Application of the Minimax Theorem in Xiangqi

Mathematical formulation of the Minimax theorem: max_{x in X} min_{y in Y} f(x, y) = min_{y in Y} max_{x in X} f(x, y)

In Xiangqi, X is the set of pure strategies for Red, Y is the set of pure strategies for Black, and f(x, y) is Red’s payoff under the strategy pair (x, y).

Intuitive meaning of the Minimax theorem: If Red chooses a strategy first and Black chooses afterward, the final result equals the result if Black chooses first and Red chooses afterward. This symmetry ensures the rationality of Minimax search results on the game tree.

Nash Equilibrium and Xiangqi Engines

Each position in Xiangqi can be viewed as a subgame, and the engine’s search process is the process of finding an approximate Nash equilibrium across all subgames.

In game theory, Nash equilibrium is a strategy profile where each player’s strategy is the optimal response to other players’ strategies. Under perfect information conditions in Xiangqi, Nash equilibrium reduces to subgame perfect equilibrium.

Modern engines like Pikafish, by combining search and evaluation, can find high-quality approximate solutions of the game tree within constrained time, approaching subgame perfect equilibrium.

Volume XIX: Detailed Piece Evaluation Table for Xiangqi Engines

Piece Value Table

The following are the default evaluation values of various Xiangqi pieces in NNUE training (in units of one-hundredth of a pawn). These values reflect the value of each piece type under average positions after NNUE network training:

Positional Value of Pieces in the Palace

The palace is one of the most important areas in Xiangqi. The following are typical positional value principles:

Positional value of the General/King: Higher in the center file (cols 4-5); lower in corner positions (col 3 or 6); extremely low when exposed.

Positional value of Advisors: Highest in their original positions; decreases after moving.

Positional value of Elephants: Higher in defensive positions; lower in offensive positions; extremely low when the elephant-eye is blocked.

Value Changes of Pawns in Different Positions

The value of pawns changes across different game phases and positions:

Opening phase: Pawn value is extremely low (approximately 0.3-0.5 pawn units). Pawns that haven’t crossed the river are about 0.3, crossed pawns about 0.8-1.0.

Middlegame phase: Pawn value begins to increase. Pawns that haven’t crossed the river are about 0.5-0.8, crossed pawns about 1.5-2.5, protected connected pawns about 2.0-3.0.

Endgame phase: Pawn value significantly increases. Pawns that haven’t crossed the river are about 0.8-1.0, crossed pawns about 2.0-3.5, protected connected pawns about 3.0-4.5, pawns near the King about 4.0-5.0.

Principles of pawn value distribution: Higher value closer to the opponent’s King; protected pawns have higher value than isolated pawns; connected pawns have higher value than single pawns; central pawns have higher value than edge pawns.