SUST012 Introductory Game Theory

1
SUST012Introductory Game Theory

• Tutorial 1

2
Collecting your email account

• For better communication, I would like you to
  fill in a list, writing down your name, email
  address.
• I know some people are auditing this course, but
  you are welcome to be added to the mailing list
  too.
• I will put the mailing list to the website also.
  Hope you can make a lot of friends via this
  programme.
• My email address emdiash_at_gmail.com

3
Combinatorial Game

• There are two players.
• There is a set, usually finite, of possible
  positions of the game.
• The rules to govern how the players move.
• Alternate moving.
• In the course we usually consider normal game
  only the player who plays the last turn wins.

4
P-position and N-position

• P positions are positions that the Previous
  player wins.
• N positions are positions that the Next player
  wins.
• To determine the P and N positions, we use the
  labeling algorithm.

5
Labeling Algorithm

• Label every terminal position as P positions.
  (Think about it it is correct from the
  definition of the P positions).
• Label every position that can be reach a
  P-position in ONE move as an N-position.
• Find those positions whose only moves are to
  labeled N-positions label such positions as
  P-positions.
• Keep repeating the above 2 steps. It is often
  able to find some special properties of the P and
  N positions.

6
Example 1

• Consider the subtraction game with S 1,3,4.
• Do you understand how to fill in the positions?
• Can you find a special property of the sequence
  of the positions? (For example, does it repeat
  itself?)

7
Example 2

• A subtraction game with S2,3,5.
• Tutorial Exercise Try to fill in the boxes with
  ?. Remember that there can be more than 1
  terminal positions.
• This time can you find a special property?

8
Homework Problem 1

• Find the set of P-positions for the subtraction
  games with subtraction sets (i) S1,3,5,7, (ii)
  S1,3,6.

9
Nim Game

• There are three piles of chips containing x1, x2,
  and x3 chips respectively.
• Two players take turns moving.
• Each move consists of selecting one of the piles
  and removing at least one chip from it (you can
  remove the whole pile).

10
Interlude Binary Representation

• For some students, they may not know what is
  binary representation. Here is a simple
  introduction
• For a number in decimal representation, say 23,
  it is equal to23 2 10 3.
• For a number in binary representation, say 1011,
  it is equal to1 23 0 22 1 21 1 20
  11.

11
Interlude Binary Representation

• Our daily-life representation is decimal
  representation. Here we have to know how to
  convert a number represented in decimal
  representation to binary representation.
• Example23 1 24 0 23 1 22 1 21
  1 20.
• So 23(10) 10111(2) in binary representation.
• The quick way to find the bolded 0s and 1s will
  be discussed during the tutorial.

12
Nim Sum and example

• Consider the Nim game with 7, 11 and 20 chips in
  3 piles.
• 7 00111(2).
• 11 01011(2).
• 20 10100(2).
• NS 11000(2).

13
The Main Theorem

• Theorem A position, (x1, x2, x3), in Nim is a
  P-position if and only if the nim-sum of its
  components is zero, x1 ? x2 ? x3 0.
• The Theorem holds for 2 piles and more than 3
  piles. The main point is that P position ltgt Nim
  Sum 0.
• I will spare the last 15 minutes to explain why
  this is true.

14
Homework Problem 2

• Find all winning moves in the game of nim, (i)
  with three piles of 12, 19, and 27 chips (ii)
  with four piles of 13, 17, 19, and 23 chips.

15
Notice

• Try to finish the 2 homework now.
• If you encounter any difficulty, try to think
  about it for a few minutes first. If you still
  cannot do it, find me and ask questions.

16
Extra Games

• Chomp
• http//www.math.ucla.edu/tom/Games/chomp.html
• I will explain the rules in class.
• It can be shown that any rectangular shape of the
  chocolate is a N position.
• I will spare the last 15 minutes to explain why
  this is true.