Anti-Chess

And

Artificial Intelligence (Major Project Part-I)

Manik Jindal

2K10/CO/054

Contents

1. Acknowledgement

2. Aims and Objectives

3. Review of Literature

a. Introduction to Logic Games

b. Solved Logic Games

c. The MinMax Algorithm

d. MinMax algorithm with Alpha-Beta pruning

4. Introduction to Anti-Chess

a. Rules

b. Similar Variants

c. Differences to Normal Chess

5. Anti-Tic-Tac-Toe

6. GUI implementation as a multiplayer Web-Application

7. KPantas Losers Chess

a. The Algorithm

b. The Heuristics

8. Future Plan of Action

9. References

10. Appendix I

11. Appendix II

12. Appendix III

Acknowledgement

I take this opportunity to express my profound gratitude and deep

regards to my project guide, Prof. Daya Gupta, HoD Computer

Engineering, Delhi Technological University, for her exemplary

guidance, monitoring and constant encouragement throughout the

course of this training. Her continued blessings and guidance shall carry

me a long way in the Computer Engineering field.

Aims and Objectives

1. To bridge the gap of our understanding in combinatorial game theory

about games with branching factor which varies heavily. Such as the

games employing forced-moves.

2. To research and implement an all-intelligent computer player for the

Game, Anti-Chess.

3. To make this algorithm available online as a GUI for users around the

world to play and strengthen the A.I.

4. To contribute to the research of the Anti-Chess community worldwide, a

relatively new field.

Review of Literature

Introduction to Logic Games

A Logic Game is a game in which the number of possible moves at any given

point of time in the game is bound by certain logical constraints and hence the

number of moves is finite and the next move will belong to that set of moves.

Examples of Logic games are many like Chess, Checkers, Go, Tic-Tac-Toe,

Connect 4, Reversi, etc.

The Logical Constraints incorporate the physical limitations of the game and

they robustly define and implement the rules of the game in logical definitions.

A two-player game is a game played between two independent entities (We

do not deal with Coalitions).

A Zero-Sum game is a game in which one players victory is equal to the

others loss that is there are no cooperative victories.

Cooperative or non-cooperative

A game is cooperative if the players are able to form binding commitments.

For instance the legal system requires them to adhere to their promises. In non-

cooperative games this is not possible.

Symmetric and asymmetric

Symmetric game is a game where the payoffs for playing a particular strategy

depend only on the other strategies employed, not on who is playing them. If

the identities of the players can be changed without changing the payoff to the

strategies, then a game is symmetric.

Simultaneous games are games where both players move simultaneously, or

if they do not move simultaneously, the later players are unaware of the earlier

players actions (making them effectively simultaneous).

Sequential games (or dynamic games) are games where later players have

some knowledge about earlier actions. This need not be perfect information

about every action of earlier players; it might be very little knowledge. For

instance, a player may know that an earlier player did not perform one

particular action, while he does not know which of the other available actions

the first player actually performed.

An important subset of sequential games consists of games of perfect

information. A game is one of perfect information if all players know the

moves previously made by all other players. Thus, only sequential games can

be games of perfect information, since in simultaneous games not every player

knows the actions of the others.

Perfect information is often confused with complete information, which is a

similar concept. Complete information requires that every player know the

strategies and payoffs available to the other players but not necessarily the

actions taken.

Infinitely long games

Games, as studied by economists and real-world game players, are generally

finished in finitely many moves. Pure mathematicians are not so constrained,

and set theorists in particular study games that last for infinitely many moves,

with the winner (or other payoff) not known until after all those moves are

completed.

Discrete and continuous games

Much of game theory is concerned with finite, discrete games that have a finite

number of players, moves, events, outcomes, etc. Many concepts can be

extended, however. Continuous games allow players to choose a strategy from

a continuous strategy set.

Examples of game that have finite number of possible moves but are not logical

games: Rock-Paper-Scissors, Coin flipping

Solved Logic Games

A solved game is a game whose outcome (win, lose, or draw) can be correctly

predicted from any position when each side plays optimally.

In game theory, perfect play is the behaviour or strategy of a player which

leads to the best possible outcome for that player regardless of the response by

the opponent. Based on the rules of a game, every possible final position can

be evaluated (as a win, lose or draw). By backward reasoning, one can

recursively evaluate a non-final position as identical to that of the position that

is one move away and best valued for the player whose move it is.

Thus a transition between positions can never result in a better evaluation for

the moving player and a perfect move in a position would be a transition

between positions that are equally evaluated. As an example, a perfect player

in a drawn position would always get a draw or win, never a loss. If there are

multiple options with the same outcome, perfect play is sometimes considered

the fastest method leading to a good result, or the slowest method leading to a

bad result.

A two-player game can be solved on several levels:

1. Ultra-weak

In the weakest sense, solving a game means proving whether the first

player will win, lose, or draw from the initial position, given perfect play

on both sides. This can be a non-constructive proof (possibly involving

a strategy stealing argument) that need not actually determine any moves

of the perfect play.

2. Weak

More typically, solving a game means providing an algorithm that

secures a win for one player, or a draw for either, against any possible

moves by the opponent, from the beginning of the game. That is,

producing at least one complete ideal game (all moves start to end), with

proof that each move is optimal for the player making it.

3. Strong

The strongest sense of solution requires an algorithm which can produce

perfect play (moves) from any position, i.e. even if mistakes have

already been made on one or both sides.

Given the rules of any two-person game with a finite number of positions, one

can always trivially construct a MinMax tree that would exhaustively traverse

the game tree. However, since for many non-trivial games such an algorithm

would require an infeasible amount of time to generate a move in a given

position, a game is not considered to be solved weakly or strongly unless the

algorithm can be run by existing hardware in a reasonable time. Many

algorithms rely on a huge pre-generated database, and are effectively nothing

more than that.

Example, the game of tic-tac-toe is solvable as a draw for both players with

perfect play (a result even manually determinable by schoolchildren).

The MinMax Algorithm

MinMax (sometimes MiniMax), is a decision rule used in decision theory,

game theory, statistics and philosophy for minimizing the possible loss for a

worst case (maximum loss) scenario. Alternatively, it can be thought of as

maximizing the minimum gain (maximin or MaxMin).

Originally formulated for two-player zero-sum game theory, covering both the

cases where players take alternate moves and those where they make

simultaneous moves, it has also been extended to more complex games and to

general decision making in the presence of uncertainty.

The MinMax theorem states:

For every two-person, zero-sum game with finitely many strategies, there

exists a value V and a mixed strategy for each player, such that

a. Given player 2s strategy, the best payoff possible for player 1 is V,

b. Given player 1s strategy, the best payoff possible for player 2 is

V.

Equivalently, Player 1s strategy guarantees him a payoff of V regardless of

Player 2s strategy, and similarly Player 2 can guarantee himself a payoff of

V.

The name MinMax arises because each player minimizes the maximum payoff

possible for the othersince the game is zero-sum, he also minimizes his own

maximum loss (i.e. maximize his minimum payoff).

A MinMax algorithm is a recursive algorithm for choosing the next move in

an n-player game, usually a two-player game. A value is associated with each

position or state of the game. This value is computed by means of a position

evaluation function and it indicates how good it would be for a player to reach

that position. The player then makes the move that maximizes the minimum

value of the position resulting from the opponents possible following moves.

The algorithm can be thought of as exploring the nodes of a game tree. The

effective branching factor of the tree is the average number of children of each

node (i.e., the average number of legal moves in a position). The number of

nodes to be explored usually increases exponentially with the number of plies

(it is less than exponential if evaluating forced moves or repeated positions).

The number of nodes to be explored for the analysis of a game is therefore

approximately the branching factor raised to the power of the number of plies.

It is therefore impractical to completely analyse games such as chess using the

MinMax algorithm.

The performance of the nave MinMax algorithm may be improved

dramatically, without affecting the result, by the use of alpha-beta pruning.

Other heuristic pruning methods can also be used, but not all of them are

guaranteed to give the same result as the un-pruned search.

A nave MinMax algorithm may be trivially modified to additionally return an

entire Principal Variation along with a MinMax score.

Alpha-Beta Pruning

Alphabeta pruning is a search algorithm that seeks to decrease the number of

nodes that are evaluated by the MinMax algorithm in its search tree. It is an

adversarial search algorithm used commonly for machine playing of two-

player games (Tic-tac-toe, Chess, Go, etc.). It stops completely evaluating a

move when at least one possibility has been found that proves the move to be

worse than a previously examined move. Such moves need not be evaluated

further. When applied to a standard MinMax tree, it returns the same move as

MinMax would, but prunes away branches that cannot possibly influence the

final decision.

Alpha-beta pruning gets its name from two bounds that are passed along during

the calculation, which restrict the set of possible solutions based on the portion

of the search tree that has already been seen. Specifically,

Beta is the minimum upper bound of possible solution

Alpha is the maximum lower bound of possible solutions

Thus, when any new node is being considered as a possible path to the solution,

it can only work if:

Where N is the current estimate of the value of the node.

Introduction to Anti-Chess

Anti-Chess (also known as Losing chess, the Losing Game, Giveaway chess,

Suicide chess, Killer chess, or Take-all chess) is a chess variant in which the

objective of each player is to lose all of his pieces, that is, a misre version. It is

one of the most popular of all chess variants.

The origin of the game is unknown, but believed to significantly pre-date an early

version, named Take Me, played in the 1870s. Because of the popularity of Anti-

Chess, several variations have spawned.

Rules

The rules of the game are the same as those of chess except for the following

additional rules:

Capturing is compulsory.

When more than one capture is available, the player may exercise choice.

The king has no special prerogative and accordingly:

o It may be captured like any other piece.

o There is no check or checkmate.

o There is no castling.

o Pawns may also promote to King.

In the case of stalemate, there are different rules:

o It is a win for the stalemated player (international rules).

o It is a draw.

o It is a win for the player with the fewer number of pieces, and if both

have the same number it is a draw. The type of the piece makes no

difference (FICS rules).

A player wins by losing all his pieces, or being stalemated (as detailed.) Apart

from move repetition, mutual accord and the fifty-move rule, the game is also

drawn when a win is impossible, such as if a dark-squared bishop and a light-

squared bishop are the only pieces remaining.

Because of the forced capture rule, Anti-Chess games often involve long

sequences of forced captures by one player. This means that a minor mistake

can ruin the game. Losing openings include 1.d4, 1.e4, 1.d3, 1.Nc3, 1.Nf3,

1.f4, 1.h4, 1.b4, 1.h3. Some of these openings took months of computer time

to solve, but the wins against 1.d3, 1.d4, and 1.e4 consist of a single series of

forced captures and can be played from memory by most experienced players.

Similar Variants

Losers Chess: Losing chess (Suicide chess) has been a popular chess

variant on most chess servers that have offered it ever since the early

days of the ICS. There are two major variants played online, Losers

Chess played on ICC, which actually created the variant because they

were unable to implement the free ICSs Suicide chess rules. The goal

in both games is to lose all of ones pieces, although in Losers Chess, a

player can also win by getting checkmated.

Kamikaze Chess:

o A player wins by losing all his pieces, or by checkmating the

opponent.

o The king has royal powers, and removing check takes precedence

over capturing.

o Players must lose their king last. Players must not move into

check until they have only the king left.

o Pawns promote only to queens.

Differences to Normal Chess:

1. Although the pieces move in the same way as in Chess, in Anti-Chess,

the restriction to capture a piece, if the player can, amounts to long

sequences of forced-captures.

2. Forced-Captures can turn the game around for the losing player very

quickly.

3. The End-Games are particularly varied.

4. Forced Captures lead to quicker games. An Average Anti-Chess game

may last only up to 1/10th the time taken by an average Chess game.

Anti-Tic-Tac-Toe

In the pursuit of developing Artificial Intelligence for Anti-Chess, a small

experiment was done with the traditional Tic-Tac-Toe on the same lines as Anti-

Chess, called Anti-Tic-Tac-Toe.

The experiment was undertaken to assess future challenges in developing an all-

intelligent computer player for Anti-Chess and to get a fair idea of the possible

methods to tackle the problem at hand.

The steepest-ascent hill climbing method was successfully implemented for the

game. The evaluation function is given below.

1. There are 8 lines, in which a player can win. (3 Horizontal, 3 Vertical

and 2 Diagonal).

2. Let X be the computer player, and O be the opponent, then:

a. Assign a line +10 if there is just a single X in it.

b. Assign a line +100 is there are just two Xs in it.

c. Assign a line +1000 if there are three Xs in it.

d. Assign a line -10 if there is just a single O in it.

e. Assign a line -100 if there are just two Os in it.

f. Assign a line -1000 if there are three Os in it.

g. If a line has both X and O assign it the value 0.

3. Total all the values assigned and negate the result, this gives us the

evaluation for the state.

4. Chose the next state which maximises the difference in the evaluation

value of the current state and the next state.

The Code base to the implementation is attached at Appendix I.

GUI implementation as a Web-

Application

A major challenge in the success of any game-related A.I. is that it will always

be beaten by players around the world who do not think conventionally,

especially in a game such as Anti-Chess.

To tackle this, we are implementing a web-application to be hosted on the

internet that allows real-time players to compete with our A.I. and generate

useful test cases for us to analyse.

This being pursued with the highest of enthusiasm using Googles App engine

with python and webapp2 (web-framework) to make this project up and running

as soon as possible.

The implementation is in its final stages with a handful of bugs to be removed

and features to be added. Its code base is attached in Appendix II.

KPantas Losers Chess

In the quest to implement an all-intelligent computer player for Anti-Chess, we

decided to study projects available as open-source literature. One such project is

KPantas Losers Chess.

With particular interest in the A.I. we analysed the following.

The Algorithm

1. KPantas Losers Chess implements an alpha-beta pruned MinMax

tree, up to an average depth of 8 plies.

2. The evaluation function consists of 4 factors:

a. Material Value: The innate value associated with each piece on the

board

b. Positional Value: The value of the piece at a particular position on

the board

c. Castling Value: The arrangement of the board w.r.t to a King being

Castled

d. Extra Bonus: Some particular arrangements of the board offering

known advantage to the player.

3. The final value of the state is given by (Material Value + Positional

Value)*10 + Castling Value + Extra bonus.

The Heuristics

The Heuristic values of KPantas Losers Chess is given out as position matrix

for each piece on the board. It is attached in Appendix III.

Future Plan of Action

1. To complete the GUI implementation as a web-application and host it

on the internet within a month.

2. To study various heuristics used in Chess and draw a parallel for Anti-

Chess.

3. To Study and analyse various heuristics developed, for their flaws

(both big and small) and their strong points.

4. To finally arrive at a versatile algorithm/heuristic function as an all-

intelligent computer player for Anti-Chess.

References

1. http://wikipedia.org

2. http://catalin.francu.com/nilatac/book.php

3. http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Antichess.html

4. http://ilk.uvt.nl/icga/games/losingchess/

5. http://www.jsbeasley.co.uk/vchess/losingendlit.pdf

6. http://suicidechess.ca/index.php

7. http://scidb.sourceforge.net/index.html

Appendix I

Appendix II

Appendix III