Game Theory in Chess Endgames – and Possibly More
Omer Aziz Kayhan 4372010
Game Theory in Chess Endgames- and Possibly More
Chess has some interesting and little-known rules. One of these rules is: “if the last 50 consecutive moves have been made by each player without the movement of any pawn and without the capture of any piece gives the players to call for a draw”. This effectively makes chess a finite game. Ernst Zermelo investigates games with two players, with strictly two opposing intentions ( as in both sides wants the other to lose), and considered hypothetical infinite games. The conclusion for chess was, either white or black can force a win, or either one can force a draw. Although this seems pretty straightforward, it implies that, if human error is taken out of the game, there is only one of these three possibilities for the game. Current grandmasters consider the last option to be true. If white and black play the perfect game on both sides, the game will inevitably end in a draw. It is important to point out that, even the strongest chess engines, rated at least 500 ELO points ahead than the top grandmaster, can only calculate 20-30 moves ahead. On average there are 35 moves a player can play, calculating all these moves 35^24 is obviously impossible even at 1 quadrillion calculations per second. The best chess engines also use previous game knowledge, and eliminate bad moves much faster, but one will observe even the computer after given half a minute, changes its mind on the best move possible in a given position.
Research from Stanford University by Qingyun Wu et al, demonstrates by using combinatorial game theory, how chess endgames can be mathematically evaluated, given which sides turn to play. In their analysis, games are divided into 5 categories. Pawn game endings are categorized, distances between pawns, closeness of Kings to pawns, and certain special positions are all considered in their work. By assigning certain numbers to positions, they calculate the endgame conclusion to a game mathematically, without even thinking about what is the best move. They show a way to know the endgame no matter if the best attack or defense is played, if the calculation indicates a draw, nothing can be done to force a win.
Another approach to chess theory might include authority and hubs. Let us assign a positional advantage of 1 to every piece in the beginning of the game. Chess engines grade positions and moves, on a points bases. After every position pieces can be given authority or hub points (white pieces can be authority and black pieces can be subs or vice versa does not matter) the best move will be given the most points. If Checkmate is possible with a certain combination, the piece that can initiate the checkmate can get the amount of moves it will take to mate the other king as hub points, the mating move can have infinite points so losing is not an option. The pieces itself will have a total value of all the scores of all the squares they can move to. Although this way of thinking is highly theoretical. Let us Imagine a position where white can checkmate black in 1 move, but if the wrong move is played, then black can checkmate white in 1 move. If its white’s turn, the piece which can checkmate will have infinite points because its mate in one, and also negative infinite points if the wrong move is played, adding up to an indeterminate conclusion. Assigning such authority and hub numbers to every move possible, with highest points pieces having the best moves might prove to be a practical approach to chess.
Links
http://web.stanford.edu/~wqy/research/Math%20191-%20Chess.pdf
http://www.math.harvard.edu/~elkies/FS23j.03/zermelo.pdf
https://arxiv.org/pdf/1610.07160.pdf
cool stuff.