In many game settings, the game is not explicitly given but is only accessible byplaying it.

We study the performance of optimistic regret-minimization algorithms for both minimizing regret in, and computing Nash equilibria of, zero-sum extensive-form games. In order to apply these algorithms to extensive-form games, a distance-generating function is needed. We study the use of the dilated entropy and dilated Euclidean distance functions. For the dilated Euclidean distance function we prove the first explicit bounds on the strong-convexity parameter for general treeplexes. Furthermore, we show that the use of dilated distance-generating functions enable us to decompose the mirror descent algorithm, and its optimistic variant, into local mirror descent algorithms at each information set. This decomposition mirrors the structure of the counterfactual regret minimization framework, and enables important techniques in practice, such as distributed updates and pruning of cold parts of the game tree. Our algorithms provably converge at a rate of $T^{-1}$, which is superior to prior counterfactual regret minimization algorithms. We experimentally compare to the popular algorithm CFR+, which has a theoretical convergence rate of $T^{-0.5}$ in theory, but is known to often converge at a rate of $T^{-1}$, or better, in practice. We give an example matrix game where CFR+ experimentally converges at a relatively slow rate of $T^{-0.74}$,

Dominance is a fundamental concept in game theory. In normal-form games dominated strategies can be identified in polynomial time. As a consequence, iterative removal of dominated strategies can be performed efficiently as a preprocessing step for reducing the size of a game before computing a Nash equilibrium. For imperfect-information games in extensive form, we could convert the game to normal form and then iteratively remove dominated strategies in the same way; however, this conversion may cause an exponential blowup in game size. In this paper we define and study the concept of dominated actions in imperfect-information games. Our main result is a polynomial-time algorithm for determining whether an action is dominated (strictly or weakly) by any mixed strategy in n-player games, which can be extended to an algorithm for iteratively removing dominated actions. This allows us to efficiently reduce the size of the game tree as a preprocessing step for Nash equilibrium computation. We explore the role of dominated actions empirically in "All In or Fold" No-Limit Texas Hold'em poker.

We introduce a quantitative framework for separating skill and chance in games by modeling them as complementary sources of control over stochastic decision trees. We define the Skill-Luck Index S(G) in [-1, 1] by decomposing game outcomes into skill leverage K and luck leverage L. Applying this to 30 games reveals a continuum from pure chance (coin toss, S = -1) through mixed domains such as backgammon (S = 0, Sigma = 1.20) to pure skill (chess, S = +1, Sigma = 0). Poker exhibits moderate skill dominance (S = 0.33) with K = 0.40 +/- 0.03 and Sigma = 0.80. We further introduce volatility Sigma to quantify outcome uncertainty over successive turns. The framework extends to general stochastic decision systems, enabling principled comparisons of player influence, game balance, and predictive stability, with applications to game design, AI evaluation, and risk assessment.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Using the Dafny verification system, we formally verify a range of minimax search algorithms, including variations with alpha-beta pruning and transposition tables. For depth-limited search with transposition tables, we introduce a witness-based correctness criterion and apply it to two representative algorithms. All verification artifacts, including proofs and Python implementations, are publicly available.