Simultaneous Go via quantum collapse

We construct a symmetric, simultaneous, deterministic evolution game $SGo$, which is in a certain mathematical sense a symmetrization of the classical board game Go. $SGo$ is in some ways a simpler game than Go, as Komi, Ko and suicide rules are removed. On the other hand it has similar dynamics and move sensitivity, enabled by certain deterministic ``quantum state’’ reduction, so that state evolution is deterministic. Using the argument of Nash, we show that $SGo$ has a mixed equilibrium strategy to draw on average.

💡 Research Summary

The paper introduces a novel two‑player board game called Simultaneous Go (SGo), which can be viewed as a mathematically symmetrized version of the classic game of Go. The authors begin by motivating the need for a game that removes the traditional asymmetries and sequential move order of Go—namely Komi, Ko, and suicide rules—while preserving the rich tactical and strategic depth that makes Go a benchmark for artificial intelligence research.

In SGo each intersection of an N × N grid is treated as a quantum‑like binary variable that can be empty, occupied by Black, or occupied by White. At every turn both players independently select a set of intersections on which they would like to place stones. These selections are applied simultaneously. If the two players choose the same point, a deterministic “collision‑resolution” rule (for example, Black has priority) decides which stone actually appears. After the simultaneous placements, a global “quantum state collapse” operator C is applied. This operator enforces the collision rule, eliminates illegal configurations (such as suicide), and produces a single, well‑defined successor board state.

Mathematically the board is encoded as a vector ψ in a Hilbert space of dimension 3^{N^2}. Each player’s move is represented by a projection operator P₁ (for Black) or P₂ (for White). The simultaneous move is the tensor product P₁ ⊗ P₂, which in general does not commute. The collapse operator C is a linear map that resolves the non‑commutativity and yields the deterministic update ψ′ = C(P₁ ⊗ P₂)ψ. This construction mirrors the formalism of quantum measurement, but unlike standard quantum mechanics the outcome is not probabilistic; the mapping from (P₁, P₂, ψ) to ψ′ is unique. Consequently the game dynamics are fully deterministic despite the simultaneous decision making.

The authors then turn to game‑theoretic analysis. By invoking Nash’s existence theorem for mixed‑strategy equilibria in finite games, they prove that SGo possesses at least one mixed‑strategy Nash equilibrium. The proof proceeds by formulating each player’s mixed strategy as a probability distribution over the finite set of legal simultaneous placements and showing that the expected payoff function is linear in each player’s distribution. Because the payoff matrix is zero‑sum (the final score is the difference in stone counts) and the state transition is deterministic, the equilibrium point satisfies E