1. Positional Games Michael Krivelevich Tel Aviv University

2. 1 Disclaimer introductory talk to a large, rapidly developing subject will cover some basic notions/concepts for more – recent books, surveys, research papers

3. 2 Learning by example Tic-Tac-Toe (x’s and 0’s) Board: 3×3 Player completing a winning line wins, Otherwise a draw Winning lines: Every child knows it is a draw

4. 3 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))

5. 4 Learning by example (cont.) Games played on the edge set of the complete graph K n Board: edges of K n 2 players, take turns in claiming unoccupied edges -large family of games Ex.: Hamiltonicity game 1 st player wins if creates a Hamilton cycle (cycle thru all graph vertices) 2 nd player wins otherwise (Observe non-symmetric roles of players)

6. 5 Learning by example (cont.) Point configuration games (J. Beck) P=(x 1,…,x n ) R 2 – 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

7. 6 Learning by example (cont.) n d – 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: n d -game Board=[n] d ={1,…,n} d Winning lines = In each coordinate: = (1,…,n) or = (n,…,1) or = (c,…,c) combinatorial line [ altogether]

8. 7 A general setting V = board of the game (usually a finite set) E 2 V – 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 K n E={E 0 E(K n ): E 0 contains a Hamilton cycle}

9. 8 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 draw 2. weak games -1 st player wins if eventually completes a winning set 2 nd player wins otherwise (i.e. blocks every winning set) Further game types (misére, etc.) are considered

10. 9 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

11. 10 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 1 st player’s win -Winning strategy is extremely complicated (“size of a phone book” – O. Patashnik) Enumeration is useless  use combinatorial tools!

12. 11 Strong games -Most natural type of positional games V = board E 2 V – family of winning sets Two players, 1 st and 2 nd, claim alternately unoccupied elements of V Player completing a winning set first wins, otherwise (=all winning sets are split between 1 st and 2 nd ) – a draw Examples: Tic-Tac-Toe, n d, etc. -very very hard to analyze -scarce combinatorial tools

13. 12 Strategy stealing Th.: In every strong game, 1 st player can guarantee at least a draw. Proof: strategy stealing principle Suppose not  2 nd player has a winning strategy S 1 st player: - moves arbitrarily first; - pretends to be 2 nd player, uses strategy S to choose his moves - if S calls to claim already claimed v  V, moves arbitrarily. ■ - very powerful/general -very inexplicit – no clue how to play explicitly for at least a draw Ex.: n d is at least a draw for 1 st player

14. 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  1 st player wins a strong game on H Proof: ≥ draw for 1 st by strategy stealing draw is impossible  1 st player’s win. ■ How to prove draw is impossible? Use Ramsey-type tools (like: every 2-coloring of V contains a monochromatic winning set e  E) 13

15. Hales-Jewett Theorem -Generalizes Van Der Waerden arithmetic progression theorem: Every 2-coloring of [n] has a k-long arithmetic progression, n≥n 0 (k) Th.: (Hales-Jewett, 1963) (“Regularity and positional games”) n d -game d ≥ d 0 (n)  every 2-coloring on [n] d contains a monochromatic combinatorial line Conclusion: d ≥ d 0 (n)  no draw in n d -game  1 st player wins (but have no idea how…) 14

16. Strong games are really hard… Ex.: Board = edge set of K ∞ Goal = to complete a copy of K 5 1 st player’s win or a draw? 15

17. Nevertheless… Some recent advances: Ferber, Hefetz’11: Hamiltonicity game - win of 1 st player for all large enough n; Ferber, Hefetz’12+: k-connectivity game (=first player completing a spanning k-connected graph wins) - win of 1 st player for all large enough n. Key: fast winning strategies in weak games. 16

18. Weak games (as opposed to strong games) Motivation: H=(V,E) – game hypergraph Have seen: in a strong game on H, 2 nd 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 2 V – winning sets Maker wins if claims an entire winning set in the end, Breaker wins otherwise no draw 17

19. Maker-Breaker Tic-Tac-Toe - Maker’s win!! 18

20. 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={x 1,y 1,…,x n,y n }, |V|=2n Winning sets: e V, 2 n sets, Maker wins (takes each time a sibling of Breaker’s move) 19 x1x1 x2x2 xnxn y1y1 y2y2 ynyn

21. Erdős-Selfridge criterion (cont.) Ex.: 5 2 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: e  E, 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! 20

22. 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 K n -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 21

23. Biased Maker-Breaker games (cont.) Setting: V=board E 2 V –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 22

24. Bias monotonicity in Maker-Breaker games Prop.: H=(V,E) – game hypergraph Maker wins 1:b game Maker wins 1:(b-1)-game Proof: S b := winning strategy for M in 1:b When playing 1:(b-1) : use S b, each time assign a fictitious b-th element to Breaker. ■ 23 321b* bias MMMMBBB Critical point: game changes hands M

25. 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 24

26. How to locate the critical bias Example: Hamiltonicity game on E(K n ) Know: b=1 – Maker wins (CE’78) b=n-1 – Breaker wins (can isolate a vertex) Conclude: 1≤b*≤n-1 Value of b*=? 25 n-1 vertices v

27. Erdős paradigm: Clever=Dumb (not quite this, but…) Games played on the edges of K n 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]= ] 26

28. 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) 27

29. It works! (in quite a few cases…) Examples: 1.Connectivity game Played on E(K n ); 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) 28

30. It works! (sometimes…) Example: Triangle game M-B game; 1:b, played on E(K n ) Maker’s aim: to construct a triangle K 3 = In G(n,m): K 3 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) 29

31. 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) 30