Google DeepMind’s Research Lets an LLM Rewrite Its Own Game Theory Algorithms — And It Outperformed the Experts
Designing algorithms for Multi-Agent Reinforcement Learning (MARL) in imperfect-information games — scenarios where players act sequentially and cannot see each other’s private information, like poker — has historically relied on manual iteration. Researchers identify weighting schemes, discounting rules, and equilibrium solvers through intuition and trial-and-error. Google DeepMind researchers proposes AlphaEvolve, an LLM-powered evolutionary coding agent that replaces that manual process with automated search. The research team applies this framework to two established paradigms: Counterfactual Regret Minimization (CFR) and Policy Space Response Oracles (PSRO). In both cases, the system discovers new algorithm variants that perform competitively against or better than existing hand-designed state-of-the-art baselines. All experiments were run using the OpenSpiel framework. Background: CFR AND PSRO CFR is an iterative algorithm that decomposes regret minimization across information sets. At each iteration it accumulates ‘counterfactual regret’ — how much a player would have gained by playing differently — and derives a new policy proportional to positive accumulated regret. Over many iterations, the time-averaged strategy converges to a Nash Equilibrium (NE). Variants like DCFR (Discounted CFR) and PCFR+ (Predictive CFR+) improve convergence by applying specific discounting or predictive update rules, all developed through manual design. PSRO operates at a higher level of abstraction. It maintains a population of policies for each player, builds a payoff tensor (the meta-game) by computing expected utilities for every combination of population policies, and then uses a meta-strategy solver to produce a probability distribution over the population. Best responses are trained against that distribution and added to the population iteratively. The meta-strategy solver — how the population distribution is computed — is the central design choice that the paper targets for automated discovery. All experiments use an exact best response oracle (computed via value iteration) and exact payoff values for all meta-game entries, removing Monte Carlo sampling noise from the results. THE AlphaEvolve FRAMEWORK AlphaEvolve is a distributed evolutionary system that uses LLMs to mutate source code rather than numeric parameters. The process: a population is initialized with a standard implementation (CFR+ as the seed for CFR experiments; Uniform as the seed for both PSRO solver classes). At each generation, a parent algorithm is selected based on fitness; its source code is passed to an LLM (Gemini 2.5 Pro) with a prompt to modify it; the resulting candidate is evaluated on proxy games; valid candidates are added to the population. AlphaEvolve supports multi-objective optimization — if multiple fitness metrics are defined, one is randomly selected per generation to guide parent sampling. The fitness signal is negative exploitability after K iterations, evaluated on a fixed set of training games: 3-player Kuhn Poker, 2-player Leduc Poker, 4-card Goofspiel, and 5-sided Liars Dice. Final evaluation is done on a separate test set of larger, unseen games. For CFR, the evolvable search space consists of three Python classes: RegretAccumulator, PolicyFromRegretAccumulator, and PolicyAccumulator. These govern regret accumulation, current policy derivation, and average policy accumulation respectively. The interface is expressive enough to represent all known CFR variants as special cases. For PSRO, the evolvable components are TrainMetaStrategySolverand EvalMetaStrategySolver— the meta-strategy solvers used during oracle training and during exploitability evaluation. Discovered Algorithm 1: VAD-CFR The evolved CFR variant is Volatility-Adaptive Discounted CFR (VAD-CFR). Rather than the linear averaging and static discounting used in the CFR family, the search produced three distinct mechanisms: Volatility-adaptive discounting. Instead of fixed discount factors α and β applied to cumulative regrets (as in DCFR), VAD-CFR tracks the volatility of the learning process using an Exponential Weighted Moving Average (EWMA) of the instantaneous regret magnitude. When volatility is high, discounting increases so the algorithm forgets unstable history faster; when volatility drops it retains more history. The EWMA decay factor is 0.1, with base α = 1.5 and base β = −0.1. Asymmetric instantaneous boosting. Positive instantaneous regrets are multiplied by a factor of 1.1 before being added to cumulative regrets. This asymmetry is applied to the instantaneous update, not the accumulated history, making the algorithm more reactive to currently good actions. Hard warm-start with regret-magnitude weighting. Policy averaging is postponed entirely until iteration 500. The regret accumulation process continues normally during this phase. Once accumulation begins, policies are weighted by a combination of temporal weight and instantaneous regret magnitude — prioritizing high-information iterations when constructing the average strategy. The 500-iteration threshold was generated by the LLM without knowledge of the 1000-iteration evaluation horizon. VAD-CFR is benchmarked against standard CFR, CFR+, Linear CFR (LCFR), DCFR, PCFR+, DPCFR+, and HS-PCFR+(30) across 1000 iterations with K = 1000. Exploitability is computed exactly. On the full 11-game evaluation, VAD-CFR matches or surpasses state-of-the-art performance in 10 of the 11 games, with 4-player Kuhn Poker as the sole exception. ALSO DISCOVERED: AOD-CFR An earlier trial on a different training set (2-player Kuhn Poker, 2-player Leduc Poker, 4-card Goofspiel, 4-sided Liars Dice) produced a second variant, Asymmetric Optimistic Discounted CFR (AOD-CFR). It uses a linear schedule for discounting cumulative regrets (α transitions from 1.0 → 2.5 over 500 iterations, β from 0.5 → 0.0), sign-dependent scaling of instantaneous regret, trend-based policy optimism via an Exponential Moving Average of cumulative regrets, and polynomial policy averaging with an exponent γ scaling from 1.0 → 5.0. The research team reports it achieves competitive performance using more conventional mechanisms than VAD-CFR. Discovered Algorithm 2: SHOR-PSRO The evolved PSRO variant is Smoothed Hybrid Optimistic Regret PSRO (SHOR-PSRO). The search produced a hybrid meta-solver that constructs a meta-strategy by linearly blending two components at every internal solver iteration: σ_ORM (Optimistic Regret Matching): Provides regret-minimization stability. Gains are computed, optionally normalized and diversity-adjusted, then used to update cumulative regrets via regret matching. A momentum term is applied to payoff gains. σ_Softmax (Smoothed Best Pure Strategy): A Boltzmann distribution over pure strategies biased toward high-payoff modes. A temperature parameter controls concentration — lower temperature means the distribution is more concentrated on the best pure strategy. σ_hybrid = (1 − λ) · σ_ORM + λ · σ_Softmax The training-time solver uses a dynamic annealing schedule over the outer PSRO iterations. The blending factor λ anneals from 0.3 → 0.05 (shifting from

