[PPT] - Maker-Breaker Games: Building a Big Chain in a Poset Daniel W. PowerPoint Presentation

SLIDE 1

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

Daniel W. Cranston

Virginia Commonwealth University dcranston@vcu.edu

Joint with Bill Kinnersley, Kevin Milans, Greg Puleo, Douglas West VCU Discrete Math Seminar 05 March 2010

SLIDE 2

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.

SLIDE 3

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 winning subset (i.e. trying to make that subset). Breaker tries to stop him.

SLIDE 4

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 winning subset (i.e. trying to make that subset). Breaker tries to stop him. Game ends when Maker succeeds or all the elements are chosen.

SLIDE 5

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 winning subset (i.e. trying to make that subset). Breaker tries to stop him. Game ends when Maker succeeds or all the elements are chosen. Examples

SLIDE 6

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 winning subset (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 aims for Hamiltonian Circuit from E(Kn)

SLIDE 7

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 winning subset (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 aims for Hamiltonian Circuit from E(Kn) ◮ Maker aims for E(Kq) from E(Kn)

SLIDE 8

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 winning subset (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 aims for Hamiltonian Circuit from E(Kn) ◮ Maker aims for E(Kq) from E(Kn) ◮ Maker aims for a k-term AP from {1, 2, . . . , n}

SLIDE 9

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 winning subset (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 aims for Hamiltonian Circuit from E(Kn) ◮ Maker aims for E(Kq) from E(Kn) ◮ Maker aims for a k-term AP from {1, 2, . . . , n} ◮ Maker and Breaker play Hex

SLIDE 10

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 winning subset (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 aims for Hamiltonian Circuit from E(Kn) ◮ Maker aims for E(Kq) from E(Kn) ◮ Maker aims for a k-term AP from {1, 2, . . . , n} ◮ Maker and Breaker play Hex

We want to find the threshold where the game switches from a Breaker win to a Maker win.

SLIDE 11

Subset Lattices

Theorem

In the subset lattice Ln, Maker can get a chain of size n that misses only the top element.

SLIDE 12

Subset Lattices

Theorem

In the subset lattice Ln, Maker can get a chain of size n that misses only the top element. {} {a} {b} {c} {d} {a, b, c, d}

SLIDE 13

Subset Lattices

Theorem

In the subset lattice Ln, Maker can get a chain of size n that misses only the top element. {} {a} {b} {c} {d} {a, b, c, d}

SLIDE 14

Subset Lattices

Theorem

In the subset lattice Ln, Maker can get a chain of size n that misses only the top element. {} {a} {b} {c} {d} {a, b, c, d}

SLIDE 15

Subset Lattices

Theorem

In the subset lattice Ln, Maker can get a chain of size n that misses only the top element. {} {a} {b} {c} {d} {a, b, c, d}

SLIDE 16

Subset Lattices

Theorem

In the subset lattice Ln, Maker can get a chain of size n that misses only the top element. {} {a} {b} {c} {d} {a, b, c, d}

SLIDE 17

Subset Lattices

Theorem

In the subset lattice Ln, Maker can get a chain of size n that misses only the top element. {} {a} {b} {c} {d} {a, b, c, d}

SLIDE 18

Subset Lattices

Theorem

In the subset lattice Ln, Maker can get a chain of size n that misses only the top element. {} {a} {b} {c} {d} {a, b, c, d}

SLIDE 19

Subset Lattices

Theorem

In the subset lattice Ln, Maker can get a chain of size n that misses only the top element. {} {a} {b} {c} {d} {a, b, c, d}

SLIDE 20

Subset Lattices

Theorem

In the subset lattice Ln, Maker can get a chain of size n that misses only the top element. {} {a} {b} {c} {d} {a, b, c, d}

SLIDE 21

Subset Lattices

Theorem

In the subset lattice Ln, Maker can get a chain of size n that misses only the top element. {} {a} {b} {c} {d} {a, b, c, d}

Corollary

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

SLIDE 22

Product of Two Chains

Theorem

If P is the product of two chains, each of size s, then Maker can build a chain in P of size at least 3

2s

− 1.

SLIDE 23

Product of Two Chains

Theorem

If P is the product of two chains, each of size s, then Maker can build a chain in P of size at least 3

2s

− 1.

(1,1)

(s,s)

SLIDE 24

Product of Two Chains

Theorem

If P is the product of two chains, each of size s, then Maker can build a chain in P of size at least 3

2s

− 1.

(1,1)

(s,s)

SLIDE 25

Product of Two Chains

Theorem

If P is the product of two chains, each of size s, then Maker can build a chain in P of size at least 3

2s

− 1.

(1,1)

(s,s)

SLIDE 26

Product of Two Chains

Theorem

If P is the product of two chains, each of size s, then Maker can build a chain in P of size at least 3

2s

− 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.

SLIDE 27

Product of Two Chains

Theorem

If P is the product of two chains, each of size s, then Maker can build a chain in P of size at least 3

2s

− 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: 1

2s

+ (s − 1) =

3

2s

− 1.

SLIDE 28

Product of Two Chains

Theorem

If P is the product of two chains, of sizes s1,s2 with s1 ≥ s2, then Maker can build a chain in P of size at least 1

2s1

+ s2 − 1.

(1,1)

(s1, s2)

Maker’s Strategy

If Breaker plays a green, then Maker plays its pair. Otherwise, Maker plays a blue, if he can. Thus: 1

2s1

+ s2 − 1.

SLIDE 29

Product of Two Chains (cont’d)

Theorem

If P is the product of two chains, of sizes s1,s2 with s1 ≥ s2, then Breaker can hold Maker to a chain of size at most 1

2s1

+ s2 − 1.

(1,1)

(s1, s2)

SLIDE 30

Product of Two Chains (cont’d)

Theorem

If P is the product of two chains, of sizes s1,s2 with s1 ≥ s2, then Breaker can hold Maker to a chain of size at most 1

2s1

+ s2 − 1.

(1,1)

(s1, s2)

Breaker’s Strategy

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

SLIDE 31

Product of Two Chains (cont’d)

Theorem

If P is the product of two chains, of sizes s1,s2 with s1 ≥ s2, then Breaker can hold Maker to a chain of size at most 1

2s1

+ s2 − 1.

(1,1)

(s1, s2)

X X X X X X X X X X X X X X X X X X X

Breaker’s Strategy

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

SLIDE 32

Product of Two Chains (cont’d)

Theorem

If P is the product of two chains, of sizes s1,s2 with s1 ≥ s2, then Breaker can hold Maker to a chain of size at most 1

2s1

+ s2 − 1.

(1,1)

(s1, s2)

X X X X X X X X X X X X X X X X X X X

Breaker’s Strategy

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

2s1

=

1

2s1

+ s2 − 1.

SLIDE 33

Product of d Chains

Theorem

If 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 − 1

2s1

.

SLIDE 34

Product of d Chains

Theorem

If 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 − 1

2s1

.

(1,1,1) •

(s1, s2, s3)

SLIDE 35

Product of d Chains

Theorem

If 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 − 1

2s1

.

(1,1,1) •

(s1, s2, s3)

SLIDE 36

Product of d Chains

Theorem

If 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 − 1

2s1

.

(1,1,1) •

(s1, s2, s3)

Maker’s Strategy

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

SLIDE 37

Product of d Chains

Theorem

If 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 − 1

2s1

.

(1,1,1) •

(s1, s2, s3)

Maker’s Strategy

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

SLIDE 38

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

SLIDE 39

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

SLIDE 40

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

SLIDE 41

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 42

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 43

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 44

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 45

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 46

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 47

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 48

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 49

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 50

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 51

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 52

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 53

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 54

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

SLIDE 55

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

Proposition

A greedy strategy for Walker loses at most ⌊k/3⌋ levels.

SLIDE 56

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

Proposition

A greedy strategy for Walker loses at most ⌊k/3⌋ levels. ℓW + ℓB = k

SLIDE 57

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

Proposition

A greedy strategy for Walker loses at most ⌊k/3⌋ levels. ℓW + ℓB = k ℓW = tW and ℓB ≤ 1

2tB and tW = tB

SLIDE 58

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

Proposition

A greedy strategy for Walker loses at most ⌊k/3⌋ levels. ℓW + ℓB = k ℓW = tW and ℓB ≤ 1

2tB and tW = tB

ℓB ≤ 1

2tB = 1 2tW = 1 2ℓW = 1 2(k − ℓB)

SLIDE 59

Walker-Blocker on the Wedge

The wedge W d

k is {(x1, . . . , xd)|xi ≥ 0 and xi < k}.

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

Theorem

In W 2

k , Walker can get ⌈2k/3⌉ levels, and no more.

Walker gets 7 levels in W 2

10

Proposition

A greedy strategy for Walker loses at most ⌊k/3⌋ levels. ℓW + ℓB = k ℓW = tW and ℓB ≤ 1

2tB and tW = tB

ℓB ≤ 1

2tB = 1 2tW = 1 2ℓW = 1 2(k − ℓB)

ℓB ≤ 1

3k

SLIDE 60

Angel-Devil game

Angel-Devil Game

SLIDE 61

Angel-Devil game

Angel-Devil Game

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

SLIDE 62

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).

SLIDE 63

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).

SLIDE 64

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).

SLIDE 65

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).

SLIDE 66

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).

SLIDE 67

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).

SLIDE 68

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).

SLIDE 69

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?

SLIDE 70

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 [M´ ath´ e, Kloster 2006]: Yes!

SLIDE 71

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 [M´ ath´ e, Kloster 2006]: Yes!

Theorem

In the wedge W 24

k , Walker can get all levels.

SLIDE 72

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 [M´ ath´ e, Kloster 2006]: Yes!

Theorem

In the wedge W 24

k , Walker can get all levels.

SLIDE 73

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 [M´ ath´ e, Kloster 2006]: Yes!

Theorem

In the wedge W 14

k , Walker can get all levels.

SLIDE 74

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 [M´ ath´ e, Kloster 2006]: Yes!

Theorem

In the wedge W 14

k , Walker can get all levels.

Question: What about W 3

k through W 13 k ?

SLIDE 75

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 [M´ ath´ e, Kloster 2006]: Yes!