Paradoxes in Logic, Mathematics and Computer Science

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Lecture 2

• More Finite Games

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All games considered in Lectures 1 and 2 have the
following features

• Two Players This contrasts with one-player
  games, e.g. Sudoku.
• Win-Lose A player wins iff the other loses.
  (except for chess, which can be modified to be a
  win-lose game). Other games could be win-win.
• Perfect Information The positions and rules of
  the game are known to both players.
• No Chance The rules of the games are
  deterministic, unlike card games.
• Finite Game The game does not last forever.
• Finite Options The players options are finite.

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An Illustrative Example

• Board The 0-1 list
• Two Players Left (L) and Right (R). L moves
  first, then R, then L, then R.
• Rules 1) Each player splits the list into 2
  equal halves and removes one of them. 2)
  When only one cell is left, L wins iff the
  cell contains 1.
• Question How should L play?

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The Max-Min Tree

• This is a tree of Game Positions.
• The root is the initial position.
• The children of a node indicate the possible
  options of the player who plays next.
• The leaves are the terminal (end) positions.
• Leaves are labeled with the gain for L.
• The gains of the children carry over to the gain
  of the parent by the following rule For each
  node n,
• If L plays, then gain(n) max(gains(children(n)))
  .
• If R plays, then gain(n) min(gains(children(n)))
  .

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Exercises

• For the following games
• 1) Construct a max-min game tree and
• 2) Decide if L has a winning strategy.
• The Chocolate Game with 7 squares.
• Nim (3 heaps) from the position (1,2,3)
• 3?3 Tic-Tac-Toe (X-O). Here L starts by placing X
  on a 3?3 grid, R places O, etc.. The winner is
  the first one occupying a row, a column or a
  diagonal. Note This is quite a large tree,
  which could be reduced using symmetry. Also, you
  can draw different subtrees on separate pages.

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Theorem In a finite win-lose game, exactly one
of the players has a winning strategy.

• We claim that one of the players can always win,
  no matter how the other plays.
• Proof
• Its impossible that both players have winning
  strategies.
• If L (say) does not have a winning strategy, a
  winning strategy for R is to just play in a way
  that prevents L from a winning strategy.
• Since the game eventually ends, R wins.

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Theorem In a finite win-lose game, exactly one
of the players has a winning strategy.

• Another (rigorous) Proof Since the game is
  finite, its Max-Min tree has a finite height.
• By induction on the height of that tree, we can
  show that at any node n
• 1) L has a winning strategy iff gain(n) 1.
• 2) R has a winning strategy iff gain(n) 0.
• Its also obvious (by induction again) that
  gain(n) 0 or 1. Thus, exactly one of L or R has
  a winning strategy.

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The Hex Game

• Board An 11?11 rhombus of hexagons.
• Two opposing sides are colored red and the others
  blue.
• Two Players Red (R) and Blue (L). Red moves
  first, etc..
• Rules
• 1) Each player places a red/blue stone on a cell.
• 2) The first player who forms a connected path
  linking his/her opposing sides wins.

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Initial Position
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Final Position Red wins
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Theorem Red can always win.

• Proof
• The game never ends in a tie (draw), i.e. Red
  wins iff Blue loses.
• Exactly one of them has a winning strategy.
• If Blue has a winning strategy, then Red can
  steal it by first making an arbitrary move, then
  following Blues strategy. When Red needs to
  place a stone on an occupied place, she makes
  another arbitrary move, etc..

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What about Chess?

• Position An 8?8 chess board with a configuration
  of the pieces.
• Two Players White and Black. White moves first,
  etc..
• Initial Position

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Can the players play forever?

• Answer No.
• One of the rules states that the game ends in a
  draw, if the same board position recurred 3
  times. Since the number of positions is finite,
  some positions will recur infinitely often.
• Also, another rule states that a draw occurs, if
  at least fifty moves (by each side) have passed
  with no pawn being moved and no capture being
  made.

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Two Open Questions

• Question Does White have a winning strategy?
• Answer Unknown.
• Question Does Black have a strategy that results
  into a win or tie?
• Answer Unknown.
• However Exactly one of the above questions has
  the answer yes.

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How should the players play?

• Chess is too complicated
• Number of legal chess positions ? 1043
• Size of the game tree ? 10120
• Contrast with the number of atoms in the universe
  (between 4?1079 and 1081)
• No simple mathematical (known) theory.
• What we can do is to search the tree of moves up
  to depth n, where for professional chess n ? 8.

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• Thank you for listening.
• Wafik