scholarly journals Manifolds associated to simple games

2016 ◽  
Vol 25 (12) ◽  
pp. 1642003 ◽  
Author(s):  
Pavel Galashin ◽  
Gaiane Panina
Keyword(s):  
Simplicial Complex ◽  
Simple Game ◽  
Dimensional Manifold ◽  
Simple Games

We describe a way of producing an [Formula: see text]-dimensional manifold [Formula: see text] starting with an Alexander self-dual simplicial complex [Formula: see text] on [Formula: see text] vertices (or, in another terminology, by a simple game with constant sum with [Formula: see text] players). The construction presents [Formula: see text] explicitly, by describing its regular cellulation.

NEW APPORTIONMENT METHODS USING SIMPLE GAMES

2010 ◽  
Vol 12 (03) ◽  
pp. 211-222
Author(s):  
EVAN SHELLSHEAR
Keyword(s):  
Simple Game ◽  
Simple Games ◽  
New Methods ◽  
Apportionment Methods

This paper investigates the suitability of new apportionment methods based on the idea of preserving the coalition function of the simple game generated by the populations of the states of some country. The new methods fill a gap in the literature concerning apportionment methods based on winning coalitions. The main results in this paper show that the new apportionment methods do not satisfy desirable properties such as house monotonicity, quota, etc.

scholarly journals Bounds for Owen's Multilinear Extension

2007 ◽  
Vol 44 (4) ◽  
pp. 852-864 ◽  
Author(s):  
Josep Freixas
Keyword(s):  
Game Theory ◽  
Simple Game ◽  
Practical Interest ◽  
Simple Games ◽  
Theoretical Interest ◽  
Complex Component ◽  
Multilinear Extension ◽  
Voting System ◽  
Minimal Winning Coalitions ◽  
Minimal Blocking

Owen's multilinear extension (MLE) of a game is a very important tool in game theory and particularly in the field of simple games. Among other applications it serves to efficiently compute several solution concepts. In this paper we provide bounds for the MLE. Apart from its self-contained theoretical interest, the bounds offer the means in voting system studies of approximating the probability that a proposal is approved in a particular simple game having a complex component arrangement. The practical interest of the bounds is that they can be useful for simple games having a tedious MLE to evaluate exactly, but whose minimal winning coalitions and minimal blocking coalitions can be determined by inspection. Such simple games are quite numerous.

scholarly journals Minimal winning coalitions and orders of criticality

2021 ◽  
Author(s):  
Michele Aleandri ◽  
Marco Dall’Aglio ◽  
Vito Fragnelli ◽  
Stefano Moretti
Keyword(s):  
Simple Game ◽  
Simple Games ◽  
Hitting Sets ◽  
The Family ◽  
Minimal Winning Coalitions ◽  
Minimal Blocking

AbstractIn this paper, we analyze the order of criticality in simple games, under the light of minimal winning coalitions. The order of criticality of a player in a simple game is based on the minimal number of other players that have to leave so that the player in question becomes pivotal. We show that this definition can be formulated referring to the cardinality of the minimal blocking coalitions or minimal hitting sets for the family of minimal winning coalitions; moreover, the blocking coalitions are related to the winning coalitions of the dual game. Finally, we propose to rank all the players lexicographically accounting the number of coalitions for which they are critical of each order, and we characterize this ranking using four independent axioms.

scholarly journals An issue based power index

2020 ◽  
Author(s):  
Qianqian Kong ◽  
Hans Peters
Keyword(s):  
Linear Order ◽  
Power Index ◽  
Simple Game ◽  
Power Indices ◽  
Weighted Sum ◽  
Simple Games ◽  
Linear Orders ◽  
Transfer Property ◽  
Power Change ◽  
Equal Power

Abstract An issue game is a combination of a monotonic simple game and an issue profile. An issue profile is a profile of linear orders on the player set, one for each issue within the set of issues: such a linear order is interpreted as the order in which the players will support the issue under consideration. A power index assigns to each player in an issue game a nonnegative number, where these numbers sum up to one. We consider a class of power indices, characterized by weight vectors on the set of issues. A power index in this class assigns to each player the weighted sum of the issues for which that player is pivotal. A player is pivotal for an issue if that player is a pivotal player in the coalition consisting of all players preceding that player in the linear order associated with that issue. We present several axiomatic characterizations of this class of power indices. The first characterization is based on two axioms: one says how power depends on the issues under consideration (Issue Dependence), and the other one concerns the consequences, for power, of splitting players into several new players (no advantageous splitting). The second characterization uses a stronger version of Issue Dependence, and an axiom about symmetric players (Invariance with respect to Symmetric Players). The third characterization is based on a variation on the transfer property for values of simple games (Equal Power Change), besides Invariance with respect to Symmetric Players and another version of Issue Dependence. Finally, we discuss how an issue profile may arise from preferences of players about issues.

scholarly journals Simple Games of Information Transmission

2012 ◽  
Vol 34 (2) ◽  
Author(s):  
Bernd Lahno
Keyword(s):  
Information Transmission ◽  
Simple Game ◽  
Strategic Analysis ◽  
Direct Impact ◽  
Simple Games ◽  
Game Theoretic ◽  
The Will

AbstractCommunication is an inherently strategic matter. This paper introduces simple game theoretic models of information transmission to identify different forms of uncertainty which may pose a problem of trust in testimony. Strategic analysis suggests discriminating between trust in integrity, trust in competence, trust in (the will to invest) effort and trust in honesty. Whereas uncertainty about the sender's honesty or integrity may directly influence a rational receiver's readiness to rely on sender's statements, neither uncertainty about the competence of a sender nor uncertainty about his willingness to invest effort has any direct impact on rational reliance on its own. In this regard, trust in honesty and trust in integrity appear to be more basic than trust in competence or effort.

scholarly journals Bounds for Owen's Multilinear Extension

2007 ◽  
Vol 44 (04) ◽  
pp. 852-864 ◽  
Author(s):  
Josep Freixas
Keyword(s):  
Game Theory ◽  
Simple Game ◽  
Practical Interest ◽  
Simple Games ◽  
Theoretical Interest ◽  
Complex Component ◽  
Multilinear Extension ◽  
Voting System ◽  
Minimal Winning Coalitions ◽  
Minimal Blocking

Owen's multilinear extension (MLE) of a game is a very important tool in game theory and particularly in the field of simple games. Among other applications it serves to efficiently compute several solution concepts. In this paper we provide bounds for the MLE. Apart from its self-contained theoretical interest, the bounds offer the means in voting system studies of approximating the probability that a proposal is approved in a particular simple game having a complex component arrangement. The practical interest of the bounds is that they can be useful for simple games having a tedious MLE to evaluate exactly, but whose minimal winning coalitions and minimal blocking coalitions can be determined by inspection. Such simple games are quite numerous.

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

10.37236/1245 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Art M. Duval
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Pure Case ◽  
Definition Of ◽  
Algebraic Shifting

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the $h$-triangle, a doubly-indexed generalization of the $h$-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the $h$-triangle of a simplicial complex $K$ if and only if $K$ is sequentially Cohen-Macaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible $h$-triangles.

The Nature of Time

2018 ◽  
Author(s):  
Donald C. Williams
Keyword(s):  
Large Scale ◽  
Time Travel ◽  
Dimensional Manifold ◽  
Nature Of Time ◽  
Manifold Theory ◽  
The Universe ◽  
Theoretical Cost ◽  
Do So

This chapter provides a fuller treatment of the pure manifold theory with an expanded discussion of competing doctrines. It is argued that competing doctrines fail to account for the extensive and/or transitory aspect(s) of time, or they do so at great theoretical cost. The pure manifold theory accounts for the extensive aspect of time because it admits a four-dimensional manifold and it accounts for the transitory aspect of time because it hypothesizes that the increase of entropy is the thing that is ‘felt’ in veridical cases of felt passage. A four-dimensionalist theory of time travel is outlined, along with a sketch of large-scale cosmological traits of the universe.

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