Stackelberg Oligopoly TU-Games: Characterization and Nonemptiness of the Core

2017 ◽  
Vol 19 (04) ◽  
pp. 1750020 ◽  
Author(s):  
Dongshuang Hou ◽  
Aymeric Lardon ◽  
T. S. H. Driessen
Keyword(s):  
Demand Function ◽  
Necessary And Sufficient Condition ◽  
Function Form ◽  
Tu Games ◽  
Marginal Costs ◽  
The Core ◽  
Dynamic Setting ◽  
Necessary And Sufficient ◽  
Second Period ◽  
Characteristic Function Form

In this paper, we consider the dynamic setting of Stackelberg oligopoly TU-games in [Formula: see text]-characteristic function form. Any deviating coalition produces an output at a first period as a leader and then, outsiders simultaneously and independently play a quantity at a second period as followers. We assume that the inverse demand function is linear and that firms operate at constant but possibly distinct marginal costs. First, we show that the core of any Stackelberg oligopoly TU-game always coincides with the set of imputations. Second, we provide a necessary and sufficient condition, depending on the heterogeneity of firms’ marginal costs, under which the core is nonempty.

scholarly journals Improved design and analysis of practical minimizers

2020 ◽  
Vol 36 (Supplement_1) ◽  
pp. i119-i127
Author(s):  
Hongyu Zheng ◽  
Carl Kingsford ◽  
Guillaume Marçais
Keyword(s):  
Randomized Algorithm ◽  
Supplementary Information ◽  
Necessary And Sufficient Condition ◽  
Asymptotically Optimal ◽  
The Core ◽  
Computation Efficiency ◽  
Bioinformatics Software ◽  
Reference Implementation ◽  
Improved Design ◽  
Necessary And Sufficient

Abstract Motivation Minimizers are methods to sample k-mers from a string, with the guarantee that similar set of k-mers will be chosen on similar strings. It is parameterized by the k-mer length k, a window length w and an order on the k-mers. Minimizers are used in a large number of softwares and pipelines to improve computation efficiency and decrease memory usage. Despite the method’s popularity, many theoretical questions regarding its performance remain open. The core metric for measuring performance of a minimizer is the density, which measures the sparsity of sampled k-mers. The theoretical optimal density for a minimizer is 1/w, provably not achievable in general. For given k and w, little is known about asymptotically optimal minimizers, that is minimizers with density O(1/w). Results We derive a necessary and sufficient condition for existence of asymptotically optimal minimizers. We also provide a randomized algorithm, called the Miniception, to design minimizers with the best theoretical guarantee to date on density in practical scenarios. Constructing and using the Miniception is as easy as constructing and using a random minimizer, which allows the design of efficient minimizers that scale to the values of k and w used in current bioinformatics software programs. Availability and implementation Reference implementation of the Miniception and the codes for analysis can be found at https://github.com/kingsford-group/miniception. Supplementary information Supplementary data are available at Bioinformatics online.

scholarly journals Improved design and analysis of practical minimizers

2020 ◽  
Author(s):  
Hongyu Zheng ◽  
Carl Kingsford ◽  
Guillaume Marçais
Keyword(s):  
Randomized Algorithm ◽  
Necessary And Sufficient Condition ◽  
Optimal Density ◽  
Asymptotically Optimal ◽  
The Core ◽  
Computation Efficiency ◽  
Bioinformatics Software ◽  
Reference Implementation ◽  
Improved Design ◽  
Necessary And Sufficient

AbstractMotivationMinimizers are methods to sample k-mers from a sequence, with the guarantee that similar set of k-mers will be chosen on similar sequences. It is parameterized by the k-mer length k, a window length w and an order on the k-mers. Minimizers are used in a large number of softwares and pipelines to improve computation efficiency and decrease memory usage. Despite the method’s popularity, many theoretical questions regarding its performance remain open. The core metric for measuring performance of a minimizer is the density, which measures the sparsity of sampled k-mers. The theoretical optimal density for a minimizer is 1/w, provably not achievable in general. For given k and w, little is known about asymptotically optimal minimizers, that is minimizers with density O(1/w).ResultsWe derive a necessary and sufficient condition for existence of asymptotically optimal minimizers. We also provide a randomized algorithm, called the Miniception, to design minimizers with the best theoretical guarantee to date on density in practical scenarios. Constructing and using the Miniception is as easy as constructing and using a random minimizer, which allows the design of efficient minimizers that scale to the values of k and w used in current bioinformatics software programs.AvailabilityReference implementation of the Miniception and the codes for analysis can be found at https://github.com/kingsford-group/[email protected]

U-CYCLES IN n-PERSON TU-GAMES WITH ONLY 1, n - 1 AND n-PERSON PERMISSIBLE COALITIONS

2006 ◽  
Vol 08 (03) ◽  
pp. 355-368 ◽  
Author(s):  
JUAN CARLOS CESCO ◽  
ANA LUCÍA CALÍ
Keyword(s):  
Characterization Theorem ◽  
Tu Game ◽  
Transfer Scheme ◽  
Tu Games ◽  
The Core ◽  
Core Of A Game ◽  
Necessary And Sufficient

It has been recently proved that the non-existence of certain type of cycles of pre-imputation, fundamental cycles, is equivalent to the balancedness of a TU-games (Cesco (2003)). In some cases, the class of fundamental cycles can be narrowed and still obtain a characterization theorem. In this paper we prove that existence of maximal U-cycles, which are related to a transfer scheme designed for computing a point in the core of a game, is condition necessary and sufficient for a TU-game be non-balanced, provided n - 1 and n-person are the only coalitions with non-zero value. These games are strongly related to games with only 1, n - 1 and n-person permissible coalitions (Maschler (1963)).

NULL PLAYERS OUT? LINEAR VALUES FOR GAMES WITH VARIABLE SUPPORTS

1999 ◽  
Vol 01 (03n04) ◽  
pp. 301-314 ◽  
Author(s):  
JEAN J. M. DERKS ◽  
HANS H. HALLER
Keyword(s):  
Cooperative Games ◽  
Symmetry Property ◽  
Sufficient Conditions ◽  
Function Form ◽  
Individual Values ◽  
Null Player ◽  
Main Emphasis ◽  
Weak Symmetry ◽  
Necessary And Sufficient ◽  
Characteristic Function Form

The paper studies the consequences of the Null Player Out (NPO) property for single-valued solutions on the class of cooperative games in characteristic function form. We allow for variable player populations (supports or carriers). A solution satisfies the NPO property, if elimination of a null player does not affect the payoffs of the other players. Our main emphasis lies on individual values. For linear values satisfying the null player property and a weak symmetry property, necessary and sufficient conditions for the NPO property are derived.

DETERMINING THE NUCLEOLUS OF COMPROMISE STABLE GAMES

2015 ◽  
Vol 92 (3) ◽  
pp. 488-495 ◽  
Author(s):  
DONGSHUANG HOU ◽  
THEO DRIESSEN
Keyword(s):  
Characteristic Function ◽  
Coalitional Game ◽  
Coalitional Games ◽  
Function Form ◽  
Dual Representation ◽  
Tu Games ◽  
Clan Games ◽  
Big Boss Games ◽  
Characteristic Function Form

The main goal is to illustrate that the so-called indirect function of a cooperative game in characteristic function form is applicable to determine the nucleolus for a subclass of coalitional games called compromise stable transferable utility (TU) games. In accordance with the Fenchel–Moreau theory on conjugate functions, the indirect function is known as the dual representation of the characteristic function of the coalitional game. The key feature of a compromise stable TU game is the coincidence of its core with a box prescribed by certain upper and lower core bounds. For the purpose of the determination of the nucleolus, we benefit from the interrelationship between the indirect function and the prekernel of coalitional TU games. The class of compromise stable TU games contains the subclasses of clan games, big boss games and $1$- and $2$-convex $n$-person TU games. As an adjunct, this paper reports the indirect function of clan games for the purpose of determining their nucleolus.

scholarly journals Convexity of b-matching Games

2020 ◽  
Author(s):  
Soh Kumabe ◽  
Takanori Maehara
Keyword(s):  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Kidney Exchange ◽  
The Core ◽  
Emptiness Problem ◽  
Matching Games ◽  
Matching Game ◽  
Necessary And Sufficient

The b-matching game is a cooperative game defined on a graph. The game generalizes the matching game to allow each individual to have more than one partner. The game has several applications, such as the roommate assignment, the multi-item version of the seller-buyer assignment, and the international kidney exchange. Compared with the standard matching game, the b-matching game is computationally hard. In particular, the core non-emptiness problem and the core membership problem are co-NP-hard. Therefore, we focus on the convexity of the game, which is a sufficient condition of the core non-emptiness and often more tractable concept than the core non-emptiness. It also has several additional benefits. In this study, we give a necessary and sufficient condition of the convexity of the b-matching game. This condition also gives an O(n log n + m α(n)) time algorithm to determine whether a given game is convex or not, where n and m are the number of vertices and edges of a given graph, respectively, and α(・) is the inverse-Ackermann function. Using our characterization, we also give a polynomial-time algorithm to compute the Shapley value of a convex b-matching game.

scholarly journals CORE CONCEPTS FOR DYNAMIC TU GAMES

2005 ◽  
Vol 07 (01) ◽  
pp. 43-61 ◽  
Author(s):  
LAURENCE KRANICH ◽  
ANDRÉS PEREA ◽  
HANS PETERS
Keyword(s):  
Cooperative Games ◽  
Sufficient Conditions ◽  
Finite Sequence ◽  
Essential Difference ◽  
Tu Games ◽  
The Core ◽  
Dynamic Setting ◽  
Core Concepts ◽  
Provided Examples

This paper is concerned with the question of how to define the core when cooperation takes place in a dynamic setting. The focus is on dynamic cooperative games in which the players face a finite sequence of exogenously specified TU-games. Three different core concepts are presented: the classical core, the strong sequential core and the weak sequential core. The differences between the concepts arise from different interpretations of profitable deviations by coalitions. Sufficient conditions are given for nonemptiness of the classical core in general and of the weak sequential core for the case of two players. Simplifying characterizations of the weak and strong sequential core are provided. Examples highlight the essential difference between these core concepts.

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