A New Variation of Hat Guessing GamesTengyu Ma Xiaoming Sun Huacheng YuInstitute for

Interdisciplinary Information Sciences Tsinghua University

Institute for Advanced Study,

Tsinghua University

Institute for Interdisciplinary

Information SciencesTsinghua University

3 cooperative players each is assigned a hat of

color red or blue each can only see others’ hat guess own color or pass players win if: at least one

correct and no wrong guess goal : to maximize winning

probability

Hat guessing puzzle

Hat guessing puzzle strategy1: only a pre-

specified player guesses randomly winning prob. =

strategy2: if other two have same color, guess the opposite, otherwise pass. winning prob. =

is optimal

pass

pass

cooperative players: ◦coordinate a strategy initially

assigned a blue or red hat◦uniformly and independently

guess a color or pass winning condition:

◦at least correct guesses and no wrong guess

goal: to maximize winning prob.

General hat guessing game

case is well studied by [?], [?].. Observation 1: randomized strategy

does not help Observation 2: related to the minimum

-dominating set of

Previous Study

Definition: A -dominating set for a graph is a subset of , such that every vertex not in has at least neighbors in

win! losepass

pass pass

pass

reduce -DS to strategy design

win! losepass

pass pass

pass

winning point losing point

reduce -DS to strategy design(2)

win! losepass

pass pass

pass

winning point has at least losing points as neighbors

reduce -DS to strategy design(3)

all losing points ◦ is -dominating set of ◦winning prob. =

reduction can be done vice versa by counting argument:

◦ winning prob.

Simple Facts

Theorem: ◦There exists a -dominating set of size ,

as long as is an integer, for large enough (.

◦It follows that there exists a strategy of the hat guessing games with winning prob.

theorem is not true for small ◦example:

Main Theorem

Perfect -dominating set

{0,1 }𝑛∖𝐷𝐷

each has neighbors in

each has neighbors in

𝑉 1𝑉 2

each has neighbors in

each has neighbors in

-regular partition of

-DS of -RP of possible -RP of :

◦the parameters are of the following form

possible -DS corresponds to the case

easy case

hard case ,

Easy and hard cases

from the cases to -- nontrivial, [?] from

to

From easy to hard

solve the case from given -RP of :

Hard cases: idea and example(1)

𝑉 1

𝑉 2

000100

010 110

111011

001 101

now construct -partition for for each sys. of equations over , the collection of solutions of

◦ is an independent set

Hard cases: idea and example (2)

{0,1 }6=𝑠𝑜𝑙 (𝐸000)∪𝑠𝑜𝑙 (𝐸001)∪…∪𝑠𝑜𝑙(𝐸¿¿111)¿Hard cases: idea and example(3)

𝑉 1

𝑉 2

𝑠𝑜𝑙(𝐸011)

𝑠𝑜𝑙(𝐸001) 𝑠𝑜𝑙(𝐸101)

𝑠𝑜𝑙(𝐸100 )

𝑠𝑜𝑙(𝐸010)

𝑠𝑜𝑙(𝐸000)

𝑠𝑜𝑙(𝐸000)

𝑠𝑜𝑙(𝐸110)

𝑉 1

𝑉 2

𝑠𝑜𝑙(𝐸011)

𝑠𝑜𝑙(𝐸001) 𝑠𝑜𝑙(𝐸101)

𝑠𝑜𝑙(𝐸100 )

𝑠𝑜𝑙(𝐸010)

𝑠𝑜𝑙(𝐸000)

𝑠𝑜𝑙(𝐸000)

𝑠𝑜𝑙(𝐸110)

find a perfect matching in cut each black set by an additional eqn. for and use eqn.:

6 = 2 * the index of the different bit

𝑉 1

𝑉 2

𝑠𝑜𝑙(𝐸011)

𝑠𝑜𝑙(𝐸001) 𝑠𝑜𝑙(𝐸101)

𝑠𝑜𝑙(𝐸100 )

𝑠𝑜𝑙(𝐸010)

𝑠𝑜𝑙(𝐸000)

𝑠𝑜𝑙(𝐸000)

𝑠𝑜𝑙(𝐸110)

find a perfect matching in cut each black set by an additional eqn. for and use eqn.:

2 = 2 * the index of the different bit

𝑉 1

𝑉 2

𝑠𝑜𝑙(𝐸011)

𝑠𝑜𝑙(𝐸001) 𝑠𝑜𝑙(𝐸101)

𝑠𝑜𝑙(𝐸100 )

𝑠𝑜𝑙(𝐸010)

𝑠𝑜𝑙(𝐸000)

𝑠𝑜𝑙(𝐸000)

𝑠𝑜𝑙(𝐸110)

all the grey points , . ◦ is a -RP of

this idea is extendable to general cases

Main contribution:◦foy any odd , and , when , there exists a -

regular partition of ◦particularly, it follows that for large

enough , there exists -dominating set of size , as long as is integer.

Recap

Thank You!

Reference