Game Theory chapter 4 from book Game Theory A Playful Introduction

1. Hackenbush &
Partizan Games
Alyaa Muhi

2. Outline
 Hackenbush
 Hackenbush Game
 Negation
 Proposition 4.2
 Integer Positions
 Fractional Positions
 Dyadic Numbers and Positions
 Proposition 4.6
 Birthdays of Dyadic Numbers
 Dyadic Positions
 Lemma 4.11
 Theorem 4.12
 Example

3. Hackenbush
 Is a normal-play game played between Louise and Richard.
 The game consists of a graph drawn with black and gray
edges, some of which are attached to the ground.
 bLack-gRay Hackenbush is an example of a game with 2
players, Left and Right.
 In bLack- gRay Hackenbush, Left deletes any bLack edge,
and Right deletes any gRay edge.
 Whichever player cannot move, loses.

4. Hackenbush Game

5. Hackenbush Game

6. Hackenbush Game

7. Hackenbush Game

8. Hackenbush Game

9. Hackenbush Game

10. Hackenbush Game

11. Hackenbush Game

12. Hackenbush Game

13. Hackenbush Game

14. Hackenbush Game

15. Hackenbush Game

16. Hackenbush Game

17. Hackenbush Game

18. Hackenbush Game

19. Hackenbush Game

20. Hackenbush Game

21. Negation
Figure 4.2. A position and its negative
 In Hackenbush , negation arises naturally since we can reverse the roles
of the players in any position 𝛼 by switching the colors of all the edges.
 This new position is denoted −𝛼 and called negative.

22. Proposition 4.2
 If 𝛼 and 𝛽 are Hackenbush positions, then
(1) −(− 𝛼 ) ≡ 𝛼 ,
(2) 𝛼 + (− 𝛼 ) ≡•0 , and
(3) 𝛽 + (− 𝛼 ) ≡•0 implies 𝛼 ≡ 𝛽 .

23. Integer Positions
 For every positive integer 𝑛, define •𝑛 to be the
Hackenbush position consisting of 𝑛 isolated black edges.
 For a negative integer −𝑚 , we define •(−𝑚) to be the
Hackenbush position consisting of 𝑚 isolated gray edges
Figure 4.3. Some integral positions

24. Figure 4.4. Negation and •
Integer Positions

25. Integer Positions
Figure 4.5. Sum and •

26. Fractional Positions
Figure 4.8. Is 𝛽 worth
1
4 to Louise?Figure 4.7. Is 𝛼 worth
1
2
to Louise?
 Hackenbush position 𝛼 that gives Louise an advantage of
1
2
• 𝛼 +𝛼=1 Louise an advantage of 1
• 𝛼 +𝛼+ •(-1)= •(0)= p

27. Fractional Positions
 For every positive integer 𝑘, define •
1
2k to be the Hackenbush position
Figure 4.9. The position •
1
2k

28. Dyadic Numbers and Positions
 Any number that can be expressed as a fraction where the denominator
is a power of 2 (and the numerator is an integer).
So for instance,
17
32
and ,
531
64
are dyadic numbers , but
2
7
and 𝜋 are not.
 The binary expansion of a number 𝑞 is a representation of 𝑞 as
a sum of distinct powers . 𝟖𝟑
𝟔𝟒
=
𝟏
𝟔𝟒
(83)
=
𝟏
𝟔𝟒
(64 + 19)
=
𝟏
𝟔𝟒
(64 + 16 + 3)
=
𝟏
𝟔𝟒
(64 + 16 + 2 + 1)
= 1 +
𝟏
𝟒
+
𝟏
𝟑𝟐
+
𝟏
𝟔𝟒

29. Proposition 4.6
 Every dyadic number has a unique (finite) binary expansion.
Figure 4.10. Sets of numbers
• Let 𝑞=
𝑛
2𝑘
be a dyadic number
• binary expansion of 𝑛 =2d1 + 2d2 + ⋯ + 2dl then binary
expansion of q =2d1-k + 2d2-k + ⋯ + 2dl-k .
• Every dyadic number m
2𝑘 and negative −𝑚
2𝑘
. for a pair of
dyadic numbers, m
2𝑘 and n
2𝑘 the sum is
likewise another dyadic number
2hm+2k 𝑛
2k+h ,This property
is known as closure .

30. In general rule: On day 0 the number 0 is born. If 𝑎1 < 𝑎1 <
⋯ < 𝑎ℓ are the numbers born on days 0, 1, … , 𝑛, then on
day 𝑛 +1 the following new numbers are born:
 The largest integer which is less than 𝑎 1 ,
 The smallest integer which is greater than 𝑎 ℓ ,
 The number (𝑎i + 𝑎 i +1 )/2 for every 1 ≤ 𝑖 ≤ ℓ −1 .
Birthdays of Dyadic Numbers

31. Birthdays of Dyadic Numbers
Figure 4.11. Birthdays of some dyadic numbers

32. Dyadic Positions
Figure 4.12. Some dyadic positions
 For every dyadic number 𝑞 >0 with binary expansion 2 𝑑1 +2𝑑2 +⋯+2 𝑑ℓ (here 𝑑1 > 𝑑2>
⋯ > 𝑑ℓ are integers which may be positive or negative), we define the position.
•𝑞 = •2 𝑑1 + •2 𝑑2 + ⋯ + •2 𝑑ℓ .

33.  Theorem If 𝑝 ,𝑞 are dyadic numbers, then
(1) −(•𝑝 ) ≡ •(−𝑝),
(2) (•𝑝 ) + (•𝑞 ) ≡ •(𝑝+𝑞) .
 Observation. For a dyadic number 𝑞
we have
𝑞 is type { L 𝑖 𝑓 𝑞>0,
P 𝑖 𝑓 𝑞=0,
R 𝑖 𝑓 𝑞<0.
Dyadic Positions

34. The Simplicity Principle
This new principle provides a procedure to determine a
dyadic position equivalent to a given one under certain
assumptions.
Figure 4.15. Position notation

35. The Simplicity Principle
Some basic properties about positions of the form •
1
2k . Using
our position notation with dyadic positions, for every 𝑘 >0 we
have.
•
1
2k = {•0 ∣ •1, •
1
2
,…, •
1
2k−1 } .
So, if a player moves •
1
2k to •𝑐, then either Richard moved and 𝑐 is
at least larger than •
1
2k, or Louise moved and 𝑐 is smaller. This
principle immediately yields the following.

36. Lemma 4.11
Let 𝑐=
1
2k with 𝑘 ≥1 and suppose a player moves the position •𝑐 to the new
position •𝑐′.
(1) If Louise moved, then 𝑐 ′ ≤ 𝑐−
1
2k .
(2) If Richard moved, then 𝑐 ′ ≥ 𝑐+
1
2k .
We are now ready to introduce the Simplicity Principle that we will use to prove that
certain positions in partizan games are equivalent to dyadic positions.

37. Theorem 4.12
(The Simplicity Principle). Consider a position in a partizan
game given by 𝛾 = {𝛼 1 , … , 𝛼 𝑚 ∣ 𝛽 1 , … , 𝛽 𝑛 } and suppose
𝛼 𝑖 ≡ •𝑎 𝑖 for 1 ≤ 𝑖 ≤ 𝑚,
𝛽 𝑗 ≡ •𝑏 𝑗 for 1 ≤ 𝑗 ≤ 𝑛.
If there do not exist 𝑎 𝑖 and 𝑏 𝑗 with 𝑎 𝑖 ≥ 𝑏 𝑗 , then 𝛾 ≡ •𝑐 where 𝑐
is the oldest number larger than all of 𝑎 1 , … , 𝑎 𝑚 and smaller
than all of 𝑏 1 , … , 𝑏 𝑛 .

38. Example
Here we use the Simplicity Principle to find a dyadic position equivalent to a chair-
shaped Hackenbush position.
Figure 4.16. Applying the Simplicity Principle in Hackenbush

39. Example
Figure 4.17. Domineering positions equivalent to dyadic positions

40. Example
Figure 4.18. Cut-Cake positions equivalent to dyadic positions