{{Short description|Type of sequential game}} A '''strong positional game''' (also called '''Maker-Maker game''') is a kind of positional game.<ref name=":0">{{Cite Positional Games 2014}}</ref>{{Rp|9–12}} Like most positional games, it is described by its set of ''positions'' (<math>X</math>) and its family of ''winning-sets'' (<math>\mathcal{F}</math>- a family of subsets of <math>X</math>). It is played by two players, called First and Second, who alternately take previously untaken positions.
In a strong positional game, the winner is the first player who holds all the elements of a winning-set. If all positions are taken and no player wins, then it is a draw. Classic Tic-tac-toe is an example of a strong positional game.
== First player advantage == In a strong positional game, Second cannot have a winning strategy. This can be proved by a strategy-stealing argument: if Second had a winning strategy, then First could have stolen it and win too, but this is impossible since there is only one winner.<ref name=":0" />{{Rp|9}} Therefore, for every strong-positional game there are only two options: either First has a winning strategy, or Second has a drawing strategy.
An interesting corollary is that, if a certain game does not have draw positions, then First always has a winning strategy.
== Comparison to Maker-Breaker game == Every strong positional game has a variant that is a Maker-Breaker game. In that variant, only the first player ("Maker") can win by holding a winning-set. The second player ("Breaker") can win only by preventing Maker from holding a winning-set.
For fixed <math>X</math> and <math>\mathcal{F}</math>, the strong-positional variant is strictly harder for the first player, since in it, he needs to both "attack" (try to get a winning-set) and "defend" (prevent the second player from getting one), while in the maker-breaker variant, the first player can focus only on "attack". Hence, ''every winning-strategy of First in a strong-positional game is also a winning-strategy of Maker in the corresponding maker-breaker game''. The opposite is not true. For example, in the maker-breaker variant of Tic-Tac-Toe, Maker has a winning strategy, but in its strong-positional (classic) variant, Second has a drawing strategy.<ref name="kruczek2010">{{cite journal|last=Kruczek|first=Klay|author2=Eric Sundberg|year=2010|title=Potential-based strategies for tic-tac-toe on the integer latticed with numerous directions|journal=The Electronic Journal of Combinatorics|volume=17|pages=R5}}</ref>
Similarly, the strong-positional variant is strictly easier for the second player: ''every winning strategy of Breaker in a maker-breaker game is also a drawing-strategy of Second in the corresponding strong-positional game'', but the opposite is not true.
=== The extra-set paradox === Suppose First has a winning strategy. Now, we add a new set to <math>\mathcal{F}</math>. Contrary to intuition, it is possible that this new set will now destroy the winning strategy and make the game a draw. Intuitively, the reason is that First might have to spend some moves to prevent Second from owning this extra set.'''<ref name="beck08">{{cite book|title=Combinatorial Games: Tic-Tac-Toe Theory|title-link=Combinatorial Games: Tic-Tac-Toe Theory|last1=Beck|first1=József|date=2008|publisher=Cambridge University Press|isbn=978-0-521-46100-9|location=Cambridge|author-link1=József Beck}}</ref><ref>{{Cite report |author=Christian Vogt |date=2025-07-14 |title=Towards Understanding the Extra Set Paradox |url=https://www.researchgate.net/publication/393648037_Towards_Understanding_the_Extra_Set_Paradox}}</ref>'''
The extra-set paradox does not appear in Maker-Breaker games.
== Examples ==
=== The clique game === The '''clique game''' is an example of a strong positional game. It is parametrized by two integers, n and N. In it:
* <math>X</math> contains all edges of the complete graph on {1,...,N}; * <math>\mathcal{F}</math> contains all cliques of size n.
According to Ramsey's theorem, there exists some number R(n,n) such that, for every N > R(n,n), in every two-coloring of the complete graph on {1,...,N}, one of the colors must contain a clique of size n.
Therefore, by the above corollary, when N > R(n,n), First always has a winning strategy.<ref name=":0" />{{Rp|10}}
=== Multi-dimensional tic-tac-toe === Consider the game of tic-tac-toe played in a ''d''-dimensional cube of length ''n''. By the Hales–Jewett theorem, when ''d'' is large enough (as a function of ''n''), every 2-coloring of the cube-cells contains a monochromatic geometric line.
Therefore, by the above corollary, First always has a winning strategy.
== Open questions == Besides these existential results, there are few constructive results related to strong-positional games. For example, while it is known that the first player has a winning strategy in a sufficiently large clique game, no specific winning strategy is currently known.<ref name=":0" />{{Rp|11–12}}
== References == {{Reflist}}
Category:Positional games