners; on routine-driving val14 it ranks among the best approaches and is the safest, posting the best collision and time-to-collision scores. On Waymo Open Sim Agents realism the same recipe outperforms other demonstration-free methods and is competitive with the strongest reference-anchored self-play method. One stack serves both roles: driving policies across dynamics for cars and trucks, and sim agents that jointly control vehicles, pedestrians, and cyclists.

Robust motion planning in dense traffic requires autonomous vehicles to interact in rare and safety-critical scenarios that are underrepresented in naturalistic driving data. Although adversarial training offers a feasible solution, existing methods often rely on external scenario generators, heuristic perturbations, or simulator-heavy rollouts, which makes them difficult to integrate with modern autoregressive planners. Here, we cast adversarially robust planner learning as a constrained min-max game and propose Adversarial World Modeling (AWM), a theoretically grounded multi-agent self-play fine-tuning framework. Since solving the exact game is intractable, AWM introduces a principled decoupled solver. In the inner minimization, the planner's predictive world model is converted into a role-conditioned adversary that learns sparse, scene-adaptive attack coalitions via counterfactual credit assignment. In the outer maximization, the ego planner optimizes a regret-aware robust best response against the frozen AWM, utilizing tail-risk weighting and reference-anchored trust regions to improve hard-case recovery while preserving nominal driving behavior. Experiments on the nuPlan and InterPlan benchmarks demonstrate that our method generates transferable adversarial interactions and yields a robust planner that achieves competitive closed-loop performance in both nominal and highly interactive long-tail scenarios. Theoretical analysis justifies the decoupled solver and the main optimization components.

AlphaZero has demonstrated that a neural-guided Monte Carlo Tree Search can achieve superhuman performance, but strong play does not necessarily imply perfect play. We study this gap in two oracle-evaluable domains with contrasting structure: Connect Four, a solved partisan game with exact game-theoretic values, and Chomp, an impartial game whose optimal play is governed by Grundy-number structure. Under a unified self-play $+$ MCTS pipeline, we compare vanilla AlphaZero, a multi-frame variant (limited to Chomp), and an AlphaZero Auxiliary Loss (AZAL) that adds oracle-derived policy supervision. We find that vanilla AlphaZero achieves strong play across both domains but cannot preserve the exact trajectories required for optimal play: in Connect Four, it fails to maintain the optimal line of play, while in Chomp, it fails to consistently restore the $g=0$ invariant. On rectangular Chomp boards, multi-frame inputs alone do not remove this gap. Nevertheless, AZAL substantially improves oracle consistency across multi-seeded full-game traces and sampled-state evaluations. On Chomp, AZAL reaches perfect full-game oracle consistency on 10x11 and high but not complete consistency on 9x10; on Connect Four, AZAL improves oracle-match rate and delays the first oracle mistake, but does not reach perfect play.

Reinforcement learning agents for imperfect-information card games are only as strong as the opponents they train against, and they are hard to grade, since they beat a random opponent over 99 percent of the time and only tie copies of themselves. So we build a strong, fixed, rule-based expert for Gin Rummy and use it only as a yardstick, never for training. It beats every agent we trained 70 to 99 percent of the time. Across more than a hundred runs, we isolate what makes a lightweight agent stronger. Trust region updates, a well-aimed reward, a curriculum of tougher opponents, warm starting, and keeping the best checkpoint all help, and stacking them lifts a self-play champion from about 30 to 36 percent against the expert. Several ideas did not pay off. Short-term and longer-term reward shaping, learned state embeddings, imitation and DAgger, and a live large language model opponent were each unhelpful, too slow, or too heavy to train at scale. Comparing MLP, convolutional, set-based, attention, and recurrent encoders shows that extra capacity does little to break the ceiling, suggesting the limit is information rather than network size. We add standard baselines (neural fictitious self-play and information set Monte Carlo search) and confirm the approach carries over to Leduc Hold'em, where the optimum is computable. The result is a lightweight, game-agnostic recipe that trains competitive agents without training on the expert, for any game a small model can handle, reported with robust statistics and released as a reusable package.

We present FootsiesGym, an open-source environment for learning in a non-trivial two-player, zero-sum, imperfect-information game. Built on HiFight's minimalist 2D fighting game Footsies, it isolates the cyclic, non-transitive strategic interactions of fighting game neutral play while remaining simple enough for efficient analysis. We provide a vectorized simulator that enables high-throughput training on standard hardware, making the environment accessible and reproducible. We describe the design of the environment, benchmark several reinforcement learning algorithms, and discuss open research directions it enables. The code is available at https://github.com/como-research/FootsiesGym.

Training a language model against its own reference-free judgments (the premise of self-rewarding, self-play, and LLM-as-a-judge pipelines) assumes a model's verdict on a shown answer tracks correctness. We show it fails structurally: conditioned on a candidate, a judge scores plausibility, not correctness, leaving false-positive basins a policy learns to exploit. We measure this with a hidden-anchor audit: a held-out, cross-source exact-match check the judge never sees. On GSM8K with Qwen3 policies, self-play drives the judge's pass rate from 0.72 to 0.94 while true accuracy stays at 0.20 (three seeds). This reward hacking is not white-box gaming: the errors transfer across judge families (Qwen, Llama, Gemma) and scales, a strict three-judge ensemble still accepts 55% of them, and no plausibility-scoring defense closes the basin. The decisive variable is whether the judge commits an answer of its own before using the candidate: committing first drops the false-positive rate from 0.719 to 0.012, blind solving lifts discrimination to 0.96, and used as the training reward the de-anchored channel keeps false positives at zero, preventing the basin rather than only detecting it. A falsifiable bound (the gap is at most 1 - accuracy) predicts which regimes are exposed. The full arc replicates without training under best-of-N selection in code and competition math, and with a Gemma policy.

Code repair is an important capability for language models (LMs): given a buggy program and unit tests, an LM must produce a fixed program that passes the tests. Because code repair data is limited, we aim to scale supervision by using an LM to generate bug--fix tasks. We propose generator--fixer self-play, in which a single model is trained with reinforcement learning to generate bugs and fix them. As the fixer improves, the generator adapts to produce more difficult bugs, yielding an automatic curriculum. To test whether this curriculum generalizes, we introduce BugSourceBench, a repair benchmark spanning realistic bug sources: bugs in human-written code, LM-generated code, and human-edited LM-generated code. On BugSourceBench, we find that self-play drifts toward difficult but unrealistic bugs, improving on synthetic bugs but degrading on human-authored ones. We propose Anchored Self-Play (ASP), which anchors self-play with a small reference set by adding a code-embedding similarity reward for generation and mixing reference bugs into fixer training. Across bug sources, ASP achieves the best fix rates, improving average fix rate over standard self-play by $+24\%$ relative / $+7.0$ pp absolute, with gains on bugs from both LMs and humans.

We investigate the problem of learning useful policy representations (embeddings) in two-player zero-sum imperfect-information games. We make three contributions: First, we introduce methods of creating datasets of policies for a given game. Second, we propose methods to learn policy representations. Third, we introduce downstream tasks to evaluate the effectiveness of such representations. We evaluate each dataset method, embedding method, and downstream task on Kuhn and Leduc Poker. Although our methods are very basic, we demonstrate that useful behavioral representations are present in the learned embeddings. To our knowledge, this work is among the first to systematically compare self-supervised learning techniques for learning policy representations in games. Our code is available at https://github.com/VitamintK/ssl-project for others to extend.

When two companies bid to buy the same target, no one knows exactly what the target is worth. Each bidder pays for due diligence: costly, imperfect homework that sharpens its own private estimate before it bids. How much of that homework is worth buying? We build a simple computer model of the bidding contest and let it teach itself to bid well by playing against itself, the way a game engine learns chess. The economic question, how much diligence pays for itself, and the computational question, when the contest becomes too complex to solve exactly, are both controlled by a single thing: how many pieces of private information a bidder carries. Our main finding is that the right amount of diligence is modest and finite. It falls as diligence gets more expensive, and it falls further when both sides are doing their homework, because competition erodes the value of knowing more. We also test a recent claim from AI research: that simple, general self-play methods can rival the specialized, expensive algorithms usually built for games like these. Running on an ordinary laptop with no costly frontier AI, we find the simple methods are the best of the self-learning approaches, though purpose-built exact methods still win whenever the game is small enough to solve outright. The simple methods earn their keep only once the game grows too large to solve exactly, which is the regime real deals live in, and there we show they still find strong bidding strategies. The contribution is threefold: a cheap, reproducible way to study deal-making under uncertainty; a concrete, model-based answer to how much due diligence is worth buying; and evidence about when lightweight, general-purpose AI is good enough to replace specialized methods. We release all the games, code, and experiments.

Open-source game theory studies agents whose behavior may depend on one another's decision procedures, but most existing models use discrete or symbolic programs. We introduce parametric open-source games, a continuous analogue of program equilibria in which players choose parameter vectors and semantics maps convert the full parameter profile into mixed actions in an underlying finite game. We establish equilibrium existence results, derive an exact coupling threshold at which selfish gradient ascent in symmetric $2\times2$ games switches from defection toward cooperation, and give a one-dimensional boundary test for parametric program Nash equilibria. We further extend the framework to a neural semantics class whose first-order cooperation condition is governed by the ratio of cross-player to self-player sensitivity. Across canonical games, the framework shows how access to internal parameterizations can qualitatively reshape learning dynamics and equilibrium structure, and how sufficiently strong open-source coupling can steer selfish optimization toward cooperative outcomes.

Recent work has established that regularized policy gradient methods such as PPO, when used in self-play, can match or exceed specialized game-theoretic algorithms for solving two-player zero-sum imperfect-information games. The uniform distribution has emerged as a strong policy regularization target for this purpose, but it regularizes equally toward all actions regardless of their viability. We introduce EMAgnet, which instead regularizes toward an exponential moving average (EMA) of the last-iterate policy's parameters, providing an adaptive regularization target that evolves with the agent's improving strategy. We evaluate EMAgnet on both standard two-player zero-sum benchmarks and modified benchmarks with exploration challenges and large numbers of strictly dominated strategies. Relative to PPO self-play with uniform-magnet regularization under both linear and power-law annealing schedules, EMAgnet achieves lower exploitability in the majority of tested environments, with consistent performance gains across games containing strictly dominated strategies.

We present a superhuman AI agent for Generals.io, a real-time strategy game that requires both long-horizon planning and short-term tactics under strong imperfect information. Trained for four days on 4x NVIDIA H200 GPUs, our agent reaches #1 on the public 1v1 leaderboard of over 5,000 human players, leading the second-ranked player by the same margin that separates second place from 25th, and beats the two top-ranked humans head-to-head with a combined 199-70 record across 269 ladder matches. A key enabler is a JAX-native simulator that reaches tens of millions of frames per second on a single GPU, roughly a 10,000x speedup over the prior simulator. On top of this, we train a vision transformer policy end-to-end by self-play with a policy-gradient loop and sparse win/loss reward, using top-advantage sample filtering and an exponential moving average of the policy parameters. Taken together, our findings highlight what matters, and what does not, once a fast simulator removes the data bottleneck.

Self-play with naive gradient ascent cycles in two-player zero-sum games: the last iterate orbits the equilibrium. Modern methods restore last-iterate convergence by regularizing toward a reference policy -- MMD a fixed one (reaching only the regularized equilibrium), R-NaD a periodic snapshot (the engine of DeepNash). We study GARIP, which anchors to the running average, and isolate what the choice of reference controls. Our central result is a mechanism: collapse tracks the peak lag of the reference, and among causal convex averages of a fixed mean lag the running average (flat profile, peak $=$ mean) uniquely minimizes that peak, while a snapshot's sawtooth has peak $= 2\times$ mean (a one-line theorem). Two consequences follow. Convergence: we prove local last-iterate convergence at constant anchor strength -- the anchor scales the base map's rotation by $1-β$, crossing the stability boundary and turning a recurrent base into a contraction (global convergence is conjectured at small $β$; we characterize a large-$β$ consensus failure). Robustness: GARIP matches R-NaD's peak performance -- on matrix games, the Coin Game, and the board games Connect Four/Othello, both moving references are far more robust than fixed-magnet and magnet-free baselines -- but is the better hyperparameter default; we report it both ways: over the full grid collapse rates are statistically indistinguishable, yet at conventional parameterizations a matched-mean-lag setting collapses in 0/40 vs 10/40 seeds (a snapshot matches it only by knowing to shorten $K$). The boundaries: an anticipatory (negative-weight) reference does better still on the stale side, and the advantage appears only where naive self-play cycles (five deep self-play loops). All experiments are pure JAX and reproducible.

End-to-end autonomous driving models are typically trained on offline human-demonstration datasets that provide limited state coverage and often no closed-loop feedback, making them prone to compounding errors when deployed in closed-loop and brittle to long-tail agent interactions. To overcome these limitations, we propose an alternative strategy for training end-to-end driving models: large-scale self-play directly from pixels in simulation. While prior self-play approaches have shown promising transfer to real-world driving, they typically assume vectorized Bird's-Eye-View (BEV) observations that are incompatible with end-to-end policies operating directly on sensor observations. To this end, we introduce Gigapixel, a high-throughput batched driving simulator with perspective rendering, enabling scalable self-play directly from pixel observations. Rather than targeting compute-costly photorealistic sensor simulation, Gigapixel renders a simplified bounding-box world that preserves essential scene structure while achieving throughput at 50k agent steps per second. Since direct pixel-space self-play RL is prohibitively sample-inefficient at end-to-end model scale, we propose self-play DAgger training: we train pixel-based policies in self-play via on-policy distillation from a privileged RL teacher. To bridge the sim-to-real gap, we subsequently transfer the self-play trained policies to real-world sensor data through lightweight perception adaptation. Policies trained in Gigapixel and adapted to real-world sensor data achieve competitive performance on the HUGSIM and NAVSIM-v2 benchmarks without human trajectory supervision. Moreover, scaling self-play training yields proportional gains in policy performance, establishing self-play as a practical and scalable strategy for training end-to-end models.

End-to-end autonomous driving has achieved state-of-the-art performance on benchmarks and real-world deployments. Its standard training recipe, however, is expensive across all stages: collecting and labeling millions of driving frames is costly, and closed-loop RL on images is bottlenecked by the per-step cost of photorealistic rendering plus a forward pass through a large vision backbone. Self-play in vectorized simulators changes the economics: millions of rollout steps per second, and a state distribution naturally rich in collisions, near-misses, and recoveries that no driving log contains. Our approach exploits this asymmetry by decoupling learning to drive from learning to see. We pretrain a single policy by self-play, then align its latent space with a pretrained vision backbone, through the action KL divergence and a batch-relational low-rank structural loss. The action target comes from the self-play policy, so alignment never supervises against a logged trajectory: a paired dataset of (image, scene-state) frames suffices, with no need for the curated expert demonstrations that imitation pretraining is built on. On photorealistic 3D Gaussian splatting closed-loop scenarios, the resulting end-to-end policy matches or exceeds prior end-to-end methods.

The Lenstra-Lenstra-Lovász (LLL) algorithm is a seminal contribution to computer science used for lattice basis reduction, yet its polynomial-time outputs produce bases that are far from optimal as the dimension grows. We show that deep reinforcement learning can discover strictly superior, generalizable reduction strategies by interacting with the primitive action space of LLL. We formulate lattice reduction as a single-player Markov Decision Process (MDP) and train a deep residual network using an AlphaZero-style self-play pipeline augmented with adaptive-horizon MCTS (Monte Carlo Tree Search), which couples multi-step network predictions with an entropy-gated expansion mechanism. The resulting policy, DeltaStar, is trained exclusively on small $8$-dimensional $q$-ary lattices and requires fewer primitive row operations than LLL. Crucially, it generalizes zero-shot to unseen moduli and higher dimensions up to $n=32$ without retraining.

Self-play reinforcement learning has recently emerged as a way to train driving policies without any human data. It uses cheap, large-scale simulations to substitute expensive, large-scale human driving demonstrations. A key limitation of this approach is that policies trained through pure self-play can learn effective but alien driving conventions incompatible with people. Previous works attempt to mitigate such behavioral misalignments through extensive reward engineering and domain randomization, which are brittle and labor-intensive. Instead of completely discarding human demonstrations, our method treats them as a regularization objective on top of a minimal safe goal-reaching reward. Like the spice in a good stew, we find that a little human data goes a long way: our method uses only 30 minutes of human demonstrations, 2500x fewer than comparable imitation learning approaches. Resulting policies coordinate with held-out human trajectories and complete training in 15 hours on a single consumer-grade GPU. Videos and full source code are available at https://spiced-self-play.com/.

Learning a single policy that reaches a goal with high geometric precision while interacting safely with nearby agents poses conflicting objectives. Precision favors commitment to a fixed geometric plan, whereas interaction requires immediate deviation when another agent intrudes, causing policies optimized for one objective to often fail at the other. We study this problem in the context of reactive autonomous parking, where multiple vehicles must reach assigned slots with sub-meter terminal accuracy while remaining responsive to neighboring vehicles throughout the maneuver. We propose CoPark, a multi-agent self-play RL approach built on a residual-policy architecture. A precomputed offline plan provides a fixed action prior, while a residual head learns the reactive corrections. The residual policy learns behaviors under self-play, where data and scripting fall short, while the fixed prior holds the slot-frame geometry that pure policies struggle to reach reliably. The key design is a partner-threat-modulated, channel-asymmetric release of the prior. A continuous threat signal shifts authority of the longitudinal channel to the residual head to enable yielding, while the lateral channel remains anchored to the precomputed reference to preserve sub-meter slot alignment. A closed-loop refinement layer corrects residual terminal error from action-grid discretization. We train our policy on six parking lots and evaluate zero-shot on our new reactive-parking benchmark spanning Dragon Lake Parking (DLP) and DeepScenario Open 3D (DSC3D). CoPark achieves ~70-85% success with only 3-6% collision rate, substantially outperforming classical, imitation-learning, and large-scale RL baselines. Importantly, the results demonstrate emergent interaction behaviors such as reverse-yielding, mid-maneuver yielding, tight-corridor passing, and queuing.

While Large Language Models (LLMs) excel in code generation, they remain prone to replicating subtle yet critical vulnerabilities endemic to their training data. Current alignment techniques, such as Supervised Fine-Tuning (SFT) and Reinforcement Learning (RL), typically apply coarse-grained optimization at the sequence level. This approach often fails to address the localized nature of security flaws, where a single incorrect token choice can compromise an entire program. To bridge this gap, we introduce Tree-like Self-Play (TSP), a framework that reframes secure code generation as a fine-grained sequential decision process. Unlike standard methods that blindly maximize likelihood, TSP constructs a decision tree where the model explores branching trajectories--generating both secure "golden paths" and vulnerable variants. By treating code generation as a self-play game, the model learns to strictly discriminate against its own localized errors. This provides a dense, on-policy learning signal that forces self-correction precisely at the critical decision nodes where vulnerabilities typically emerge. Our experiments demonstrate that TSP fundamentally enhances model reliability. In Python security benchmarks, TSP boosts CodeLlama-7B's pass rate (SPR@1) to 75.8%, significantly outperforming SFT (57.0%) and unstructured self-play baselines. Crucially, TSP induces robust out-of-distribution generalization: the model not only reduces vulnerabilities in unseen categories (CWEs) by 24.5% but also successfully transfers security principles learned from C/C++ to diverse languages, including Python, Go, and JavaScript. This suggests that TSP does not merely memorize patches, but internalizes abstract, language-agnostic security logic.

Self-play, a type of training algorithm that enables a model to self-improve, has recently shown promising empirical results in the context of formal theorem proving using Large Language Models (LLMs). (Dong & Ma, 2025) instantiate self-play with two cooperating agents: a prover, which proves theorems, and a conjecturer, which generates new theorems as a curriculum to the prover. In this paper, we provide a theoretical framework for understanding the self-improvement capabilities of self-play algorithms for theorem proving. First, we formalize the set of theorems as a graph, with nodes as theorems and edges between pairs of theorems with similar semantics. We introduce a set of primitive assumptions that characterize the guarantees of a trained prover and how a conjecturer can access the structure of the graph. Second, we show that if the underlying graph of theorems is well-connected, then a prover-conjecturer system, where the conjecturing algorithm is based on a reversible random walk, is sufficient to grow the set of proved theorems exponentially. Third, motivated by an issue encountered empirically by self-play algorithms, where the conjecturer tends to generate artificially complex and non-fundamental theorems, we propose a diversity measure for a training distribution of theorems generated by a conjecturer and an improved conjecturing algorithm that locally maximizes this diversity measure, by computing the diffusion similarity between neighboring theorems in the theorem graph. Finally, we describe a method to compute the diffusion similarity by using contrastive learning to embed nodes into Euclidean space and then computing the inner-product between embeddings.

Aligning Large Language Models (LLMs) with human preferences is often formulated via Direct Preference Optimization (DPO). However, the standard Bradley-Terry instantiation of DPO is limited in modeling common departures from transitivity in human preferences. To address this, recent work has introduced Self-Play Preference Optimization (SPPO), which iteratively refines the policy by training on self-generated win-lose pairs. Our investigation, however, reveals a critical instability in SPPO: the optimization is prone to policy degeneration when the preference oracle assigns overly confident wins to semantically indistinguishable responses. To mitigate this, we propose S-SPPO, a dual-space semantic calibration framework comprising: i) Supervision Calibration via semantic gating, which anneals win rate targets toward the maximum-entropy baseline as semantic overlap increases; and ii) Representation Calibration via latent repulsion to enforce geometric diversity to prevent manifold collapse and maintain latent diversity between chosen and rejected samples. Theoretically, we show that the calibration preserves the constant-sum game structure, facilitating convergence to a Nash Equilibrium. Empirically, S-SPPO avoids the performance degradation seen in prior methods, achieving 52.19% win rate and 47.46% length-controlled win rate on AlpacaEval 2.0 with Llama-3-8B, without using additional human-annotated preferences during training. The code will be available at https://github.com/xiwenc1/s-sppo.

Self-play can train language models without external supervision. However, existing methods require rule-checkable answers, leaving open-ended tasks dependent on curated prompts or frontier-model judges. We introduce SCOPE, a data-free self-play framework for open-ended tasks that co-evolves two policies: a Challenger that generates document-grounded tasks, and a Solver that answers them through multi-turn retrieval. A frozen copy of the initial model serves as the self-judge, which writes task-specific rubrics from the source document and grades Solver responses against them. Across three 7-8B instruction-tuned models (Qwen2.5, Qwen3, OLMo-3), SCOPE improves open-ended performance by up to +10.4 points on eight benchmarks and matches or exceeds GRPO_data trained on ~9K curated prompts. Although trained only on open-ended tasks, SCOPE also improves held-out short-form QA by up to +13.8 points on seven held-out benchmarks, surpassing GRPO_data on all three models. Ablations show that co-evolving the Challenger is necessary to keep tasks near the Solver's frontier, that gains arise from improvements in both retrieval and synthesis with the relative contribution varying by task, and that rubric generation quality is the bottleneck for self-judging.

We study two-level autoresearch for cooperation: an outer-loop AI agent autonomously redesigns the inner-loop pipeline of an LLM policy-synthesis system for multi-agent Sequential Social Dilemmas (SSDs). A researcher agent $\mathcal{R}$ (run as a coding agent) reads the inner-loop source code, edits system prompts, feedback functions, helper libraries, and iteration logic, runs evaluations, and decides what to keep, following the autoresearch paradigm. Across two games (Cleanup and Gathering), two policy-synthesizer LLMs, and two welfare objectives (utilitarian efficiency and Rawlsian maximin), the researcher reliably exceeds hand-designed baselines, sharply tightens run-to-run variance, and outperforms prompt-only optimization. The discovered pipelines are objective-dependent: only under maximin does the researcher inject an explicit fairness mechanism into synthesizer pipelines, a class of mechanism that is absent from its own objective-agnostic system prompt and from every efficiency-optimized pipeline. This supports an information-design reading in which the researcher chooses what to reveal to the boundedly rational synthesizer as a function of the welfare objective. Code at https://github.com/vicgalle/autoresearch-social-dilemmas.

A central challenge in game theory and learning systems such as GANs is understanding which algorithms can efficiently compute equilibria across the heterogeneous landscape of games. Equilibrium computation is typically studied solver by solver and game class by game class, yielding strong local guarantees but a fragmented view of solver behaviour. Existing discrete taxonomies often provide an incomplete account of where algorithms succeed. We study this problem through a solver-game map linking games to effective solver dynamics. Classical theory identifies isolated regions of this map but provides limited insight into intermediate or overlapping regimes, suggesting that solvability is governed by latent structural properties defining a continuous solver-aligned geometry of games. We formalise this perspective through structure-aware solver synthesis. A learned structure recogniser maps each game to a low-dimensional solver-aligned representation, and a policy maps this representation to effective primitive mechanisms, adapting solver behaviour across regimes. This reveals regions where particular solver dynamics are effective and where mixtures of primitives are required rather than a single dominant solver. A bounded residual acts as a local corrector and diagnostic signal for incomplete solver bases or representations. The framework yields both an adaptive solver and an analytical lens: games with similar optimisation dynamics cluster together, revealing continuous regions of algorithmic validity and overlapping solver behaviour. Empirically, we show that fixed primitives exhibit systematic regime mismatch, while the learned representation organises game space into a structured cartography aligned with solver behaviour. These results suggest viewing equilibrium computation as the joint problem of learning solver mechanisms and mapping the geometry of solvability.

Reinforcement learning (RL) has emerged as an effective paradigm for improving the reasoning capability of vision-language models (VLMs). However, RL-based optimization typically depends on costly high-quality annotations that are difficult to scale. Existing unsupervised alternatives may drift toward biased solutions due to weak visual grounding and the lack of reliable verification signals. We propose a self-evolving post-training framework, DUEL, where supervision emerges from adversarial interactions between two policies initialized from the same pretrained VLM. A Challenger generates an image-grounded true claim together with a minimally perturbed hard-negative counterpart, while a Solver verifies both claims against the image, encouraging fine-grained visual discrimination under near-neighbor semantics. To stabilize optimization, we introduce a length-normalized log-likelihood reward that preserves informative optimization signals beyond binary outcome supervision and improves learning stability under sparse feedback. Experiments show that DUEL consistently improves visual reasoning and robust discrimination without additional human annotations, external reward models, or image editing tools.

Imperfect-information multiplayer games test whether agents can act under hidden information, sparse rewards, and non-stationary opponents. We study these challenges in Big 2, a four-player imperfect-information card game. We develop a self-play RL framework for Big 2 that enables controlled comparisons between policy-gradient and value-approximating agents. Under a common environment, input representation, training budget, and evaluation protocol, PPO outperforms Monte Carlo Q approximation, SARSA, and Q-learning against random, greedy, and heuristic Big 2 opponents. We further find that moderate entropy regularization improves PPO by preventing the policy from becoming overly deterministic, and that current-policy self-play provides a stronger finite-budget curriculum than checkpoint self-play or fixed-opponent training. Together, these results show that Big 2 is a useful controlled setting for studying deep RL under imperfect information, multiplayer interaction, delayed rewards, and variable action sets.

Autonomous systems have achieved superhuman performance in isolation or simulation, yet they remain brittle in shared, dynamic real-world spaces. This failure stems from the dominant single-agent paradigm for physical applications, where other actors are ignored or treated as environmental noise, preventing effective coordination. Here we show that multi-agent reinforcement learning provides the essential safety scaffolding required for real-world interaction. Using high-speed quadrotor racing as a high-stakes testbed, we train agents to navigate complex aerodynamic interactions and strategic maneuvering with a variable number of racers. Through league-based self-play, agents evolve sophisticated anticipatory behaviors, including proactive collision avoidance, overtaking, and handling multi-agent physical interactions, including aerodynamic downwash. Our agents outperform a champion-level human pilot in multi-player races at speeds exceeding 22 m/s, while simultaneously reducing collision rates by 50 % compared to state-of-the-art single-agent baselines. Crucially, training with diverse artificial agents enables zero-shot generalization to safer human interaction. These results suggest that the path to robust robotic co-existence lies not in isolated safety constraints, but in the rigorous demands of multi-agent interaction. Multimedia materials are available at: https://rpg.ifi.uzh.ch/marl

Self-play reinforcement learning trains language models on their own generated tasks, co-evolving a proposer and solver without human labels. Recent systems report strong reasoning gains, but collapse and instability are widely observed and poorly understood. The dominant response treats this as a reward-design problem. We argue instead that self-play stability is governed by two distinct levers: a data-level gate that decides which proposer-generated tasks enter the training pool, and the reward signal that updates the policy on tasks already admitted. Through controlled experiments on a Python output-prediction task and a deterministic-DSL twin task that strips pretraining priors, output ambiguity, and executor noise, we find the two levers are asymmetric. A strict gate is sufficient for stability under every reward variant we test, including a self-consistency reward with no access to ground truth; while no reward variant is sufficient once the gate is removed. This asymmetry exposes a counter-intuitive coupling we call the Grounded Proposer Paradox: a proposer with ground-truth access accelerates collapse faster than an ungrounded one when paired with a self-consistency solver, by concentrating training on clean tasks that form the fastest path to a spurious self-consistent attractor. Replacing the binary gate with a continuous strictness parameter $\varepsilon$ further reveals a two-stage phase transition: training-side metrics decouple at low $\varepsilon$, while validation accuracy holds until $\varepsilon$ is much higher. Data-level gating, not reward calibration, is the binding constraint on self-play stability.

Geospatial reasoning requires solving image-grounded problems over the complex spatial structure of a scene. However, developing this capability is hindered by the cost of annotating a vast and combinatorial question space. We propose GeoX, a self-play framework that acquires spatial logic through executable programs that yield verifiable rewards, without relying on large-scale human-curated data Given a satellite or aerial image, our framework employs a single multimodal policy that proposes spatial problems as executable programs and solves them under three reasoning modes-abduction, deduction, and induction-over spatial primitives and an image understanding tool. A verifier executes each program to covert a reward signal that jointly optimizes the two roles via reinforcement learning. GeoX consistently improves its base VLMs by up to 5.5 points on average, matching or exceeding conventional baselines trained on millions of curated data. Along-side the proposed method, we release a benchmark for geospatial understanding accumulated through self-play.

Competitive multi-agent reinforcement learning in imperfect-information games requires agents to act under partial observability and against adversarial opponents, necessitating stochastic policies. While self-play reinforcement learning with Proximal Policy Optimization (PPO) has achieved strong empirical success, its standard advantage estimator, generalized advantage estimation, suffers from additional variance due to the sampling of stochastic future actions. This variance is amplified in equilibrium self-play because of the stochastic nature of the equilibrium policy and persists even when the critic is exact. We address this bottleneck by introducing $Q$-boosting, a variance-reduced advantage estimator based on a centralized action-value critic, and propose Variance-Reduced Policy Optimization (VRPO), incorporating this new estimator. The algorithm replaces sampled multi-step backups with a multi-step Expected SARSA$(λ)$ trace, computing policy expectations at each step to average out action-sampling noise, while retaining PPO's clipped objective and on-policy actor updates. Empirically, VRPO consistently achieves strong performance from mid-sized to large-scale games including Dou Dizhu and Heads-Up No-Limit Texas Hold'em.

Standard RLHF relies on transitive scalar rewards, failing to capture the cyclic nature of human preferences. While some approaches like the General Preference Model (GPM) address this, we identify a theoretical limitation: their implicit formulation entangles hierarchy with cyclicity, failing to guarantee dominant solutions. To address this, we propose the Hybrid Reward-Cyclic (HRC) model, which utilizes game-theoretic decomposition to explicitly disentangle preferences into orthogonal transitive (scalar) and cyclic (vector) components. Complementing this, we introduce Dynamic Self-Play Preference Optimization (DSPPO), which treats alignment as a time-varying game to progressively guide the policy toward the Nash equilibrium. Synthetic data experiments further validate HRC's structural superiority in mixed transitive--cyclic settings, where HRC converges faster and achieves higher accuracy than GPM. Experiments on RewardBench 2 demonstrate that HRC consistently improves over both BT and GPM baselines (e.g., +1.23% on Gemma-2B-it). In particular, its superior performance in the Ties domain empirically validates the model's robustness in handling complex, non-strict preferences. Extensive downstream evaluations on AlpacaEval 2.0, Arena-Hard-v0.1, and MT-Bench confirm the efficacy of our framework. Notably, when using Gemma-2B-it as the base preference model, HRC+DSPPO achieves a peak length-controlled win-rate of 44.75% on AlpacaEval 2.0 and 46.8% on Arena-Hard-v0.1, significantly outperforming SPPO baselines trained with BT or GPM. Our code is publicly available at https://github.com/lab-klc/Hybrid-Reward-Cyclic.

We introduce PopuLoRA, a population-based asymmetric self-play framework for reinforcement learning with verifiable rewards (RLVR) post-training of LLMs. Teachers and students are specialised LoRA adapters on a shared frozen base: teachers propose problems, matched students solve them under a programmatic verifier, and cross-evaluation between sub-populations replaces the self-calibration that limits single-agent self-play. A family of LoRA weight-space evolution operators (mutations and crossovers that produce same-rank population members in seconds) serves as the replacement step of a population-based training loop at 7B scale. We instantiate PopuLoRA on top of Absolute Zero Reasoner and compare it against a per-adapter compute-matched single-agent baseline. Where the single agent self-calibrates to generating easy problems it can reliably solve, the population enters a co-evolutionary arms race: teachers produce increasingly complex problems, student solve rates oscillate, and problem-space coverage keeps expanding throughout training. Despite lower training-time reward, the population mean outperforms the baseline on three code benchmarks (HumanEval+, MBPP+, LiveCodeBench) and seven math benchmarks (AIME 24/25, AMC 23, MATH-500, Minerva, GSM8K, OlympiadBench), and even the weakest member of the population beats the baseline on aggregate.

Finding approximate equilibria for large-scale imperfect-information competitive games such as StarCraft, Dota, and CounterStrike remains computationally infeasible due to sparse rewards and challenging exploration over long horizons. In this paper, we propose a multi-agent starting-state sampling strategy designed to substantially accelerate online exploration in regularized policy-gradient game methods for two-player zero-sum (2p0s) games. Motivated by an assumption that offline demonstrations from skilled humans can provide good coverage of high-level strategies relevant to equilibrium play, we propose the initialization of reinforcement learning data collection at intermediate states sampled from offline data to facilitate exploration of strategically relevant subgames. Referring to this method as Data-Augmented Game Starts (DAGS), we perform experiments using synthetic datasets and analytically tractable, long-horizon control variants of two-player Kuhn Poker, Goofspiel, and a counterexample game designed to penalize biased beliefs over hidden information. Under fixed computational budgets, DAGS enables regularized policy gradient methods to achieve lower exploitability in games with significantly more challenging exploration. We show that augmenting starting state distributions when solving imperfect information games can lead to biased equilibria, and we provide a straightforward mitigation to this in the form of multi-task observation flags. Finally, we release a new set of benchmark environments that drastically increase exploration challenges and state counts in existing OpenSpiel games while keeping exploitability measurements analytically tractable.

We study the problem of learning Nash equilibria in offline two-player zero-sum Markov games. While existing approaches often rely on explicit pessimism to address distribution shift, we show that KL regularization alone suffices to stabilize learning and guarantee convergence. We first introduce Regularized Offline Sequential Equilibrium (ROSE), a theoretical framework that achieves a fast $\widetilde{\mathcal{O}}(1/n)$ convergence rate under \textit{unilateral concentrability}, improving over the standard $\widetilde{\mathcal{O}}(1/\sqrt{n})$ rates in unregularized settings. We then propose Sequential Offline Self-play Mirror Descent (SOS-MD), a practical model-free algorithm based on least-squares value estimation and iterative self-play updates. We prove that the last iterate of SOS-MD attains the same $\widetilde{\mathcal{O}}(1/n)$ statistical rate up to a vanishing optimization error of order $\widetilde{\mathcal{O}}(1/\sqrt{T})$ in the number of self-play iterations $T$.

We present Seirênes, a self-play RL framework that transforms contextual interference from a failure mode of LLM reasoning into an internal training signal for co-evolving more resilient reasoners. While RL with verifiable rewards has significantly advanced reasoning capabilities, models can still exhibit fragility when encountering non-idealized contexts: scenarios characterized by superfluous information, tangential instructions, or incidental correlations that differ from the clean distributions typical of standard benchmarks. Seirênes harnesses this vulnerability through a parameter-shared and adversarial self-play loop. Within this framework, a single model is trained to both construct plausible yet distracting contexts that expose its own reasoning blind spots, and solve problems by discerning the essential task from these perturbations to recover the core underlying logic. By pitting these competing objectives against each other, Seirênes compels the model to move beyond superficial pattern matching and anchors its capabilities in robust underlying reasoning. This continuous interaction sustains an informative co-evolutionary curriculum as the model improves. Across seven mathematical reasoning benchmarks and model scales from 4B to 30B, Seirênes achieves average gains of +10.2, +9.1, and +7.2 points. Besides, distracting contexts produced by the 4B Seirênes model reduce the accuracy of top-tier closed-source models (GPT and Gemini) by roughly 4--5 points, revealing Seirênes' general ability to uncover reasoning models' blind spots.

Recent Autonomous Driving (AD) works such as GigaFlow and PufferDrive have unlocked Reinforcement Learning (RL) at scale as a training strategy for driving policies. Yet such policies remain disconnected from established benchmarks, leaving the performance of large-scale RL for driving on standardized evaluations unknown. We present BehaviorBench -- a comprehensive test suite that closes this gap along three axes: Evaluation, Complexity, and Behavior Diversity. In terms of Evaluation, we provide an interface connecting PufferDrive to nuPlan, which, for the first time, enables policies trained via RL at scale to be evaluated on an established planning benchmark for autonomous driving. Complementarily, we offer an evaluation framework that allows planners to be benchmarked directly inside the PufferDrive simulation, at a fraction of the time. Regarding Complexity, we observe that today's standardized benchmarks are so simple that near-perfect scores are achievable by straight lane following with collision checking. We extract a meaningful, interaction-rich split from the Waymo Open Motion Dataset (WOMD) on which strong performance is impossible without multi-agent reasoning. Lastly, we address Behavior Diversity. Existing benchmarks commonly evaluate planners against a single rule-based traffic model, the Intelligent Driver Model (IDM). We provide a diverse suite of interactive traffic agents to stress-test policies under heterogeneous behaviors, beyond just using IDM. Overall, our benchmarking analysis uncovers the following insight: despite learning interactive behaviors in an emergent manner, policies trained via pure self-play under standard reward functions overfit to their training opponents and fail to generalize to other traffic agent behaviors. Building on this observation, we propose a hybrid planner that combines a PPO policy with a rule-based planner.

Self-evolving LLMs excel in verifiable domains but struggle in open-ended tasks, where reliance on proxy LLM judges introduces capability bottlenecks and reward hacking. To overcome this, we introduce G-Zero, a verifier-free, co-evolutionary framework for autonomous self-improvement. Our core innovation is Hint-$δ$, an intrinsic reward that quantifies the predictive shift between a Generator model's unassisted response and its response conditioned on a self-generated hint. Using this signal, a Proposer model is trained via GRPO to continuously target the Generator's blind spots by synthesizing challenging queries and informative hints. The Generator is concurrently optimized via DPO to internalize these hint-guided improvements. Theoretically, we prove a best-iterate suboptimality guarantee for an idealized standard-DPO version of G-Zero, provided that the Proposer induces sufficient exploration coverage and the data filteration keeps pseudo-label score noise low. By deriving supervision entirely from internal distributional dynamics, G-Zero bypasses the capability ceilings of external judges, providing a scalable, robust pathway for continuous LLM self-evolution across unverifiable domains.

While recent self-training approaches have reduced reliance on human-labeled data for aligning LLMs, they still face critical limitations: (i) sensitivity to synthetic data quality, leading to instability and bias amplification in iterative training; (ii) ineffective optimization due to a diminishing gap between positive and negative responses over successive training iterations. In this paper, we propose Team-based self-Play with dual Adaptive Weighting (TPAW), a novel self-play algorithm designed to improve alignment in a fully self-supervised setting. TPAW adopts a team-based framework in which the current policy model both collaborates with and competes against historical checkpoints, promoting more stable and efficient optimization. To further enhance learning, we design two adaptive weighting mechanisms: (i) a response reweighting scheme that adjusts the importance of target responses, and (ii) a player weighting strategy that dynamically modulates each team member's contribution during training. Initialized from a SFT model, TPAW iteratively refines alignment without requiring additional human supervision. Experimental results demonstrate that TPAW consistently outperforms existing baselines across various base models and LLM benchmarks. Our code is publicly available at https://github.com/lab-klc/TPAW.

Closed-loop traffic simulation requires agents that are both scalable and behaviorally realistic. Recent self-play reinforcement learning approaches demonstrate strong scalability, but their equilibrium strategies fail to capture the socially aware behaviors of real human drivers. We propose a hierarchical architecture that goes beyond self-play by combining high-level multi-agent interaction reasoning with low-level continuous trajectory realization. Specifically, a Stackelberg-style Multi-Agent Reinforcement Learning (MARL) module generates interaction-aware intention commands. These commands condition a low-level continuous motion module, translating the strategic intent into physically consistent, scene-responsive control sequences. To mitigate distribution shift in closed-loop deployment, we introduce a hybrid co-training scheme combining MARL with auxiliary recovery supervision. Experiments on a SUMO-based urban network demonstrate that the proposed framework achieves superior control smoothness and safety compared to self-play and passive imitation baselines, while maintaining competitive traffic efficiency.

Poker is an imperfect information game that has served as a long-standing benchmark for decision-making under uncertainty. To maximize utility beyond the Nash equilibrium, an agent can deviate from Nash-equilibrium policies to exploit suboptimal play. We introduce AlphaExploitem, which extends the competitive RL poker agent AlphaHoldem by using a hierarchical transformer encoder that enables reasoning over previously played hands and modifying the training procedure with the inclusion of a diverse pool of exploitable opponents to facilitate learning to exploit. We train and evaluate AlphaExploitem on two standard benchmarks for imperfect-information games. Empirically, AlphaExploitem successfully exploits weak play by both in- and out-of-distribution opponents, without losing performance against NE opponents.