Solution Concepts for Games with General Coalitional Structure (Replaces CentER DP 2011-025)

We introduce a theory on marginal values and their core stability for cooperative games with arbitrary coalition structure. The theory is based on the notion of nested sets and the complex of nested sets associated to an arbitrary set system and the M-extension of a game for this set. For a set system being a building set or partition system, the corresponding complex is a polyhedral complex, and the vertices of this complex correspond to maximal strictly nested sets. To each maximal strictly nested set is associated a rooted tree. Given characteristic function, to every maximal strictly nested set a marginal value is associated to a corresponding rooted tree as in [9]. We show that the same marginal value is obtained by using the M-extension for every permutation that is associated to the rooted tree. The GC-solution is defined as the average of the marginal values over all maximal strictly nested sets. The solution can be viewed as the gravity center of the image of the vertices of the polyhedral complex. The GC-solution differs from the Myerson-kind value defined in [2] for union stable structures. The HS-solution is defined as the average of marginal values over the subclass of so-called half-space nested sets. The NT-solution is another solution and is defined as the average of marginal values over the subclass of NT-nested sets. For graphical buildings the collection of NT-nested sets corresponds to the set of spanning normal trees on the underlying graph and the NT-solution coincides with the average tree solution. We also study core stability of the solutions and show that both the HS-solution and NT-solution belong to the core under half-space supermodularity, which is a weaker condition than convexity of the game. For an arbitrary set system we show that there exists a unique minimal building set containing the set system. As solutions we take the solutions for this building covering by extending in a natural way the characteristic function to it by using its Möbius inversion.Core;polytope;building set;nested set complex;Möbius inversion;permutations;normal fan;average tree solution;Myerson value

Competitive Equilibria in Economies with Multiple Divisible and Indivisible Commodities and No Money

A general equilibrium model is considered with multiple divisible and multiple indivisible commodities.In models with indivisibles it is always assumed that an indivisible commodity, called money, is present that is used to transfer the value of certain amounts of indivisible goods.For these economies with a finite number of divisible and indivisible goods and money and without producers it is well understood that a general equilibrium exists if the individual demands and supplies for the indivisibele goods belong to a same class of discrete convexity.In this paper we a model with multiple divisible and multiple undivisible commodities, in which none of the divisible goods may serve as money.Moreover, there are a finite number of producers owning a non-increasing returns to scale technology.One of the producesrs is assumed to have a linear production technology in order to produce divisible goods. Individual endowments being sufficienly large for production and discrete convexity guarantees the existence of a competitive equilibrium.indivisible commodities;divisible commodities;discrete convexity;competitive equilibrium

Competitive Equilibria in Economies with Multiple Divisible and Multiple Divisible Commodities

In this paper we consider a general equilibrium model with a finite number of divisible and indivisible commodities.In models with indivisibilities it is typically assumed that there is only one perfectly divisible good, which serves as money.The presence of money in the model is used to transfer the value of certain amounts of indivisible goods.For such economies with one divisible commodity Danilov et al. showed the existence of a general equilibrium if the individual demands and supplies belong to a same class of discrete convexity.For economies with multiple divisible goods and money van der Laan et al. proved existence of a general equilibrium if the divisible goods are produced out of money using a linear production technology and no other producers are present in the model.In the models to be presented in this paper we allow for multiple divisible commodities and a finite number of producers with non-increasing returns to scale technologies.Convexity is replaced by pseudoconvexity, while the indivisible parts of individual demands and supply should belong to some class of discrete convexity.In the first model money is present.Money is strictly desired by the consumers like in the other models, is indispensable for production and enough money should be present in the economy.To guarantee existence of a general equilibrium individual demands and supplies should be products of divisible and indivisible parts.In the second model there is no money, but at least one linear production technology is present in order to produce the divisible goods.Individual endowments being sufficienly large for production guarantee the existence of a competitive equilibrium.

A General Existence Thorem of Zero Points

Let X be a non-empty, compact, convex set in Rn and ° an upper semi-continuous mapping from X to the collection of non-empty, compact, convex subsets of Rn.It is well known that such a mapping has a stationary point on X, i.e. there exists a point in X satisfying that its image under ° has a non-empty intersection with the normal cone of X at the point.In case for every point in X it holds that the intersection of the image under ° with the normal cone of X at the point is either empty orcontains the origin 0n , then ° must have a zero point on X, i.e. there exists a point in X satisfying that 0n lies in the image of the point.Another well-known condition for the existence of a zero point follows from Ky Fan's coincidence theorem, which says that if for every point the intersection of the image with the tangent cone of X at the point is non-empty, the mapping must have a zero point.In this paper we extend all these existence results by giving a general zero point existence theorem, of which the two results are obtained as special cases.We also discuss what kind of solutions may exist when no further conditions are stated on the mapping °.Finally, we show how our results can be used to establish several new intersection results on a compact, convex set

A General Existence Thorem of Zero Points

Let X be a non-empty, compact, convex set in Rn and ° an upper semi-continuous mapping from X to the collection of non-empty, compact, convex subsets of Rn.It is well known that such a mapping has a stationary point on X, i.e. there exists a point in X satisfying that its image under ° has a non-empty intersection with the normal cone of X at the point.In case for every point in X it holds that the intersection of the image under ° with the normal cone of X at the point is either empty orcontains the origin 0n , then ° must have a zero point on X, i.e. there exists a point in X satisfying that 0n lies in the image of the point.Another well-known condition for the existence of a zero point follows from Ky Fan's coincidence theorem, which says that if for every point the intersection of the image with the tangent cone of X at the point is non-empty, the mapping must have a zero point.In this paper we extend all these existence results by giving a general zero point existence theorem, of which the two results are obtained as special cases.We also discuss what kind of solutions may exist when no further conditions are stated on the mapping °.Finally, we show how our results can be used to establish several new intersection results on a compact, convex set.stationary point;zero point

Multivariate risks and depth-trimmed regions

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this abstract axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from the multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship.Comment: 26 pages. Substantially revised version with a number of new results adde