Maker-Breaker Games: Building a Big Chain in a Poset

Maker-Breaker Games:Building a Big Chain in a Poset

Daniel W. CranstonVirginia Commonwealth University

[email protected]

Joint with Bill Kinnersley, Kevin Milans, Greg Puleo, Douglas West

VCU Discrete Math Seminar05 March 2010

Maker-Breaker Games (in General)

Maker-Breaker Game:Two players, Maker and Breaker, alternate turns. On each turn,the player chooses a not-yet-picked element from a base set.

Maker tries to collect all the elements in at least one winningsubset (i.e. trying to make that subset). Breaker tries to stop him.Game ends when Maker succeeds or all the elements are chosen.

Examples

I Maker aims for Hamiltonian Circuit from E (Kn)

I Maker aims for E (Kq) from E (Kn)

I Maker aims for a k-term AP from {1, 2, . . . , n}I Maker and Breaker play Hex

We want to find the thresholdwhere the game switches froma Breaker win to a Maker win.


Maker-Breaker Game:Two players, Maker and Breaker, alternate turns. On each turn,the player chooses a not-yet-picked element from a base set.Maker tries to collect all the elements in at least one winningsubset (i.e. trying to make that subset). Breaker tries to stop him.

Game ends when Maker succeeds or all the elements are chosen.

Examples






Maker-Breaker Game:Two players, Maker and Breaker, alternate turns. On each turn,the player chooses a not-yet-picked element from a base set.Maker tries to collect all the elements in at least one winningsubset (i.e. trying to make that subset). Breaker tries to stop him.Game ends when Maker succeeds or all the elements are chosen.

Examples







Examples







Examples







Examples







Examples



I Maker aims for a k-term AP from {1, 2, . . . , n}

I Maker and Breaker play Hex




Examples







Examples





Subset LatticesTheoremIn the subset lattice Ln, Maker can get a chain of size nthat misses only the top element.

{}

{a} {b} {c} {d}

{a, b, c, d}


{}

{a} {b} {c} {d}

{a, b, c, d}


{}

{a} {b} {c} {d}

{a, b, c, d}


{}

{a} {b} {c} {d}

{a, b, c, d}


{}

{a} {b} {c} {d}

{a, b, c, d}


{}

{a} {b} {c} {d}

{a, b, c, d}


{}

{a} {b} {c} {d}

{a, b, c, d}


{}

{a} {b} {c} {d}

{a, b, c, d}


{}

{a} {b} {c} {d}

{a, b, c, d}


{}

{a} {b} {c} {d}

{a, b, c, d}


{}

{a} {b} {c} {d}

{a, b, c, d}

Corollary

In the poset Ln, Maker can get a maximum size chain.

Product of Two Chains

TheoremIf P is the product of two chains, each of size s, thenMaker can build a chain in P of size at least

⌈32s⌉− 1.

(1,1)•

(s,s)•

Maker’s Strategy

If Breaker plays a green, then Maker plays its pair. Otherwise,Maker plays a blue, if he can. Thus:

⌈12s⌉

+ (s − 1) =⌈

32s⌉− 1.



⌈32s⌉− 1.

(1,1)•

(s,s)•

Maker’s Strategy


⌈12s⌉

+ (s − 1) =⌈

32s⌉− 1.



⌈32s⌉− 1.

(1,1)•

(s,s)•

Maker’s Strategy


⌈12s⌉

+ (s − 1) =⌈

32s⌉− 1.



⌈32s⌉− 1.

(1,1)•

(s,s)•

Maker’s Strategy

If Breaker plays a green, then Maker plays its pair. Otherwise,Maker plays a blue, if he can.

Thus:⌈

12s⌉

+ (s − 1) =⌈

32s⌉− 1.



⌈32s⌉− 1.

(1,1)•

(s,s)•

Maker’s Strategy

If Breaker plays a green, then Maker plays its pair. Otherwise,Maker plays a blue, if he can.

Thus:⌈

12s⌉

+ (s − 1) =⌈

32s⌉− 1.



⌈32s⌉− 1.

(1,1)•

(s,s)•

Maker’s Strategy


⌈12s⌉

+ (s − 1) =⌈

32s⌉− 1.


TheoremIf P is the product of two chains, of sizes s1,s2 with s1 ≥ s2, thenMaker can build a chain in P of size at least

⌈12s1⌉

+ s2 − 1.

(1,1)•

(s1, s2)

Maker’s Strategy


⌈12s1⌉

+ s2 − 1.

Product of Two Chains (cont’d)

TheoremIf P is the product of two chains, of sizes s1,s2 with s1 ≥ s2, thenBreaker can hold Maker to a chain of size at most

⌈12s1⌉

+ s2 − 1.

(1,1)•

•(s1, s2)

XXX X X X X

X X XXX X

XX X X X X

Breaker’s Strategy

Pair elements “length-wise”. Whatever element Maker plays,Breaker plays its pair. So: (s1 + s2 − 1)−

⌊12s1⌋

=⌈

12s1⌉

+ s2 − 1.



⌈12s1⌉

+ s2 − 1.

(1,1)•

•(s1, s2)

XXX X X X X

X X XXX X

XX X X X X


Pair elements “length-wise”. Whatever element Maker plays,Breaker plays its pair.

So: (s1 + s2 − 1)−⌊

12s1⌋

=⌈

12s1⌉

+ s2 − 1.



⌈12s1⌉

+ s2 − 1.

(1,1)•

•(s1, s2)

XXX X X X X

X X XXX X

XX X X X X


Pair elements “length-wise”. Whatever element Maker plays,Breaker plays its pair.

So: (s1 + s2 − 1)−⌊

12s1⌋

=⌈

12s1⌉

+ s2 − 1.



⌈12s1⌉

+ s2 − 1.

(1,1)•

•(s1, s2)

XXX X X X X

X X XXX X

XX X X X X


Pair elements “length-wise”. Whatever element Maker plays,Breaker plays its pair. So: (s1 + s2 − 1)−

⌊12s1⌋

=⌈

12s1⌉

+ s2 − 1.

Product of d ChainsTheoremIf P is the product of d chains, with sizes s1 ≥ · · · ≥ sd ,then a maximum chain in P has size S =

∑si − (d − 1).

Maker can build a chain in P of size at least S −⌊

12s1⌋.

(1,1,1) •

• (s1, s2, s3)

Maker’s Strategy

Like before, but Maker’s old green pairs now become green Lks.


∑si − (d − 1).


12s1⌋.

(1,1,1) •

• (s1, s2, s3)

Maker’s Strategy



∑si − (d − 1).


12s1⌋.

(1,1,1) •

• (s1, s2, s3)

Maker’s Strategy



∑si − (d − 1).


12s1⌋.

(1,1,1) •

• (s1, s2, s3)

Maker’s Strategy



∑si − (d − 1).


12s1⌋.

(1,1,1) •

• (s1, s2, s3)

Maker’s Strategy


Walker-Blocker on the Wedge

The wedge W dk is {(x1, . . . , xd)|xi ≥ 0 and

∑xi < k}.

Walker-Blocker: same as Maker-Breaker, but nowWalker must get the elements of his chain in increasing order.

TheoremIn W 2

k , Walker can get d2k/3e levels, and no more.

Walker gets 7 levels in W 210



∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2





∑xi < k}.


TheoremIn W 2



Proposition

A greedy strategy for Walkerloses at most bk/3c levels.



∑xi < k}.


TheoremIn W 2



Proposition


`W + `B = k



∑xi < k}.


TheoremIn W 2



Proposition


`W + `B = k

`W = tW and `B ≤ 12 tB and tW = tB



∑xi < k}.


TheoremIn W 2



Proposition


`W + `B = k


`B ≤ 12 tB = 1

2 tW = 12`W = 1

2(k − `B)



∑xi < k}.


TheoremIn W 2



Proposition


`W + `B = k


`B ≤ 12 tB = 1

2 tW = 12`W = 1

2(k − `B)

`B ≤ 13k

Angel-Devil game

Angel-Devil Game

Angel: Move from (x , y) to (x1, y1)if |x − x1| ≤ 2 and |y − y1| ≤ 2.Devil: Burn one point (x , y).

Question [Conway 1982]:Can the angel move forever?

Answer [Mathe, Kloster 2006]: Yes!

TheoremIn the wedge , Walker can get all levels.

Question: What about W 3k through W 13

k ?

Conjecture

In the wedge W 3k , Walker can get all the levels.

Angel-Devil game

Angel-Devil GameAngel: Move from (x , y) to (x1, y1)if |x − x1| ≤ 2 and |y − y1| ≤ 2.

Devil: Burn one point (x , y).





k ?

Conjecture


Angel-Devil game

Angel-Devil GameAngel: Move from (x , y) to (x1, y1)if |x − x1| ≤ 2 and |y − y1| ≤ 2.Devil: Burn one point (x , y).





k ?

Conjecture


Angel-Devil game






k ?

Conjecture


Angel-Devil game






k ?

Conjecture


Angel-Devil game






k ?

Conjecture


Angel-Devil game






k ?

Conjecture


Angel-Devil game






k ?

Conjecture


Angel-Devil game






k ?

Conjecture


Angel-Devil game






k ?

Conjecture


Angel-Devil game






k ?

Conjecture


Angel-Devil game




TheoremIn the wedge W 24

k , Walker can get all levels.


k ?

Conjecture


Angel-Devil game







k ?

Conjecture


Angel-Devil game







k ?

Conjecture


Angel-Devil game







k ?

Conjecture


Angel-Devil game







k ?

Conjecture


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Maker-Breaker Games: Building a Big Chain in a Poset

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