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### Deciding the Winner of an Arbitrary Finite Poset Game Is PSPACE-Complete [chapter]

Daniel Grier
2013 Lecture Notes in Computer Science
A poset game is a two-player game played over a partially ordered set (poset) in which the players alternate choosing an element of the poset, removing it and all elements greater than it.  ...  The first player unable to select an element of the poset loses. Polynomial time algorithms exist for certain restricted classes of poset games, such as the game of Nim.  ...  There are also polynomial time algorithms for some two-level poset games.  ...

### Combinatorial Game Complexity: An Introduction with Poset Games [article]

Stephen A. Fenner, John Rogers
2015 arXiv   pre-print
In this introduction we develop the fundamentals of combinatorial game theory and focus for the most part on poset games, of which Nim is perhaps the best-known example.  ...  Poset games have been the object of mathematical study for over a century, but little has been written on the computational complexity of determining important properties of these games.  ...  Figure 2 shows the numeric value of two simple, two-level black-white poset games. Figure 2 : The numerical values of two simple black-white poset games.  ...

### Pebblings

Henrik Eriksson
1995 Electronic Journal of Combinatorics
The analysis of chessboard pebbling by Fan Chung, Ron Graham, John Morrison and Andrew Odlyzko is strengthened and generalized, first to higher dimension and then to arbitrary posets.  ...  Thus, level zero contains the origin only, level one has n points, level two n(n + 1)=2 points etc.  ...  Conversely, if a shot count poset has some level with only two nodes on it, then the quadrangle containg these nodes may be expanded into a dihedral interval.  ...

### Chain-making games in grid-like posets

Daniel W. Cranston, William B. Kinnersley, Kevin G. Milans, Gregory J. Puleo, Douglas B. West
2012 Journal of Combinatorics
We study the Maker-Breaker game on the hypergraph of chains of fixed size in a poset.  ...  When d = 2, the maximum that Walker can guarantee is only 2/3 of the levels, and 2/3 is asymptotically achievable in the product of two equal chains.  ...  For the chain game and ordered chain game on these posets, we prove two main results. Theorem 1.1.  ...

### Chain-making games in grid-like posets [article]

Daniel W. Cranston, William B. Kinnersley, Kevin G. Milans, Gregory J. Puleo, Douglas B. West
2011 arXiv   pre-print
We study the Maker-Breaker game on the hypergraph of chains of fixed size in a poset.  ...  When d=2, the maximum that Walker can guarantee is only 2/3 of the levels, and 2/3 is asymptotically achievable in the product of two equal chains.  ...  For the chain game and ordered chain game on these posets, we prove two main results. Theorem 1.1.  ...

### On-line Chain Partitioning Approach to Scheduling [article]

Bartłomiej Bosek
2018 arXiv   pre-print
An on-line chain partitioning algorithm receives the points of the poset from some externally determined list.  ...  Kierstead presented an algorithm using (5^w-1)/4 chains to cover each poset of width w. Felsner proved that width 2 posets can be partitioned on-line into 5 chains.  ...  Antichains L 1 , . . . , L w may be seen as levels of the poset P. Two consecutive levels L i−1 , L i determine our chains α i and β i as described in J3.  ...

### The Ordered Join of Impartial Games [article]

Mišo Gavrilović, Alexander Thumm
2021 arXiv   pre-print
Inspired by the theory of poset games, we introduce a new compound of impartial combinatorial games and provide a complete analysis in the spirit of the Sprague-Grundy theory.  ...  For example, the ordered join S i 1 is the same game as the poset game on S. Therefore, each poset game has a natural description as a nontrivial ordered join. 3.2. Different Shapes.  ...  For the two other cases there exist unique maximal decompositions. Recursively decomposing the posets S i , one obtains a hierarchical representation of S.  ...

### Infinitely Split Nash Equilibrium Problems in Repeated Games [article]

Jinlu Li
2017 arXiv   pre-print
In this paper, we introduce the concept of infinitely split Nash equilibrium in repeated games in which the profile sets are chain-complete posets.  ...  Then by using a fixed point theorem on posets in , we prove an existence theorem. As an application, we study the repeated extended Bertrant duopoly model of price competition.  ...  on posets to study the solvability of split Nash equilibrium problems for dual games.  ...

### The Combinatorics and Absoluteness of Definable Sets of Real Numbers

Zach Norwood
2022 Bulletin of Symbolic Logic

### Combinatorial Aspects of the Card Game War [article]

Tanya Khovanova, Atharva Pathak
2022 arXiv   pre-print
This paper studies a single-suit version of the card game War on a finite deck of cards.  ...  We introduce several combinatorial objects related to the game: game graphs, win-loss sequences, win-loss binary trees, and game posets. We show how these objects relate to each other.  ...  Example 5 .Figure 3 : 53 Figure 3: Game poset for WL-putback on W/LW/LL Figure 4 : 4 Figure 4: Game poset for WL-putback on W/W W/LW W W/LLLLLL Figure 5 : 7 . 57 Figure 5: Game poset for WL-putback  ...

### Forcing indestructibility of set-theoretic axioms [article]

Bernhard Koenig
2006 arXiv   pre-print
Define δ = sup(ht"S), we have two cases: Case 1: if cf(δ) = κ then T δ is a non-stationary level of the tree T .  ...  These new developments were heading into two different directions, on the one hand there was the development of semiproper forcing in  which lead to the Semiproper Forcing Axiom and later to Martin's  ...

### Directive trees and games on posets

Tetsuya Ishiu, Yasuo Yoshinobu
2001 Proceedings of the American Mathematical Society
We show that for any infinite cardinal κ, every (κ+1)-strategically closed poset is κ + -strategically closed if and only if κ holds. This extends previous results of Velleman, et.al.  ...  Introduction In this paper we study a property of posets called 'strategic closure', characterized in terms of games on posets, which have been studied by Jech [J1] , [J2] , Foreman [F] , Veličkovič  ...  Note that this theorem says almost nothing in the case κ = ω, since Player II trivially wins in games of length ω 1 on any σ-closed posets.  ...
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