Beyond the Shannon Number

Chess has long been the canonical example of combinatorial explosion in artificial intelligence. Since Claude Shannon's 1950 paper, we have known that the number of possible chess games grows exponentially—on the order of 10¹²°. Exhaustive search is impossible. The real question has always been: how can a machine make strong decisions in a space that large?

For decades, the answer was structured search. Engines like Stockfish relied on minimax with alpha–beta pruning, sophisticated move ordering, transposition tables, and highly tuned evaluation functions. Strength came from searching deeper and pruning smarter.

Then, in 2017, AlphaZero, developed by DeepMind, disrupted the landscape. Using a deep neural network trained purely through self-play and guided by Monte Carlo Tree Search, it defeated the then-leading version of Stockfish 28–0 (with 72 draws). The result was striking: instead of exploring more nodes, AlphaZero explored fewer—but far more selectively, guided by learned priors.

That comparison became symbolic. It was widely interpreted as “neural beats classical.” But the reality is more nuanced.

The match was played under specific conditions, and Stockfish at the time did not yet use neural evaluation. In the years since, classical engines have evolved. Modern Stockfish integrates NNUE (Efficiently Updatable Neural Networks) into its evaluation function, effectively becoming a hybrid system: classical alpha–beta search powered by neural positional understanding. Today, it is once again the strongest engine in practical competition.

The real story, therefore, is not a victory of one paradigm over another. It is about how different architectures reduce exponential complexity. This paper revisits Shannon's original framework and uses the AlphaZero–Stockfish contrast as a case study to understand what “best move” actually means under bounded computational resources.

Rather than focusing on who won a match, the goal is to analyze how modern engines transform an intractable search space into efficient decision-making.