SLIDE 1 No Normal al an and misère pla lay of
ltip ipla laye yer ga games wi with th pr preference
Games and Graphs Workshop October 23rd - 25th , 2017 University Lyon~1 Koki Suetsugu
Graduate School of Human and Environmental Studies Kyoto Univ.
SLIDE 2 Table of contents
Normal, misère and multiplayer NIM with preference
The Integration of misère NIM and multiplayer NIM
SLIDE 3 Background
- Early studies
- Normal and misère NIM
- Multiplayer game with preference
– Includes Li's theory
SLIDE 4 Nimber
(3, 2, 4) 011 010 100 101 3 ⊕ 2 ⊕ 4 = 5
・Calculate mod-2 sum of the number of
stones of each heap in binary notation without carry
SLIDE 5
Normal NIM
P-position of normal NIM:
𝑜1 ⊕ 𝑜2 ⊕. . .⊕ 𝑜𝑙= 0
SLIDE 6
Misère NIM
P-position of misère NIM:
ቊ𝑜1 ⊕ 𝑜2 ⊕. . .⊕ 𝑜𝑙= 0(∃𝑜𝑗 > 1) 𝑜1 ⊕ 𝑜2 ⊕. . .⊕ 𝑜𝑙= 1(∀𝑜𝑗 ≤ 1)
SLIDE 7 Background
- Early studies
- Normal and misère NIM
- Multiplayer game with preference
– Includes Li's theory
SLIDE 8 3-player NIM
Third player wins Second player wins First player can't win
SLIDE 9 Each player has a total “preference” ordering. If player 𝑌 has preference
- rder 𝐵 > 𝐶 then it is better
for 𝑌 that player 𝐵 moves last than player 𝐶 moves last.
※Assuming players behave optimally for her “preference”.
Preference
: > > : > > : > >
SLIDE 10
Definitions
𝑂(𝐵): Next player of player 𝐵 𝑂−1(𝐵): Previous player of player 𝐵 𝑂2 𝐵 = 𝑂 𝑂 𝐵 , 𝑂3 𝐵 = 𝑂 𝑂2 𝐵 , … 𝑂−2 𝐵 = 𝑂−1 𝑂−1 𝐵 , 𝑂−3 𝐵 = 𝑂−1 𝑂−2 𝐵 , … Note that 𝑂0 𝐵 = 𝑂𝑜 𝐵 = 𝐵.
SLIDE 11 Each player has a total “preference” ordering. If player 𝑌 has preference
- rder 𝐵 > 𝐶 then it is better
for 𝑌 that player 𝐵 moves last than player 𝐶 moves last.
※Assuming players behave optimally for her “preference”.
Preference
: > > : > > : > >
SLIDE 12 A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 13 A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 14 B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 15 B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 16 B A B B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 17 B A B B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 18 B A B B B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 19 B A B B B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 20 B C A B A B B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 21 B C A B A B B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 22 B B C A B A B B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 23 B B C A B A B B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 24 B B C A B A B B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 25 B A B C A B A B B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 26 B A B C A B A B B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 27 B A A B C A B A B B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 28 A B A A B C A B A B B A B B C A A B C A B C : 𝐵 > 𝑂(𝐵) > 𝑂2(𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂2(𝐶) : 𝐷 > 𝑂(𝐷) > 𝑂2(𝐷)
Play order:
SLIDE 29
Definitions
Let 𝐻 be a game position. Suppose that 𝑌 is the first player of 𝐻. For all player 𝑌, if player 𝑂𝑗−1(𝑌) moves last, then 𝐻 is called an 𝑗-position.
SLIDE 30
Generalized NIM Sum:⊕𝑛
𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 Example: 3 ⊕3 15 ⊕3 13 ⊕3 11 3 15 13 11 3 ⊕3 15 ⊕3 13 ⊕3 11
SLIDE 31
Generalized NIM Sum:⊕𝑛
𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 Example: 3 ⊕3 15 ⊕3 13 ⊕3 11 3 0011 15 1111 13 1101 11 1011 3 ⊕3 15 ⊕3 13 ⊕3 11
SLIDE 32
Generalized NIM Sum:⊕𝑛
𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 Example: 3 ⊕3 15 ⊕3 13 ⊕3 11 3 0011 15 1111 13 1101 11 1011 3 ⊕3 15 ⊕3 13 ⊕3 11
SLIDE 33
Generalized NIM Sum:⊕𝑛
𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 Example: 3 ⊕3 15 ⊕3 13 ⊕3 11 3 0011 15 1111 13 1101 11 1011 3 ⊕3 15 ⊕3 13 ⊕3 11 "0201"
SLIDE 34 𝑛-player normal NIM
If for all player 𝑌, her preference order is 𝑌 > 𝑂 𝑌 > ⋯ > 𝑂𝑛−1 𝑌 , then NIM position is a 0–position(𝑛–position) if and only if 𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 = "00 … 00" ※Note that this result includes the theory of two- player normal play.
S.-Y Robert Li. N-person Nim and N-person Moore's Games.
- Internat. J. Game Theory, Vol. 7, No. 1, pp.31-36, 1978.
SLIDE 35
New result
SLIDE 36 When does worst player take last stone?
normal play: 𝑜1 ⊕ 𝑜2 ⊕ ⋯ ⊕ 𝑜𝑙＝0 misère play: ቊ𝑜1 ⊕ 𝑜2 ⊕ ⋯ ⊕ 𝑜𝑙＝0(∃𝑜𝑗 > 1) 𝑜1 ⊕ 𝑜2 ⊕ ⋯ ⊕ 𝑜𝑙＝1(∀𝑜𝑗 ≤ 1) 𝑛-player normal play: 𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙＝"00 … 00" 𝑛 = 2
SLIDE 37 When does worst player take last stone?
normal play: 𝑜1 ⊕ 𝑜2 ⊕ ⋯ ⊕ 𝑜𝑙＝0 misère play: ቊ𝑜1 ⊕ 𝑜2 ⊕ ⋯ ⊕ 𝑜𝑙＝0(∃𝑜𝑗 > 1) 𝑜1 ⊕ 𝑜2 ⊕ ⋯ ⊕ 𝑜𝑙＝1(∀𝑜𝑗 ≤ 1) 𝑛-player normal play: 𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙＝0 𝑛-player misère play:
New result
𝑛 = 2 𝑛-player normal play: 𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙＝"00 … 00"
SLIDE 38
New result: 𝑛-player misère play
Theorem: Assume that for all integer 𝑘 and for all player 𝑌, her preference order is 𝑂𝑘 𝑌 > 𝑂𝑘+1 𝑌 > ⋯ > 𝑂𝑛−1 𝑌 > 𝑌 > 𝑂 𝑌 … > 𝑂𝑘−1 𝑌 , then 𝑜1, 𝑜2, … , 𝑜𝑙−1, 𝑜𝑙 is a 𝑘-position if and only if ቊ𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 = "00 … 00"(∃𝑜𝑗 > 1) 𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 = "00 … 0𝑘"(∀𝑜𝑗 ≤ 1)
SLIDE 39 : …> > > >… > > >…………… : ……………> > > >… > > >… : > > > … > > >…………………… : > > > >… >…………………… > … …
SLIDE 40
New result: 𝑛-player misère play
Theorem: Assume that for all integer 𝑘 and for all player 𝑌, her preference order is 𝑂𝑘 𝑌 > 𝑂𝑘+1 𝑌 > ⋯ > 𝑂𝑛−1 𝑌 > 𝑌 > 𝑂 𝑌 … > 𝑂𝑘−1 𝑌 , then 𝑜1, 𝑜2, … , 𝑜𝑙−1, 𝑜𝑙 is a 𝑘-position if and only if ቊ𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 = "00 … 00"(∃𝑜𝑗 > 1) 𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 = "00 … 0𝑘"(∀𝑜𝑗 ≤ 1)
SLIDE 41
This result includes two- player misère NIM by 𝑛 = 2 and 𝑘 = 1
SLIDE 42
Two-player misère NIM
Theorem: Assume that for all integer 𝑘 and for all player 𝑌, her preference order is 𝑂𝑘 𝑌 > 𝑂𝑘+1 𝑌 > ⋯ > 𝑂𝑛−1 𝑌 > 𝑌 > 𝑂 𝑌 … > 𝑂𝑘−1 𝑌 , then 𝑜1, 𝑜2, … , 𝑜𝑙−1, 𝑜𝑙 is a 𝑘-position if and only if ቊ𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 = "00 … 00"(∃𝑜𝑗 > 1) 𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 = "00 … 0𝑘"(∀𝑜𝑗 ≤ 1)
SLIDE 43
Two-player misère NIM
Theorem: Assume that for all player 𝑌, her preference order is 𝑂1 𝑌 > 𝑌 then 𝑜1, 𝑜2, … , 𝑜𝑙−1, 𝑜𝑙 is a 1-position if and only if ቊ𝑜1 ⊕2 𝑜2 ⊕2 … ⊕2 𝑜𝑙 = "00 … 00"(∃𝑜𝑗 > 1) 𝑜1 ⊕2 𝑜2 ⊕2 … ⊕2 𝑜𝑙 = "00 … 01"(∀𝑜𝑗 ≤ 1)
SLIDE 44
This result also includes multiplayer normal NIM by 𝑘 = 0
SLIDE 45
Multiplayer normal NIM
Theorem: Assume that for all integer 𝑘 and for all player 𝑌, her preference order is 𝑂𝑘 𝑌 > 𝑂𝑘+1 𝑌 > ⋯ > 𝑂𝑛−1 𝑌 > 𝑌 > 𝑂 𝑌 … > 𝑂𝑘−1 𝑌 , then 𝑜1, 𝑜2, … , 𝑜𝑙−1, 𝑜𝑙 is a 𝑘-position if and only if ቊ𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 = "00 … 00"(∃𝑜𝑗 > 1) 𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 = "00 … 0𝑘"(∀𝑜𝑗 ≤ 1)
SLIDE 46
Multiplayer normal NIM
Theorem: Assume that for all player 𝑌, her preference order is 𝑌 > 𝑂 𝑌 > ⋯ > 𝑂𝑛−1 𝑌 , then 𝑜1, 𝑜2, … , 𝑜𝑙−1, 𝑜𝑙 is a 0-position if and only if ቊ𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 = "00 … 00"(∃𝑜𝑗 > 1) 𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙 = "00 … 00"(∀𝑜𝑗 ≤ 1)
SLIDE 47 When does worst player take last stone?
normal play: 𝑜1 ⊕ 𝑜2 ⊕ ⋯ ⊕ 𝑜𝑙＝0 misère play: ቊ𝑜1 ⊕ 𝑜2 ⊕ ⋯ ⊕ 𝑜𝑙＝0(∃𝑜𝑗 > 1) 𝑜1 ⊕ 𝑜2 ⊕ ⋯ ⊕ 𝑜𝑙＝1(∀𝑜𝑗 ≤ 1) 𝑛-player normal play: 𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙＝0 𝑛-player misère play: ቊ𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙＝"00 … 00"(∃𝑜𝑗 > 1) 𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙＝"00 … 0j"(∀𝑜𝑗 ≤ 1) 𝑛 = 2 𝑘 = 0 𝑛 = 2 𝑘 = 1 𝑛-player normal play: 𝑜1 ⊕𝑛 𝑜2 ⊕𝑛 … ⊕𝑛 𝑜𝑙＝"00 … 00"
SLIDE 48
Another theorem
Theorem: Assume that for all integer 𝑘 and for each player 𝑌, her preference order is 𝑂𝑘 𝑌 > 𝑂𝑘−1 𝑌 > ⋯ > 𝑂 𝑌 > 𝑌 > 𝑂𝑛−1 𝑌 … > 𝑂𝑘+1 𝑌 , then for all integer 𝑜1, 𝑜2, … , 𝑜𝑙−1, there is an exactly one integer 𝑜𝑙 such that NIM position 𝑜1, 𝑜2, … , 𝑜𝑙−1, 𝑜𝑙 is a 𝑘-position.
SLIDE 49 Future problems
- 1. Another preferences
- 2. Another games
1. Moore’s game, LIM, WYTHOFF, Graph Games,…
