This work estimates the complexity of Uruguayan Truco, an imperfect-information two-team card game, by computing a lower bound on the total number of information sets in its extensive-form representation. We adapt the HyperLogLog (HLL) probabilistic counting algorithm to support arbitrarily long hashes, enabling its use in this new domain. This adapted version is combined with Monte Carlo rollouts to handle the intractable size of the game tree. To validate our approach, we introduce mini-Truco, a simplified and tractable variant of the original game. Our experiments show that, in expectation, this approach yields accurate estimates of the underlying set size while providing horizontal scalability across multiple nodes. After running this game-agnostic method for 100 days (totaling 4.8×105core-hours), we establish that a 20-point 2-player game of Uruguayan Truco contains at least 1.49×1012information sets, providing a concrete lower bound on the game’s complexity.

Uruguayan Truco is a positive-sum and imperfect information card game with 2, 4 and 6 player variants. Finding the Nash equilibria of such games is very hard, so we approximate them using Computational Game Theory and Deep Reinforcement Learning methods. We implement Counterfactual Regret Minimization (CFR) and some of its variants, and Deep Monte Carlo (DMC). We also propose two levels of manual abstraction to reduce the number of information sets, which are sets of indistinguishable game states. We evaluate our methods on T1K22, a dataset of 79,000 random hands of Uruguayan Truco, against two baseline agents and a human player. We find that CFR-based methods outperform DMC, especially External Sampling Monte Carlo CFR, which converges faster and achieves a higher win rate. It is remarkable that, after 2 weeks of training (totaling 4,032 core hours), starting from scratch and without using any human knowledge, the best agents defeated every baseline, with win rates significantly higher than 50%.