Plays, Values, Analysis and The Complexity of Chinese Chess

Chinese Chess (Xiangqi) and Backgammon evolved from an ancient Chinese game called Liubo that was invented some 3,500 years ago [10, pp. 3–11]. Like Liubo, Xiangqi consists of one king, five pawns, and several higher order pieces, and the game is decided by capturing the opponent’s king. Unlike in Liubo, the moves of Xiangqi are not determined by the roll of dice. The dice part of the game eventually evolved into another famous game, Backgammon. Xiangqi predates both Shogi and International Chess, and the latter are the obvious variants of the former. In this paper we will refer to International Chess as Chess and Chinese Chess as Xiangqi. Since Xiangqi and Chess are both purely strategic games (without random moves), it should conceptually be possible to decide whether there is a winning strategy for the first or the second player, or the game always ends with a draw, if both the players play perfect strategies. Such games are called combinatorial games, and combinatorial game theory (CGT) is the branch of mathematics devoted to their analysis [3]. In recent times a rich theory on how to evaluate/analyze game positions has been developed. This theory has successfully been applied to certain endgame positions of Chess [6] and Go [2]. Unfortunately, CGT cannot be directly applied to Xiangqi. Therefore in this paper, we will make a few simplifying assumptions about the game to make it better accessible to CGT. These assumptions will of course affect the actual value of a game position but we will still be able to correctly identify the winner, which is the foremost important part in analysis of combinatorial board games. In Chess, pawn endgames can be easily analyzed if there are no higher order pieces present on the board [6]. By higher order pieces we mean all pieces except pawns, e.g. Bishop, Rook, etc. In Xiangqi this is not an easy job, since pure pawn endgames are rare because higher order pieces usually survive until the end, leading to loopy games with more complex values [3]. Another problem in strategic games is to find a winning move for a given game position. Since the search space usually becomes astronomically huge with every move, finding the best move by exhaustive search is not a good idea at all. Actually, most interesting games are PSPACEor