Positional Games

Michael Krivelevich

Tel Aviv University

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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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Maker-Breaker Tic-Tac-Toe

- Maker’s win!!

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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

||

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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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Further research

Strong games

Different types of games

(misére versions of strong and weak games; biased versions)

fast wins (how long does it take a winner to win?)

games on different boards

(Ex.: Hamiltonicity games on sparse graphs; games on random graphs)

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