scholarly journals Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency

2020 ◽  
Vol 45 (3) ◽  
pp. 1056-1068
Author(s):  
Pedro Calleja ◽  
Francesc Llerena ◽  
Peter Sudhölter
Keyword(s):  
Regular System ◽  
Necessary Condition ◽  
Transferable Utility ◽  
Tu Games ◽  
Scale Covariance ◽  
Per Capita ◽  
Arbitrary Weight ◽  
Translation Covariance ◽  
Equal Surplus Division ◽  
Aggregate Monotonicity

A solution on a set of transferable utility (TU) games satisfies strong aggregate monotonicity (SAM) if every player can improve when the grand coalition becomes richer. It satisfies equal surplus division (ESD) if the solution allows the players to improve equally. We show that the set of weight systems generating weighted prenucleoli that satisfy SAM is open, which implies that for weight systems close enough to any regular system, the weighted prenucleolus satisfies SAM. We also provide a necessary condition for SAM for symmetrically weighted nucleoli. Moreover, we show that the per capita nucleolus on balanced games is characterized by single-valuedness (SIVA), translation covariance (TCOV) and scale covariance (SCOV), and equal adjusted surplus division (EASD), a property that is comparable to but stronger than ESD. These properties together with ESD characterize the per capita prenucleolus on larger sets of TU games. EASD and ESD can be transformed to independence of (adjusted) proportional shifting, and these properties may be generalized for arbitrary weight systems p to I(A)Sp. We show that the p-weighted prenucleolus on the set of balanced TU games is characterized by SIVA, TCOV, SCOV, and IASp and on larger sets by additionally requiring ISp.

Effects of Players’ Nullification and Equal (Surplus) Division Values

2018 ◽  
Vol 20 (01) ◽  
pp. 1750029 ◽  
Author(s):  
Takumi Kongo
Keyword(s):  
The Other ◽  
Equal Division ◽  
Transferable Utility ◽  
Tu Games ◽  
Axiomatic Characterizations ◽  
The Difference ◽  
Monotonicity Axioms ◽  
Equal Surplus Division

We provide axiomatic characterizations of the solutions of transferable utility (TU) games on the fixed player set, where at least three players exist. We introduce two axioms on players’ nullification. One axiom requires that the difference between the effect of a player’s nullification on the nullified player and on the others is relatively constant if all but one players are null players. Another axiom requires that a player’s nullification affects equally all of the other players. These two axioms characterize the set of all affine combinations of the equal surplus division and equal division values, together with the two basic axioms of efficiency and null game. By replacing the first axiom on players’ nullification with appropriate monotonicity axioms, we narrow down the solutions to the set of all convex combinations of the two values, or to each of the two values.

NONEMPTY CORE-TYPE SOLUTIONS OVER BALANCED COALITIONS IN TU-GAMES

2012 ◽  
Vol 14 (03) ◽  
pp. 1250018 ◽  
Author(s):  
JUAN C. CESCO
Keyword(s):  
Equal Opportunity ◽  
Axiomatic Characterization ◽  
Individual Rationality ◽  
Transferable Utility ◽  
Tu Game ◽  
Tu Games ◽  
Additional Axiom ◽  
Game Property ◽  
Minimality Property ◽  
New Framework

In this paper we introduce two related core-type solutions for games with transferable utility (TU-games) the [Formula: see text]-core and the [Formula: see text]-core. The elements of the solutions are pairs [Formula: see text] where x, as usual, is a vector representing a distribution of utility and [Formula: see text] is a balanced family of coalitions, in the case of the [Formula: see text]-core, and a minimal balanced one, in the case of the [Formula: see text]-core, describing a plausible organization of the players to achieve the vector x. Both solutions extend the notion of classical core but, unlike it, they are always nonempty for any TU-game. For the [Formula: see text]-core, which also exhibits a certain kind of "minimality" property, we provide a nice axiomatic characterization in terms of the four axioms nonemptiness (NE), individual rationality (IR), superadditivity (SUPA) and a weak reduced game property (WRGP) (with appropriate modifications to adapt them to the new framework) used to characterize the classical core. However, an additional axiom, the axiom of equal opportunity is required. It roughly states that if [Formula: see text] belongs to the [Formula: see text]-core then, any other admissible element of the form [Formula: see text] should belong to the solution too.

scholarly journals Hart–Mas-Colell consistency and the core in convex games

2021 ◽  
Author(s):  
Bas Dietzenbacher ◽  
Peter Sudhölter
Keyword(s):  
Shapley Value ◽  
Cooperative Games ◽  
Individual Rationality ◽  
Transferable Utility ◽  
Convex Games ◽  
The Core ◽  
The Shapley Value ◽  
Scale Covariance

AbstractThis paper formally introduces Hart–Mas-Colell consistency for general (possibly multi-valued) solutions for cooperative games with transferable utility. This notion is used to axiomatically characterize the core on the domain of convex games. Moreover, we characterize all nonempty solutions satisfying individual rationality, anonymity, scale covariance, superadditivity, weak Hart–Mas-Colell consistency, and converse Hart–Mas-Colell consistency. This family consists of (a) the Shapley value, (b) all homothetic images of the core with the Shapley value as center of homothety and with positive ratios of homothety not larger than one, and (c) their relative interiors.

A new axiomatization of a class of equal surplus division values for TU games

2018 ◽  
Vol 52 (3) ◽  
pp. 935-942 ◽  
Author(s):  
Xun-Feng Hu ◽  
Deng-Feng Li
Keyword(s):  
Equal Division ◽  
Tu Game ◽  
Tu Games ◽  
Special Cases ◽  
Equal Surplus Division

In this paper, we propose a variation of weak covariance named as non-singleton covariance, requiring that changing the worth of a non-singleton coalition in a TU game affects the payoffs of all players equally. We establish that this covariance is characteristic for the convex combinations of the equal division value and the equal surplus division value, together with efficiency and a one-parameterized axiom treating a particular kind of players specially. As special cases, parallel axiomatizations of the two values are also provided.

The Need for Permission, the Power to Enforce, and Duality in Cooperative Games with a Hierarchy

2016 ◽  
Vol 18 (04) ◽  
pp. 1650015 ◽  
Author(s):  
Frank Huettner ◽  
Harald Wiese
Keyword(s):  
Cooperative Game ◽  
Cooperative Games ◽  
Transferable Utility ◽  
Tu Game ◽  
Tu Games ◽  
Permission Value

A cooperative game with transferable utility (TU game) captures a situation in which players can achieve certain payoffs by cooperating. We assume that the players are part of a hierarchy. In the literature, this invokes the assumption that subordinates cannot cooperate without the permission of their superiors. Instead, we assume that superiors can force their subordinates to cooperate. We show how both notions correspond to each other by means of dual TU games. This way, we capture the idea that a superiors’ ability to enforce cooperation can be seen as the ability to neutralize her subordinate’s threat to abstain from cooperation. Moreover, we introduce the coercion value for games with a hierarchy and provide characterizations thereof that reveal the similarity to the permission value.

INTERGENERATIONAL EQUITY AND THE DISCOUNT RATE FOR POLICY ANALYSIS

2011 ◽  
Vol 16 (1) ◽  
pp. 61-93 ◽  
Author(s):  
Jean-François Mertens ◽  
Anna Rubinchik
Keyword(s):  
Growth Rate ◽  
Discount Rate ◽  
Capita Income ◽  
Cost Benefit Analysis ◽  
Solution Concept ◽  
Intergenerational Equity ◽  
Necessary Condition ◽  
Per Capita Income ◽  
Time Invariance ◽  
Per Capita

For two independent principles of intergenerational equity, the implied discount rate equals the growth rate of real per capita income, say, 2%, thus falling right into the range suggested by the U.S. Office of Management and Budget. To prove this, we develop a simple tool to evaluate small policy changes affecting several generations, by reducing the dynamic problem to a static one. A necessary condition is time invariance, which is satisfied by any common solution concept in an overlapping-generations model with exogenous growth. This tool is applied to derive the discount rate for cost–benefit analysis under two different utilitarian welfare functions: classical and relative. It is only with relative utilitarianism, and assuming time-invariance of the set of alternatives (policies), that the discount rate is well defined for a heterogeneous society at a balanced growth equilibrium, is corroborated by an independent principle equating values of human lives, and equals the growth rate of real per-capita income.

POTENTIAL IN MULTI-CHOICE COOPERATIVE TU GAMES

2008 ◽  
Vol 25 (05) ◽  
pp. 591-611 ◽  
Author(s):  
YAN-AN HWANG ◽  
YU-HSIEN LIAO
Keyword(s):  
Transferable Utility ◽  
Tu Games ◽  
Cooperative Tu Games ◽  
All Solutions ◽  
The Family ◽  
Balanced Contributions ◽  
Associated Consistency

This paper is devoted to the study of solutions for multi-choice transferable-utility (TU) games which admit a potential, such as the potential associated with a solution in the context of multi-choice TU games. Several axiomatizations of the family of all solutions that admit a potential are offered and, as a main result, it is shown that each of these solutions can be obtained by applying the weighted associated consistent value proposed in this paper to an appropriately modified game. We also characterize the weighted associated consistent value by means of the weighted balanced contributions and the associated consistency.

Solutions, Potentializability and Axiomatizations under Interval Uncertainty

2014 ◽  
Vol 22 (04) ◽  
pp. 605-613 ◽  
Author(s):  
Yu-Hsien Liao
Keyword(s):  
Shapley Value ◽  
Equivalence Theorem ◽  
Interval Uncertainty ◽  
Transferable Utility ◽  
Tu Games ◽  
The Family

In the framework of interval transferable-utility (TU) games, we propose an equivalence theorem to characterize the family of all interval solutions that admit a potential. Further, we also provide several axiomatizations of the interval Shapley value based on this equivalence theorem.

SOME EXCESS-BASED SOLUTIONS FOR COOPERATIVE GAMES WITH TRANSFERABLE UTILITY

2013 ◽  
Vol 15 (04) ◽  
pp. 1340029 ◽  
Author(s):  
KRISHNA CHAITANYA VANAM ◽  
N. HEMACHANDRA
Keyword(s):  
Cost Sharing ◽  
Cooperative Games ◽  
Closed Form Expression ◽  
Transferable Utility ◽  
Form Expression ◽  
Per Capita ◽  
Step Algorithm ◽  
Finite Step ◽  
General Game ◽  
Cost Game

For a finite player cooperative cost game, we consider two solutions that are based on excesses of coalitions. We define per-capita excess-sum of a player as sum of normalized excesses of coalitions involving this player and view it as a measure of player's dissatisfaction. So, per-capita excess-sum allocation is that imputation that minimizes the maximum per-capita excess-sums of players. We provide a closed form expression for an allocation, which is the per-capita excess-sum allocation if it is also individually rational. We propose a finite step algorithm to compute per-capita excess-sum allocation for a general game. We show that per-capita excess-sum allocation is coalitionally monotonic. Next, we consider excess-sum solution wherein a player views entire coalition's excess as a measure of dissatisfaction. This excess-sum solution also has above properties. In addition, we consider a super set of core and show that excess-sum allocation can be viewed as an imputation that is a certain center of this polyhedron. We introduce a class of cooperative games that can model cost sharing among divisions of a firm when they buy items at volume discounts. We characterize when excess-based allocations coincide with Shapley value, nucleolus, etc. in such games.

scholarly journals Poverty, Value, and Conservation

Conservation ◽  
2021 ◽  
pp. 234-254
Author(s):  
Charles Perrings
Keyword(s):  
Property Rights ◽  
Natural Resources ◽  
Biodiversity Conservation ◽  
Empirical Relation ◽  
Biodiversity Loss ◽  
Three Dimensions ◽  
Necessary Condition ◽  
The Third ◽  
Per Capita

Chapter 10 addresses the relation between conservation, income, and wealth, focusing on the problem of biodiversity. It reconsiders the current consensus that poverty alleviation will reduce biodiversity loss. It addresses three dimensions of the problem: the role of income in the demand for natural resources; the empirical relation between income and biodiversity conservation; and the link between wealth, property rights, and the incentive to conserve. The first connects poverty, population growth, and the demand for natural resources. The second shows how biodiversity conservation and per capita income are related. The third connects poverty and property rights. It shows that for rural landholders to have an incentive to conserve their land, they also need to have secure rights. A necessary condition for land conservation to be optimal by the Hotelling principle is that the rights-holder can realize the gains to be had from conserving the resource.

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