No Normal al an and misère pla lay of of mult ltip ipla laye yer ga games wi with th pr preference Games and Graphs Workshop October 23 rd - 25 th , 2017 University Lyon~1 Koki Suetsugu Graduate School of Human and Environmental Studies Kyoto Univ.
Table of contents 1. Background Normal, misère and multiplayer NIM with preference 2. Result The Integration of misère NIM and multiplayer NIM 3. Future questions
Background ● Early studies ● Normal and misère NIM ● Multiplayer game with preference – Includes Li's theory
Nimber (3, 2, 4) ・ Calculate mod-2 sum of the number of 011 stones of each heap in binary notation 010 without carry 100 101 3 ⊕ 2 ⊕ 4 = 5
Normal NIM P-position of normal NIM: 𝑜 1 ⊕ 𝑜 2 ⊕. . .⊕ 𝑜 𝑙 = 0
Misère NIM P-position of misère NIM: ቊ 𝑜 1 ⊕ 𝑜 2 ⊕. . .⊕ 𝑜 𝑙 = 0(∃𝑜 𝑗 > 1) 𝑜 1 ⊕ 𝑜 2 ⊕. . .⊕ 𝑜 𝑙 = 1(∀𝑜 𝑗 ≤ 1)
Background ● Early studies ● Normal and misère NIM ● Multiplayer game with preference – Includes Li's theory
3-player NIM First player can't win Second player wins Third player wins
Preference Each player has a total “preference” ordering. If player 𝑌 has preference : > > order 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”. : > >
Definitions 𝑂(𝐵) : Next player of player 𝐵 𝑂 −1 (𝐵) : Previous player of player 𝐵 𝑂 2 𝐵 = 𝑂 𝑂 𝐵 , 𝑂 3 𝐵 = 𝑂 𝑂 2 𝐵 , … 𝑂 −2 𝐵 = 𝑂 −1 𝑂 −1 𝐵 , 𝑂 −3 𝐵 = 𝑂 −1 𝑂 −2 𝐵 , … Note that 𝑂 0 𝐵 = 𝑂 𝑜 𝐵 = 𝐵 .
Preference Each player has a total “preference” ordering. If player 𝑌 has preference : > > order 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”. : > >
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) C Play order: A B C
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) C Play order: A B C
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) B C Play order: C A A B A C B B
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) B C Play order: C A A B A C B B
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) B C Play order: A C A A B B B A C B B
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) B C Play order: A C A A B B B A C B B
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) B C Play order: B A C A A B B B A C B B
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) B C Play order: B A C A A B B B A C B B
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) C B C Play order: A B A C A A B B B A C B B
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) C B C Play order: A B A C A A B B B A C B B
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) C B C Play order: A B A C A A B B B A C B B
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) C B C Play order: A B A C A A B B B A C B B
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) C B C Play order: A B A C A A B B B A C B B
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) A C B C Play order: A B A C A A B B B A C B B
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) A C B C Play order: A B A C A A B B B A C B B
A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B A B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) A C B C Play order: A B A C A A B B B A C B B
A A : 𝐵 > 𝑂(𝐵) > 𝑂 2 (𝐵) : 𝐶 > 𝑂(𝐶) > 𝑂 2 (𝐶) B A B : 𝐷 > 𝑂(𝐷) > 𝑂 2 (𝐷) A C B C Play order: A B A C A A B B B A C B B
Definitions Let 𝐻 be a game position. Suppose that 𝑌 is the first player of 𝐻 . For all player 𝑌 , if player 𝑂 𝑗−1 (𝑌) moves last, then 𝐻 is called an 𝑗 -position.
Generalized NIM Sum :⊕ 𝑛 𝑜 1 ⊕ 𝑛 𝑜 2 ⊕ 𝑛 … ⊕ 𝑛 𝑜 𝑙 Example: 3 ⊕ 3 15 ⊕ 3 13 ⊕ 3 11 3 15 13 11 3 ⊕ 3 15 ⊕ 3 13 ⊕ 3 11
Generalized NIM Sum :⊕ 𝑛 𝑜 1 ⊕ 𝑛 𝑜 2 ⊕ 𝑛 … ⊕ 𝑛 𝑜 𝑙 Example: 3 ⊕ 3 15 ⊕ 3 13 ⊕ 3 11 3 0011 15 1111 13 1101 11 1011 3 ⊕ 3 15 ⊕ 3 13 ⊕ 3 11
Generalized NIM Sum :⊕ 𝑛 𝑜 1 ⊕ 𝑛 𝑜 2 ⊕ 𝑛 … ⊕ 𝑛 𝑜 𝑙 Example: 3 ⊕ 3 15 ⊕ 3 13 ⊕ 3 11 3 0011 15 1111 13 1101 11 1011 3 ⊕ 3 15 ⊕ 3 13 ⊕ 3 11
Generalized NIM Sum :⊕ 𝑛 𝑜 1 ⊕ 𝑛 𝑜 2 ⊕ 𝑛 … ⊕ 𝑛 𝑜 𝑙 Example: 3 ⊕ 3 15 ⊕ 3 13 ⊕ 3 11 3 0011 15 1111 13 1101 11 1011 3 ⊕ 3 15 ⊕ 3 13 ⊕ 3 11 "0201"
𝑛 -player normal NIM If for all player 𝑌 , her preference order is 𝑌 > 𝑂 𝑌 > ⋯ > 𝑂 𝑛−1 𝑌 , then NIM position is a 0 – position( 𝑛 – position) if and only if 𝑜 1 ⊕ 𝑛 𝑜 2 ⊕ 𝑛 … ⊕ 𝑛 𝑜 𝑙 = "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.
New result
When does worst player take last stone? misère play: normal play: ቊ𝑜 1 ⊕ 𝑜 2 ⊕ ⋯ ⊕ 𝑜 𝑙 ＝ 0(∃𝑜 𝑗 > 1) 𝑜 1 ⊕ 𝑜 2 ⊕ ⋯ ⊕ 𝑜 𝑙 ＝ 0 𝑜 1 ⊕ 𝑜 2 ⊕ ⋯ ⊕ 𝑜 𝑙 ＝ 1(∀𝑜 𝑗 ≤ 1) 𝑛 = 2 𝑛 -player normal play: 𝑜 1 ⊕ 𝑛 𝑜 2 ⊕ 𝑛 … ⊕ 𝑛 𝑜 𝑙 ＝ "00 … 00"
When does worst player take last stone? misère play: normal play: ቊ𝑜 1 ⊕ 𝑜 2 ⊕ ⋯ ⊕ 𝑜 𝑙 ＝ 0(∃𝑜 𝑗 > 1) 𝑜 1 ⊕ 𝑜 2 ⊕ ⋯ ⊕ 𝑜 𝑙 ＝ 0 𝑜 1 ⊕ 𝑜 2 ⊕ ⋯ ⊕ 𝑜 𝑙 ＝ 1(∀𝑜 𝑗 ≤ 1) 𝑛 = 2 𝑛 - player misère play: 𝑛 -player normal play: 𝑛 -player normal play: New result 𝑜 1 ⊕ 𝑛 𝑜 2 ⊕ 𝑛 … ⊕ 𝑛 𝑜 𝑙 ＝ "00 … 00" 𝑜 1 ⊕ 𝑛 𝑜 2 ⊕ 𝑛 … ⊕ 𝑛 𝑜 𝑙 ＝ 0
New result: 𝑛 -player misère play Theorem: Assume that for all integer 𝑘 and for all player 𝑌 , her preference order is 𝑂 𝑘 𝑌 > 𝑂 𝑘+1 𝑌 > ⋯ > 𝑂 𝑛−1 𝑌 > 𝑌 > 𝑂 𝑌 … > 𝑂 𝑘−1 𝑌 , then 𝑜 1 , 𝑜 2 , … , 𝑜 𝑙−1 , 𝑜 𝑙 is a 𝑘 -position if and only if ቊ 𝑜 1 ⊕ 𝑛 𝑜 2 ⊕ 𝑛 … ⊕ 𝑛 𝑜 𝑙 = "00 … 00"(∃𝑜 𝑗 > 1) 𝑜 1 ⊕ 𝑛 𝑜 2 ⊕ 𝑛 … ⊕ 𝑛 𝑜 𝑙 = "00 … 0𝑘"(∀𝑜 𝑗 ≤ 1)
… … : ……………> > > >… > > >… : …> > > >… > > >…………… : > > > … > > >…………………… : > > > >… >…………………… >
New result: 𝑛 -player misère play Theorem: Assume that for all integer 𝑘 and for all player 𝑌 , her preference order is 𝑂 𝑘 𝑌 > 𝑂 𝑘+1 𝑌 > ⋯ > 𝑂 𝑛−1 𝑌 > 𝑌 > 𝑂 𝑌 … > 𝑂 𝑘−1 𝑌 , then 𝑜 1 , 𝑜 2 , … , 𝑜 𝑙−1 , 𝑜 𝑙 is a 𝑘 -position if and only if ቊ 𝑜 1 ⊕ 𝑛 𝑜 2 ⊕ 𝑛 … ⊕ 𝑛 𝑜 𝑙 = "00 … 00"(∃𝑜 𝑗 > 1) 𝑜 1 ⊕ 𝑛 𝑜 2 ⊕ 𝑛 … ⊕ 𝑛 𝑜 𝑙 = "00 … 0𝑘"(∀𝑜 𝑗 ≤ 1)
This result includes two- player misère NIM by 𝑛 = 2 and 𝑘 = 1
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