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Disclaimer
• introductory talk to a large, rapidly developing subject
• will cover some basic notions/concepts
• for more – recent books, surveys, research papers
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Learning by example
Tic-Tac-Toe (x’s and 0’s)Board: 3×3Player completing a winning line wins,Otherwise a drawWinning lines:
Every child knows it is a draw
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Learning by example (cont.)
The Game of Hex (J. Nash)
Board: rhombus of hexagons, sides of size n
Each player gets 2 opposite sides of the board
Player connecting two of his sides by a path of his hexagons wins
Nash: first player wins
(only one player wins – equivalent to Brouwer’s fixed point theorem (Gale))
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Learning by example (cont.)
Games played on the edge set of the complete graph Kn
Board: edges of Kn
2 players, take turns in claiming unoccupied edges- large family of games
Ex.: Hamiltonicity game1st player wins if creates a Hamilton cycle(cycle thru all graph vertices)2nd player wins otherwise(Observe non-symmetric roles of players)
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Learning by example (cont.)
Point configuration games (J. Beck)
P=(x1,…,xn) R2 – point configuration in the plane
(defined by pairwise distances)
Game: two players, Red and Blue, mark
alternately points of the plane
Red’s goal: to create a Red copy of P
Blue’s goal: to prevent Red from doing so
Th. (Beck): for every finite configuration P, Red can create a congruent copy of P
2
n
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Learning by example (cont.)nd – far reaching generalization of Tic-Tac-Toe
Winning lines in TTT are of the form:
In each coordinate: is either (1,2,3) or (3,2,1) or a constant
Ex:
Generalization: nd-game
Board=[n]d={1,…,n}d
Winning lines =
In each coordinate: = (1,…,n)
or = (n,…,1)
or = (c,…,c)
),,( 321 aaa),( 1
211
1 aaa 2 2 2
1 2( , )a a a3 3 3
1 2( , )a a a
),,( 321iii aaa
)3,1(1 a2 (2,2)a 3 (3,1)a
),...,( 1 naa),...,( 1 n
ii aacombinatorial line
[ altogether]2
)2( dd nn
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A general setting
V = board of the game (usually a finite set)E 2V – a family of winning sets(H=(V,E) – the hypergraph of the game)
Ex.: 1. Tic-Tac-Toe V=[3]2
E={ , , , }, |E|=8
2. Hamiltonicity game
V=edge set of Kn
E={E0 E(Kn): E0 contains a Hamilton cycle}
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Who is the winner?
Game definition is completed by defining who is the winner for every final position/game course
Two main types: 1. strong games- Player completing a winning set first wins, otherwise a
draw2. weak games- 1st player wins if eventually completes a winning set
2nd player wins otherwise (i.e. blocks every winning set)
Further game types (misére, etc.) are considered
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Positional games vs other games
Here: two player perfect information zero-sum game, alternate moves
Compare to:
1. classical game theory (von Neumann,…)- critical role of probabilistic considerations, mixed
strategies etc.
2. Nim-type games
- Game sums, algebraic considerations
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So what’s complicated about it?
- Perfect information games- For each such game, can in principle construct its game
tree, then analyze it completely using a computer…
- not that simple!Ex.: 4×4×4 game (generalization of Tic-Tac-Toe)- Known to be 1st player’s win- Winning strategy is extremely complicated
(“size of a phone book” – O. Patashnik)
Enumeration is useless use combinatorial tools!
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Strong games
- Most natural type of positional games
V = board
E 2V – family of winning sets
Two players, 1st and 2nd, claim alternately unoccupied elements of V
Player completing a winning set first wins, otherwise (=all winning sets are split between 1st and 2nd) – a draw
Examples: Tic-Tac-Toe, nd, etc.- very very hard to analyze - scarce combinatorial tools
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Strategy stealing
Th.: In every strong game, 1st player can guarantee at least a draw.
Proof: strategy stealing principle
Suppose not 2nd player has a winning strategy S
1st player: - moves arbitrarily first;- pretends to be 2nd player, uses strategy S to
choose his moves- if S calls to claim already claimed vV, moves arbitrarily. ■
- very powerful/general- very inexplicit – no clue how to play explicitly for at least a drawEx.: nd is at least a draw for 1st player
Ramsey theory comes into play
Th.: H=(V,E) – game hypergraph
Strong game played on H
H is such that there is NO drawing final position 1st player wins a strong game on H
Proof: ≥ draw for 1st by strategy stealing
draw is impossible
1st player’s win. ■
How to prove draw is impossible?
Use Ramsey-type tools
(like: every 2-coloring of V contains a monochromatic winning set eE)
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Hales-Jewett Theorem
- Generalizes Van Der Waerden arithmetic progression theorem:
Every 2-coloring of [n] has a k-long arithmetic progression, n≥n0(k)
Th.: (Hales-Jewett, 1963)
(“Regularity and positional games”)
nd-game
d ≥ d0(n) every 2-coloring on [n]d contains
a monochromatic combinatorial line
Conclusion: d ≥ d0(n)
no draw in nd-game 1st player wins
(but have no idea how…)
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Strong games are really hard…
Ex.: Board = edge set of K∞
Goal = to complete a copy of K5
1st player’s win or a draw?
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Nevertheless…
Some recent advances:• Ferber, Hefetz’11: Hamiltonicity game
- win of 1st player for all large enough n;• Ferber, Hefetz’12+: k-connectivity game
(=first player completing a spanning k-connected graph wins)
- win of 1st player for all large enough n.
Key: fast winning strategies in weak games.17
Weak games
(as opposed to strong games)
Motivation: H=(V,E) – game hypergraph
Have seen: in a strong game on H, 2nd player never wins He may as well play for a draw- will try to block every winning set
Maker-Breaker games• Two players, called Maker and Breaker, move alternately• V=board, E 2V – winning sets• Maker wins if claims an entire winning set in the end,
Breaker wins otherwise
no draw
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Erdős-Selfridge criterion
Th.: (ES’73)
H=(V,E) – game hypergraph
M-B game played on H
Assume B starts (does not change much)
If: (< ½ if Maker starts)
then Breaker has a winning strategy.
Tight: V={x1,y1,…,xn,yn}, |V|=2n
Winning sets: e V,
2n sets, Maker wins
(takes each time a sibling of Breaker’s move)
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| |2 1e
e E
| { , } | 1i ie x y
x1
x2
xn
y1
y2
yn
Erdős-Selfridge criterion (cont.)
Ex.: 52 is Breaker’s win
(a draw in the strong game)
Proof: lines, of size 5 each
12·2-5<1/2 Breaker wins
Proof idea of ES: If M,B act at random:
eE, Prob[e is Maker’s]=(1/2)|e|
= expected number of Maker’s sets
ES criterion: expectation<1
random coloring is good for B with positive probability convert a random argument into a deterministic one
(derandomization) – probabilistic considerations!21
( 2)12
2
d dn n
| |2 e
e E
Biased Maker-Breaker games
Motivation: in quite a few M-B games, Maker wins rather easily
Ex.: Hamiltonicity game
- Played on the edge set of Kn
- Maker-Breaker- Maker wins if completes a Hamilton cycle in the end
Chvátal-Erdős’78: Maker wins
…, Hefetz-Stich’09: Maker wins in n+1 moves (optimal)
Natural remedy: give Breaker a break!- Give him more power to even out the odds
biased Maker-Breaker games
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Biased Maker-Breaker games (cont.)
Setting: V=board
E 2V –winning sets
Two players: Maker, Breaker
p,q≥1 – integers (bias parameters)
Maker: claims p elements each turn
Breaker: claims q elements each turn
Very important case: p=1, q=b
1:b biased Maker-Breaker games
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Bias monotonicity in Maker-Breaker games
Prop.: H=(V,E) – game hypergraph
Maker wins 1:b game
Maker wins 1:(b-1)-game
Proof: Sb := winning strategy for M in 1:b
When playing 1:(b-1) : use Sb, each time assign a fictitious
b-th element to Breaker. ■
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321 b* bias
M M M M B B B
Critical point: game changes hands
M
Critical bias
Def.: H=(V,E) – game hypergraph
Maker vs Breaker
b*=b*(H) = critical bias of H
= max{b: Maker still wins a 1:b game on H}
Have seen: critical bias always exists; for b>b*, Breaker wins
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How to locate the critical bias
Example: Hamiltonicity game on E(Kn)
Know: b=1 – Maker wins (CE’78)
b=n-1 – Breaker wins
(can isolate a vertex)
Conclude: 1≤b*≤n-1
Value of b*=?
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n-1 vertices
v
Erdős paradigm: Clever=Dumb
(not quite this, but…)
Games played on the edges of Kn
M-B games, 1:b, P:=target graph property to reach
In the end: M has m= edges
Suppose: (clever) Maker, Breaker start playing randomly (=dumb) in the end: Maker’s graph = random graph G(n,m)
[ G(n,m) = prob. space Ω of all graphs on V={1,…,n}
with exactly m edges, |Ω|= , all graphs are equiprobable:
Pr[G]= ]
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1
2
1
|)(|
b
n
b
KE n
m
n
2
m
n
2
1
||
1
Erdős paradigm: Clever=Dumb (cont.)
Maker/Breaker → RandomMaker/RandomBreaker
Erdős: look for critical m*=m*(n,P) where property P starts appearing typically in G(n,m)
Guess: critical bias b* satisfies:
Very important/surprising; connection between positional games and probability
(recall also Erdős-Selfridge criterion)
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** 2
m
n
b
It works!
(in quite a few cases…)
Examples:
1. Connectivity game
Played on E(Kn); Maker wins iff claims a connected spanning subgraph in the end
In G(n,m): typically becomes connected at m*= nlnn
Critical bias: b*= (CE’78; Gebauer, Szabó’09)
– a perfect match!
2. Hamiltonicity game
In G(n,m): typically becomes Hamiltonian at m*= nlnn
Critical bias: b*= (CE’78; K.’11)29
2
1
n
n
ln
2
1
n
n
ln
It works! (sometimes…)
Example: Triangle game
M-B game; 1:b, played on E(Kn)
Maker’s aim: to construct a triangle K3=
In G(n,m): K3 starts appearing at m=Θ(n)
In games: critical bias= Θ(√n) (CE’78)- do not quite match- but there is a probabilistic explanation for it
(Bednarska, Łuczak’00)
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