Stable cores of large games

We give general conditions, based on the largeness of the core, under which cores of exact TU games are their unique von Neumann-Morgenstern stable sets. We show that this condition is satisfied by convex games and by nonatomic exact market games. In this way, we extend and unify earlier results existing in literature. Under some additional conditions we also prove the equivalence between the core and the Mas-Colell bargaining set.We thank Jean-Francois Mertens, Enrico Minelli, William Thomson, and two anonymous referees for helpful comments. We also thank seminar audiences at CORE, Cornell, Pescara, and Rochester. We gratefully acknowledge the financial support of the Ministero dell’Istruzione, dell’Universitá e della Ricerca.     

Cooperative games with large cores

A payoff vector in ann-person cooperative game is said to be acceptable if no coalition can improve upon it. The core of a game consists of all acceptable vectors which are feasible for the grand coalition. The core is said to be large if for every acceptable vectory there is a vectorx in the core withx?y. This paper examines the class of games with large cores.     

Symmetric solutions for symmetric games with large cores

International Journal of Game Theory - Symmetric solutions (symmetric stable sets) and their uniqueness are investigated for symmetric games when the cores are large enough to have intersections...     

Cores and retracts

Theretracts (idempotent, isotone self-maps) of an ordered set are naturally ordered as functions. In this note we characterize the possible ways that one retract can cover another one. This gives some insight into the structure of the ordered set of retracts and leads to a natural generalization of the core of an ordered set.Supported by NSERC Operating Grant 41702.     

On the convergence of population protocols when population goes to infinity

Population protocols have been introduced as a model of sensor networks consisting of very limited mobile agents with no control over their own movement. A population protocol corresponds to a collection of anonymous agents, modeled by finite automata, that interact with one another to carry out computations, by updating their states, using some rules.Their computational power has been investigated under several hypotheses but always when restricted to finite size populations. In particular, predicates stably computable in the original model have been characterized as those definable in Presburger arithmetic.We study mathematically the convergence of population protocols when the size of the population goes to infinity. We do so by giving general results, that we illustrate through the example of a particular population protocol for which we even obtain an asymptotic development.This example shows in particular that these protocols seem to have a rather different computational power when a huge population hypothesis is considered.     

Proper cores and dominance cores of fuzzy games

Many kinds of proper cores and dominance cores of fuzzy games are proposed in this paper. We also consider two similar concepts of payoff of a fuzzy coalition that are called the true payoff and quasi-payoff. The different concepts of proper cores and dominance cores will be proposed based on the true payoff and quasi-payoff of a fuzzy coalition. Some mild sufficient conditions are provided in this paper to guarantee the equalities of different proper cores and dominance cores.     

Light tails: all summands are large when the empirical mean is large

It is well known that for a fixed number of independent identically distributed summands with light tail, large values of the sample mean are obtained only when all the summands take large values. This paper explores this property as the number of summands tends to infinity. It provides the order of magnitude of the sample mean for which all summands are in some interval containing this value and it also explores the width of this interval with respect to the distribution of the summands in their upper tail. These results are proved for summands with log-concave or nearly log concave densities. Making use of some extension of the Erdös-Rényi law of large numbers it also explores the forming of aggregates in a sequence of i.i.d. random variables. As a by product the connection is established between large exceedances of the local slope of a random walk on growing bins and the theory of extreme order statistics.     

Consistent variable selection in large panels when factors are observable

In this paper we develop an econometric method for consistent variable selection in the context of a linear factor model with observable factors for panels of large dimensions. The subset of factors that best fit the data is sequentially determined. Firstly, a partial R² rule is used to show the existence of an optimal ordering of the candidate variables. Secondly, We show that for a given order of the regressors, the number of factors can be consistently estimated using the Bayes information criterion. The Akaike will asymptotically lead to overfitting of the model. The theory is established under approximate factor structure which allows for limited cross-section and serial dependence in the idiosyncratic term. Simulations show that the proposed two-step selection technique has good finite sample properties. The likelihood of selecting the correct specification increases with the number of cross-sections both asymptotically and in small samples. Moreover, the proposed variable selection method is computationally attractive. For K potential candidate factors, the search requires only 2K regressions compared to 2^K for an exhaustive search.     

Fuzzy cores and fuzzy balancedness

We study the relation between the fuzzy core and balancedness for fuzzy games. For regular games, this relation has been studied by Bondareva (Problemy Kibernet 10:119–139, 1963) and Shapley (Naval Res Logist Q 14: 453–460, 1967). First, we gain insight in this relation when we analyse situations where the fuzzy game is continuous. Our main result shows that any fuzzy game has a non-empty core if and only if it is balanced. We also consider deposit games to illustrate the use of the main result.     

Hyperpolygon spaces and their cores

Given an -tuple of positive real numbers , Konno (2000) defines the hyperpolygon space , a hyperkähler analogue of the Kähler variety parametrizing polygons in with edge lengths . The polygon space can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, is the hyperkähler quiver variety defined by Nakajima. A quiver variety admits a natural -action, and the union of the precompact orbits is called the core. We study the components of the core of , interpreting each one as a moduli space of pairs of polygons in with certain properties. Konno gives a presentation of the cohomology ring of ; we extend this result by computing the -equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.

     

Fitting cores and supersolvable groups

Let A be a group. What can be said about the group B to ensure that A and the normal product AB belong to the same prescribed class of groups? Results in this direction are given for the classes of supersolvable groups, absolutely solvable groups and Lagrange groups.     

Strange expectations and simultaneous cores

Let \(\gcd (a,b)=1\). J. Olsson and D. Stanton proved that the maximum number of boxes in a simultaneous (a, b)-core is

$$\begin{aligned} \max _{\lambda \in {\mathrm {core}}(a,b)} (\mathsf{size}(\lambda )) = \frac{(a^2-1)(b^2-1)}{24} \end{aligned}$$

and that this maximum is achieved by a unique core. P. Johnson combined Ehrhart theory with the polynomial method to prove D. Armstrong’s conjecture that the expected number of boxes in a simultaneous (a, b)-core is

$$\begin{aligned} \mathop {\mathbb {E}}\limits _{\lambda \in {\mathrm {core}}(a,b)}\left( \mathsf{size}(\lambda )\right) = \frac{(a-1)(b-1)(a+b+1)}{24}. \end{aligned}$$

We extend Johnson’s method to compute the variance to be

$$\begin{aligned} \mathop {\mathbb {V}}\limits _{\lambda \in {\mathrm {core}}(a,b)}\left( \mathsf{size}(\lambda )\right) = \frac{ab(a-1)(b-1)(a+b)(a+b+1)}{1440}, \end{aligned}$$

and also prove polynomiality of all moments. By extending the definitions of “simultaneous cores” and “number of boxes” to affine Weyl groups, we give uniform generalizations of all three formulae above to simply laced affine types. We further explain the appearance of the number 24 using the “strange formula” of H. Freudenthal and H. de Vries.

     

Banach and Statistical Cores of Bounded Sequences

In this paper, we are mainly concerned with characterizing matrices that map every bounded sequence into one whose Banach core is a subset of the statistical core of the original sequence.     

Cores in random hypergraphs and Boolean formulas

We describe a technique for determining the thresholds for the appearance of cores in random structures. We use it to determine (i) the threshold for the appearance of a k‐core in a random r‐uniform hypergraph for all r, k ≥ 2, r + k > 4, and (ii) the threshold for the pure literal rule to find a satisfying assignment for a random instance of r‐SAT, r ≥ 3. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

k-convexn-person games and their cores

For any natural numbersk andn, the subclass ofk-convexn-person games is introduced. In casek=n, the subclass consists of the convexn-person games. Ak-convexn-person game is characterized in several ways in terms of the core and certain marginal worth vectors. The marginal worth vectors of a game are described in terms of an upper bound for the core and the corresponding gap function. It is shown that thek-convexity of ann-person gamev is equivalent to

1. all marginal worth vectors ofv belong to the core ofv; or
2. the core ofv is the convex hull of the set consisting of all marginal worth vectors ofv; or
3. the extreme points of the core ofv are exactly the marginal worth vectors ofv.

Examples ofk-convexn-person games are also treated.     

A large population job search game with discrete time

A job search problem is considered, in which there is a large population of jobs initially available and a large population of searchers. The ratio of the number of searchers to the number of jobs is α. Each job has an associated value from a known distribution. At each of N moments the searchers observe a job, whose value comes from the distribution of the values of currently available jobs. If a searcher accepts a job, s/he ceases searching and the job becomes unavailable. Hence, the distribution of the values of available jobs changes over time. Also, the ratio of the number of those still searching to the number of available jobs changes. The model is presented and Nash equilibrium strategies for such problems are considered. By definition, when all the population use a Nash equilibrium strategy, the optimal response of an individual is to use the same strategy. Conditions are given that ensure the existence of a unique Nash equilibrium strategy. Examples are given to illustrate the model and present different approaches to solving such problems.