A characterization of convex games by means of bargaining sets

The aim of the paper is to characterize the classical convexity notion for cooperative TU games by means of the Mas-Colell and the Davis–Maschler bargaining sets. A new set of payoff vectors is introduced and analyzed: the max-Weber set. This set is defined as the convex hull of the max-marginal worth vectors. The characterizations of convexity are reached by comparing the classical Weber set, the max-Weber set and a selected bargaining set.     

The not-quite non-atomic game: Non-emptiness of the core in large production games

We show that, in cooperative production games, when the production functions are not concave, the core may well be empty. However, as the number of players increases (subject to some regularity conditions), the relative deficit obtained by using concavified functions decreases to zero. Furthermore, differentiability of the functions will cause the absolute deficit to go to zero.     

On the core of cooperative queueing games

We consider a class of cooperative games for managing several canonical queueing systems. When cooperating parties invest optimally in common capacity or choose the optimal amount of demand to serve, cooperation leads to “single-attribute” games whose characteristic function is embedded in a one-dimensional function. We show that when and only when the latter function is elastic will all embedded games have a non-empty core, and the core contains a population monotonic allocation. We present sufficient conditions for this property to be satisfied. Our analysis reveals that in most Erlang B and Erlang C queueing systems, the games under our consideration have a non-empty core, but there are exceptions, which we illustrate through a counterexample.     

Partially ordered cooperative games: extended core and Shapley value

In this paper we analyze cooperative games whose characteristic function takes values in a partially ordered linear space. Thus, the classical solution concepts in cooperative game theory have to be revisited and redefined: the core concept, Shapley–Bondareva theorem and the Shapley value are extended for this class of games. The classes of standard, vector-valued and stochastic cooperative games among others are particular cases of this general theory. The research of the authors is partially supported by Spanish DGICYT grant numbers MTM2004-0909, HA2003-0121, HI2003-0189, MTM2007-67433-C02-01, P06-FQM-01366.     

On the bargaining set, kernel and core of superadditive games

We prove that for superadditive games a necessary and sufficient condition for the bargaining set to coincide with the core is that the monotonic cover of the excess game induced by a payoff be balanced for each imputation in the bargaining set. We present some new results obtained by verifying this condition for specific classes of games. For N-zero-monotonic games we show that the same condition required at each kernel element is also necessary and sufficient for the kernel to be contained in the core. We also give examples showing that to maintain these characterizations, the respective assumptions on the games cannot be lifted. Received: March 1998/Revised version: December 1998     

Multi-choice clan games and their core

In this paper, we consider market situations with two corners. One corner consists of a group of powerful agents with yes-or-no choices and clan behavior. The other corner consists of non-powerful agents with multi-choices regarding the extent at which cooperation with the clan can be achieved. Multi-choice clan games arise from such market situations. The focus is on the analysis of the core of multi-choice clan games. Several characterizations of multi-choice clan games by the shape of the core are given, and the connection between the convexity of a multi-choice clan game and the stability of its core is studied.      

A test of the core solution in finite strategy non-sidepayment games

This paper reports a test of the core solution in cooperative non-sidepayment games where players have finite strategy sets. Two laboratory experiments were conducted with three-person and four-person games; in both experiments, the core solution was tested competitively against the von Neumann-Morgenstern stable set and the imputation set. Predictions from these solution concepts were computed under parameters of α-effectiveness and strict preference. Results show that the frequency of outcomes falling in core is substantially higher than that observed in previous experiments (most of which involve sidepayment games). In addition, goodness-of-fit tests show that the core solution predicts the observed outcomes more accurately than do the stable set or the imputation set.     

拟凸对策的解

本文先引入拟凸对策的概念作为凸对策的推广,然后研究这种对策的各种解的性质。我们主要证得,当局中人数小于6或者对策的复盖严格凸时,谈判集与核心重合,核是单点集。另外,存在一个6人拟凸对策,其谈判集与核心不同。     

A core of voting games with improved foresight

This paper considers voting situations in which the vote takes place iteratively. If a coalition replaces the status quo a with a contestant b, then b becomes the new status quo, and the vote goes on until a candidate is reached that no winning coalition is willing to replace. It is well known that the core, that is, the set of undominated alternatives, may be empty. To alleviate this problem, Rubinstein [Rubinstein, A., 1980. Stability of decision systems under majority rule. Journal of Economic Theory 23, 150–159] assumes that voters look forward one vote before deciding to replace an alternative by a new one. They will not do so if the new status quo is going to be replaced by a third that is less interesting than the first. The stability set, that is, the set of undominated alternatives under this behavior, is always non-empty when preferences are strict. However, this is not necessarily the case when voters’ indifference is allowed. Le Breton and Salles [Le Breton, M., Salles, M., 1990. The stability set of voting games: Classification and generecity results. International Journal of Game Theory 19, 111–127], Li [Li, S., 1993. Stability of voting games. Social Choice and Welfare 10, 51–56] and Martin [Martin, M., 1998. Quota games and stability set of order d. Economic Letters 59, 145–151] extend the sophistication of the voters by having them look d votes forward along the iterative process. For d sufficiently large, the resulting set of undominated alternatives is always non-empty even if indifference is allowed. We show that it may be unduly large. Next, by assuming that other voters along a chain of votes are also rational, that is, they also look forward to make sure that the votes taking place later on will not lead to a worst issue for them, we are able to reduce the size of this set while insuring its non-emptiness. Finally, we show that a vote with sufficient foresight satisfies a no-regret property, contrarily to the classical core and the stability set.     

On the core,the Weber set and convexity in games with a priori unions

This paper deals with the concepts of core and Weber set with a priori unions à la Owen. As far as we know, the Owen approach to games with a priori unions has never been studied from the coalitional stability point of view. Thus we introduce the coalitional core and coalitional Weber set and characterize the class of convex games with a priori unions by means of the relationships between both solution concepts.     

Linear values for interior operator games

Interior operator games were introduced by Bilbao et al. (2005) as additive games restricted by antimatroids. In that paper several interesting cooperative games were shown as examples of interior operator games. The antimatroid is a known combinatorial structure which represents, in the game theory context, a dependence system among the players. The aim of this paper is to study a family of values which are linear functions and satisfy reasonable conditions for interior operator games. Two classes of these values are considered assuming particular properties.     

Coalitional games: Monotonicity and core

We characterize a monotonic core solution defined on the class of veto balanced games. We also discuss what restricted versions of monotonicity are possible when selecting core allocations. We introduce a family of monotonic core solutions for veto balanced games and we show that, in general, the per capita nucleolus is not monotonic.     

The core and the bargaining set in glove-market games

In a glove-market game, the worth of a coalition is defined as the minimum, over all commodities in the market, of the total quantity that the coalition owns of each commodity. We identify a subclass of these games for which the core and the bargaining set coincide with the set of competitive equilibrium outcomes. We present examples showing that these solution concepts differ outside that subclass. We also illustrate a peculiar behavior of the bargaining set with respect to replication of a glove-market. These examples provide a simple economic setting in which the merits of the various solution concepts may be discussed and compared.Correspondence with Michael Maschler is gratefully acknowledged.This authors work was partly done while visiting the Department of Mathematics, Massachusetts Institute of Technology. Research supported by the Fund for the Promotion of Research at the Technion.     

Minimal large sets for cooperative games

We analyze the concept of large set for a coalitional game v introduced by Martínez-de-Albéniz and Rafels (Int. J. Game Theory 33(1):107–114, 2004). We give some examples and identify some of these sets. The existence of such sets for any game is proved, and several properties of largeness are provided. We focus on the minimality of such sets and prove its existence using Zorn’s lemma. Institutional support from research grants (Generalitat de Catalunya) 2005SGR00984 and (Spanish Government and FEDER) SEJ2005-02443/ECON is gratefully acknowledged, and the support of the Barcelona Economics Program of CREA.     

The fuzzy core in games with fuzzy coalitions

In this paper, the fuzzy core of games with fuzzy coalition is proposed, which can be regarded as the generalization of crisp core. The fuzzy core is based on the assumption that the total worth of a fuzzy coalition will be allocated to the players whose participation rate is larger than zero. The nonempty condition of the fuzzy core is given based on the fuzzy convexity. Three kinds of special fuzzy cores in games with fuzzy coalition are studied, and the explicit fuzzy core represented by the crisp core is also given. Because the fuzzy Shapley value had been proposed as a kind of solution for the fuzzy games, the relationship between fuzzy core and the fuzzy Shapley function is also shown. Surprisingly, the relationship between fuzzy core and the fuzzy Shapley value does coincide, as in the classical case.     

Weighted allocation rules for standard fixed tree games

In this paper we consider standard fixed tree games, for which each vertex unequal to the root is inhabited by exactly one player. We present two weighted allocation rules, the weighted down-home allocation and the weighted neighbour-home allocation, both inspired by the painting story in Maschler et al. (1995) . We show, in a constructive way, that the core equals both the set of weighted down-home allocations and the set of weighted neighbour allocations. Since every weighted down-home allocation specifies a weighted Shapley value (Kalai and Samet (1988)) in a natural way, and vice versa, our results provide an alternative proof of the fact that the core of a standard fixed tree game equals the set of weighted Shapley values. The class of weighted neighbour allocations is a generalization of the nucleolus, in the sense that the latter is in this class as the special member where players have all equal weights.