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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.  ... 
doi:10.1007/978-3-642-39206-1_42 fatcat:e6afio2x6nbdnksg6dk5cfvrwq

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.  ... 
arXiv:1505.07416v2 fatcat:yxcmoncmpvailp27irc7e3ydbi

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.  ... 
doi:10.37236/1201 fatcat:yk5mymfacrfu3m6bdpjctkhpz4

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.  ... 
doi:10.4310/joc.2012.v3.n4.a3 fatcat:horpht2prfbulergkf36blq4aq

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.  ... 
arXiv:1108.0710v1 fatcat:vzhkzci3fbenjn4kj5gixolkzu

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.  ... 
arXiv:1804.01567v1 fatcat:v5mkcepi2ff2tjrfwwvzljl6ba

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.  ... 
arXiv:2104.13131v2 fatcat:dc5hcepnxvgl7cbuuj4u5xwbka

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 [8], 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.  ... 
arXiv:1712.08509v1 fatcat:dvwksileurez3mezpk22n67izu

The Combinatorics and Absoluteness of Definable Sets of Real Numbers

Zach Norwood
2022 Bulletin of Symbolic Logic  
AbstractThis thesis divides naturally into two parts, each concerned with the extent to which the theory of $L(\mathbf {R})$ can be changed by forcing.The first part focuses primarily on applying generic-absoluteness  ...  Joint with Itay Neeman, we improve Schindler's theorem by showing that absoluteness for $\sigma $ -closed $\ast $ ccc posets—instead of the larger class of proper posets—implies the remarkability of $\  ...  L(R)-absoluteness for -linked proper posets.  ... 
doi:10.1017/bsl.2021.55 fatcat:wfigdgvtzbez7kh4tokt2ppe2u

The restricted core of games on distributive lattices: how to share benefits in a hierarchy

Michel Grabisch, Lijue Xie
2011 Mathematical Methods of Operations Research  
Finding a solution concept is one of the central problems in cooperative game theory, and the notion of core is the most popular solution concept since it is based on some rationality condition.  ...  In many real situations, not all possible coalitions can form, so that classical TU-games cannot be used.  ...  Acknowledgment We are grateful to the two referees for pointing out many references to the literature related to this work, and for helping at improving the presentation.  ... 
doi:10.1007/s00186-010-0341-2 fatcat:f2qbhi34cfbcxcaofxmsirlgsq

Posetal Games: Efficiency, Existence, and Refinement of Equilibria in Games with Prioritized Metrics [article]

Alessandro Zanardi and Gioele Zardini and Sirish Srinivasan and Saverio Bolognani and Andrea Censi and Florian Dörfler and Emilio Frazzoli
2021 arXiv   pre-print
By contextualizing standard game theoretical notions, we provide two sufficient conditions on the preference of the players to prove existence of pure Nash Equilibria in finite action sets.  ...  We present Posetal Games as a class of games in which each player expresses a preference over the outcomes via a partially ordered set of metrics.  ...  . • We provide two sufficient conditions for the existence of a pure Nash Equilibrium (NE) in posetal games with finite action sets.  ... 
arXiv:2111.07099v1 fatcat:rvunrdkr3jbo7drkjxuigwgshm

On-line Adaptive Chain Covering of Upgrowing Posets

Bartłomiej Bosek, Piotr Micek
2005 Discrete Mathematics & Theoretical Computer Science  
International audience We analyze on-line chain partitioning problem and its variants as a two-person game. One person (Spoiler) builds an on-line poset presenting one point at time.  ...  The other one (Algorithm) assigns new point to a chain. Kierstead gave a strategy for Algorithm showing that width w posets can be on-line chain partitioned into $\frac{{5}^{w-1}}{4}$ chains.  ...  This means that their color remains unchanged through the rest of the game. On the other hand points in the dynamic part are always colored with two or more colors.  ... 
doi:10.46298/dmtcs.3473 fatcat:ncdluskzj5e6hguzesmkbx2ysq

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  ... 
arXiv:2202.00473v1 fatcat:ylza5a46kvdenjb2vkhyqnx27q

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 [15] which lead to the Semiproper Forcing Axiom and later to Martin's  ... 
arXiv:math/0605129v1 fatcat:rvnpjelpkzddvci52o7mga3cfi

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.  ... 
doi:10.1090/s0002-9939-01-06235-9 fatcat:hihcxzmusfax3iwm5bzteejwoq
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