Kernekoncepter
Poset positional games are a generalization of standard positional games that incorporate additional restrictions on the order in which board elements can be claimed. This framework enables the study of games like Connect-4 within the positional game setting.
Resumé
The content introduces poset positional games, which are a generalization of standard positional games. In a poset positional game, the board elements are structured by a partial order (a poset), and players can only claim an element if all its predecessors in the poset have already been claimed.
The key highlights and insights are:
Poset positional games extend the standard positional game framework by incorporating a poset structure on the board elements, which restricts the available moves.
The authors analyze the complexity of determining the game outcome in poset positional games, focusing on the Maker-Breaker convention.
The complexity of the problem depends on parameters of the poset, such as its height and width, as well as the structure of the winning sets.
For posets of height 2 with winning sets of size 1, the problem can be solved in polynomial time. However, for height 3 and a single winning set of size 1, the problem becomes NP-hard.
For posets of bounded width, the problem is PSPACE-complete even when the winning sets are of size 3. But it becomes polynomial-time solvable when both the width of the poset and the number of winning sets are bounded.
The authors also consider the case where the poset is a union of disjoint chains, which generalizes the game of Connect-4.