http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

1
COMP4031/COMP4631 2006-7Artificial
Intelligence for Games and PuzzlesDr. Arthur
Cater
2
Course Web Page - http//csiweb.ucd.ie/Staff/acate
r/comp4031.html

• As time progresses, Lecture Notes and Assignments
  will be added. Watch for them!
• Assessment will be partly by examination (60)
  and partly by programming assignment (2x 20).
• Assignment 1 (20 of unit) set in week 4 and
  due in week 8.
• Assignment 2 (20 of unit) set in week 7 and
  due in week 11.

3
What sort of Games and Puzzles?

• There are two broad categories of multi-player
  games
• Parlour Games of a primarily intellectual
  character
• Chess, Poker, Draughts, Backgammon, Connect-Four,
  
• Physical attributes of a player (dexterity,
  strength, steadiness, speed) have no real bearing
  on the game
• Games of a sports character
• Soccer, Tennis, finger-twitching computer
  games,
• Physical attributes are important (too)
• We will deal almost exclusively with the
  primarily intellectual games, and their close
  kin, combinatorial puzzles.

4
Is this just frivolous?

• Certainly it should be some fun interesting,
  entertaining, challenging. Further,
• Studies of game playing have led historically to
  valuable spin-offs
• In mid-17th Century, mathematicians Pascal,
  Fermat, and others laid the foundations of
  probability theory, and hence statistics, arising
  from a study of gambling games (though beaten by
  Cardan by a century)
• In 20th Century, von Neumann and others
  formulated game theory, which now has
  applications in economics and commerce - being
  used to design auctions for telecoms bandwidth
  for example, and guiding corporate takeover
  strategy
• AI for games has direct commercialisation
  possibilities
• chess machines, in-flight entertainment consoles,
  computer games with AI-driven opposition

5
Is this just frivolous? (2)

• Games provide a proving ground for AI (and
  cognitive science) theories of mentality
• perception, representation, reasoning, learning,
  modelling, risk assessment,
• There may be prizes to be won
• 1m Ing prize for Go sadly no longer available
• Prize for beating Taiwanese junior Go champion
• Prize for Arimaa
• There is fame and prestige in beating champions,
  winning tournaments
• There are still scientific open problems to be
  solved

6
Parlour Games The Three Games

• Three particular traditional games show up a
  further significant division among competitive
  parlour games
• Chess - last man standing type of game
• There are many games with this flavour. They are
  characterised by a race to achieve some goal,
  often involving capture or destruction or
  immobilisation of the opponent.
• Backgammon - the gods help those who help
  themselves type of game
• In games with this flavour, there is a chance
  element, depending usually on dice or cards. More
  skilled players will nevertheless usually win.
• Go - take the lions share type of game
• In games with this flavour, players compete for
  shares of a resource of some kind. To win you
  need not win everything, but through
  give-and-take, just win a greater share than the
  opponent.

7
Last-Man-Standing game Chess

• Chess is the dominant intellectual game in the
  west. In AI it is the most heavily researched
  game by far. After about 50 years work, a chess
  machine was developed which beat the reigning
  world champion, Garry Kasparov.
• Two players each have six kinds of piece, each
  with its own movement rules. Players alternate
  play, moving one piece at a time, sometimes
  capturing and removing an opponents piece.
• Win by checkmate, where the opponent has legal
  moves but none of them will prevent immediate
  capture of the king.
• Note a mirror-image symmetry, top-to-bottom with
  colrev - colour reversal of pieces.

8
Last-Man-Standing game Draughts / Checkers

• Draughts (Checkers) is played on a chessboard, or
  a similar board 10x10.
• A program beat the then world champion, Tinsley,
  in ill health, in the 1990s.
• There is initially just one kind of piece, which
  can move one square along diagonals in a forward
  direction. Upon reaching the far edge they are
  promoted to kings, able to go backward too.
• A piece may capture an opponent piece by hopping
  over it to a vacant square beyond. Many captures
  in one move are possible.
• Win by capturing all the opponents pieces.
• Note a rotational symmetry, with colrev.

9
Puzzle Eternity

• There are 209 playing pieces, all of different
  shapes but covering the same area. They have
  jagged edges with a small number of angles and
  straight-line lengths.
• They are assemblies of six tridrafters -
  30o-60o-90o triangles. The pieces are in reality
  all the same colour and can be used either way
  up. Each piece therefore has 12 possible
  manifestations.
• The puzzle is to fit them together to fill
  perfectly a particular shape of board - shown
  here as the lined blue dodecahedron. The board
  has mirror symmetry along 2 axes and 6-way
  rotational symmetry.

10
Game with chance element Backgammon

• White tries to move pieces anticlockwise to end
  in the bottom right, Red tries to move clockwise
  to top right. (Logically, it is just a straight
  line, wrapped over to be compact)
• Players in turn roll two dice. Each die roll
  allows one piece to be advanced the given number
  of points, to land on a point that is empty,
  occupied by friendly pieces, or occupied by only
  one enemy piece - a blot. The blot is removed
  and must begin from an imaginary off-board point
  before the starting point. This can cost moves.
• Win by getting all pieces, first into the final 6
  points, then to (or beyond) another imaginary
  off-board point beyond the end.
• Gambling for money is usual.

11
Lions-Share game Go

• Players take turns to place one stone on any
  unoccupied intersection of a (usually) 19x19
  board, which is initially empty.
• Blocks of same-colour strongly-connected stones
  are captured and removed if they are surrounded
  so that they have no adjacent empty intersection.
• Capturing stones is part of the game, but not its
  object. One wins mainly by surrounding empty
  space in which opponent stones could not survive.
• Rules are very simple, good play is very hard.
• Note 4-rotation mirror symmetry.

12

• If you are not already familiar with it, spend a
  few minutes on the 9-dot problem
• Draw four straight lines, joined end-to-end, to
  pass through all nine dots.
• (Do not ask me questions about this now! )

13
Points of difference between various games (
puzzles)

• Finger-twitching or not
• Number of players 1, 2, many
• Chance element or not
• Racing To Finish or Sharing Out
• Zero-Sum or not
• Mathematically Solved or not

• Kinds of symmetry
• Perfect Information or not
• Past moves, or current state
• Options of other players
• Simultaneous move or not
• Impartial rules or not

14
(No Transcript)
15
Last-Man-Standing game Nine Mens Morris

• Beginning with an empty board, in phase 1 players
  alternately put a new peg (man) onto an empty
  intersection. If they get 3 in a row - a mill -
  they capture an opponents man.
• When both players have placed all 9 men, phase 2
  begins. A player may slide a man to an adjacent
  connected empty intersection, capturing an enemy
  man if making a new mill.
• A player with exactly 3 men left may jump a man
  to any empty intersection, capturing if making a
  new mill. Win by reducing opponent to 2 men.
• Note 4-way rotational symmetry, combined with two
  forms of mirror symmetry, without requiring
  colrev.

16
Last-Man-Standing race game Tic-Tac-Toe
(NoughtsCrosses)

• A very simple example of a game where there is no
  capturing, merely a race to achieve an objective.
• There is often no winner a draw occurs when no
  player may make a move.
• 4-way rotational symmetry with mirror symmetry.

17
Last-Man-Standing race game Connect-Four

• Players take turns to place one piece of their
  colour on the topmost empty cell of one of seven
  columns.
• There is no capturing. Win by being first to get
  four pieces of your colour in a row, either
  vertically horizontally or diagonally.
• There is only one mirror symmetry, left-to-right.

18
Last-Man-Standing race game Go-Moku / Five In A
Row

• Players take turns to put a piece of their colour
  onto (almost) any empty point in a large grid
  sometimes infinite, sometimes a 19x19 Go board,
• The winner is the first to get five pieces in a
  row.
• Note 4-rotational mirror symmetry.

19
Last-Man-Standing race game Fox and Hounds / Fox
and Geese

• Played on an 8x8 chessboard.
• One player controls the four hounds, which can
  move one square along diagonals in a
  forwards-only direction. The other player
  controls the fox, which moves one square along
  diagonals either forwards or backwards. No two
  pieces can occupy the same square.
• Hounds win by trapping the fox. Fox wins by
  slipping through the line of hounds.
• Note there is no symmetry.
• Also the players are bound by different rules of
  movement. The rules are partisan not impartial.

20
Last-Man-Standing game Roshambo /
Rock-Paper-Scissors

• Two players simultaneously pick one of three
  options,
• Rock wins over scisssors
• Scissors wins over paper
• Paper wins over Rock
• There is arguably a 3-rotational cyclic symmetry
  here.
• This is the first multi-player game we have seen
  to not feature turn-taking.

21
Last-Man-Standing race game Nim

• Players take turns to remove 1, 2, or 3 items all
  from the same row. The one to take the last item
  is the winner.
• (Alternative rule the one to take the last item
  is the loser.)
• There is some symmetry, since all rows are
  interchangeable, even though not all positions
  generated by symmetry are reachable.
• The game is solved there is an algorithm that
  can be followed for winning.

22
Puzzles Last-Man-Standing games versus the
rules Solitaire

• Start with pieces occupying all pits except the
  central one. You may jump a piece over another
  into an empty pit, removing the piece you jumped.
• You succeed (win) if you end with just one piece
  remaining, in the central pit.
• You lose if you cannot move.
• Note 4-way rotational mirror symmetry, and also
  a symmetry between the two ends of the game the
  finish is like the start, with empty and filled
  pits interchanged.

23
Puzzle Logic puzzles

• You are given pieces of information from which
  you should be easily able to deduce pairs of
  attributes that could not go together, as well as
  pairs that do go together.
• Ultimately, you will be left with a few
  alternatives that must be enumerated in order to
  find the one consistent solution.
• There is typically a concealed symmetry, which
  can be made explicit with a tabular
  representation.

• Five men went to five separate cities on five
  different days by five modes of transport.
• Mr. Brown went on the train..
• Mr. Green went to Galway.
• The bus went on Saturday.
• .
• Who went where, how, and when?

24
Puzzle Kakuro

• There is a grid like a crossword.
• Each word across or down must be filled with
  digits 1 - 9, summing to a specified total,
  without duplicates.
• Some words can be filled only by certain
  combinations of digits eg
• 7 in 3 cells 124
• 34 in 5 cells 46789
• In easy kakuro puzzles there are several cells
  where highly-constrained words meet so you can
  get started by fixing a few cell values.

25
Game of Chance Yahtzee

• Any number can play.
• Roll 5 dice per turn, up to three times in a
  turn, keeping as many as you like from previous
  roll that turn.
• Add the pips on qualifying dice to make a score
  in one of the non-bonus categories. You may
  ultimately be forced score zero for some
  categories.
• At the end, add your scores and bonuses, player
  with highest score wins.

26
Game of chance and bluff Liar Dice

• For two or more players.
• The first rolls the five dice, keeping them
  hidden, and passes them to the next player with a
  description of their goodness. Each player may
  then secretly re-roll all some or none, and pass
  them on with an ever more impressive description.
• A player may choose not to accept the description
  given them. If the dice are at least as good as
  described, they lose otherwise the player caught
  lying loses.

27
Game of chance Russian Roulette

• Not recommended.
• There is no winner, only a loser.
• It is not, in game-theoretic terms, a zero-sum
  game.

28
Lions-Share game Mancala / Awari / Kalah

• Each player controls the pits on one side of the
  board. On your turn, pick up the stones from one
  of your pits and add one of them to each
  succeeding pit in an anticlockwise direction. If
  the last pit you reach now has 2 or 3 stones,
  remove them, and then do likewise for the
  preceding pit and so on.
• The winner is the one who captures the majority
  of the stones.
• Note rotational 2-symmetry all pieces alike.

29
Lions-Share game Othello / Reversi

• Players begin with two pieces each as shown. They
  take turns, adding one new piece of their colour.
  All enemy pieces in a vertical/diagonal/horizontal
  line between the new piece and another friendly
  piece are replaced with friendly pieces.
• In practice, 2-sided pieces are used.
• A player who cannot move misses a turn.
• Win by having more pieces than the opponent.
• 4-way rotational mirror symmetry.

30
Lions-share games of chance Poker, Bridge,
Whist, Piquet,

• The use of playing cards, shuffled and dealt in
  secret, introduces an element of chance.
• Some are strictly many-player games (poker,
  bridge), some are 2-player (Piquet, Nomination
  Whist).
• Id call Poker a lions-share game because
  although there is an outright winner of a single
  deal, there are usually many deals in a session.
  The winner is the one who wins most money
  overall.
• In Bridge, Whist, etc, there are several tricks
  to be shared out among the players or teams and
  winning is determined in terms of the share won -
  perhaps set against a contract.

31
Lions-share game Diplomacy

• 2 to 7 players control Army and Navy pieces for
  seven countries. Every other turn, players may
  lose pieces or acquire pieces depending on their
  being the last to occupy certain provinces.
• Attacks may be made on pieces occupying
  provinces, with both offence and defence
  fortified by neighbouring pieces. Moves are
  written secretly and revealed simultaneously.
  Conspiracy and treachery are both encouraged.
• The winner is the first to control more than half
  of the salient provinces.
• No symmetry alliances are important.

32
Games in the real world

• Many real-world situations and problems can be
  viewed as games of a sort
• Players have a choice of actions,
• Players have conflicting goals,
• Players may move sequentially or simultaneously,
• Alliances may prosper,
• Treachery may occur,
• Understanding of the goals of others may be
  useful in predicting their actions and planning
  actions of ones own.
• Parlour games offer environments in which various
  kinds of simplification can be made in order to
  focus attention on particular AI issues
  perception, representation, reasoning, learning,
  opponent modelling, and risk assessment.

• Stock Market
• War
• Passing legislation
• Hustling
• Cartels
• Fight for market share
• Biological evolution
• Industrial relations
• Democratic elections
• Takeover negotiations

33
Points of difference between various games (
puzzles)

• Finger-twitching or not
• Number of players 1, 2, many
• Chance element or not
• Racing To Finish or Sharing Out
• Zero-Sum or not
• Mathematically Solved or not

• Kinds of symmetry
• Perfect Information or not
• Past moves, or current state
• Options of other players
• Simultaneous move or not
• Impartial rules or not

34
The End