The construction and characterization of egalitarian solutions for multi-choice NTU games

In this article we construct a procedure to define the egalitarian solutions in the context of multi-choice non-transferable utility (NTU) games. Also, we show that in the presence of other weak axioms the egalitarian solutions are the only monotonic ones.     

An axiomatization of the egalitarian solutions

The egalitarian solutions for general cooperative games which were defined and axiomatized by Kalai and Samet, are compared to the Harsanyi solution. It is shown that axioms used by Hart to characterize the Harsanyi solution can be used to characterize the (symmetric) egalitarian solution. The only changes needed are the omission of the scale covariance axiom and the inclusion, in the domain of the solution, of games which lack a certain smoothness requirement.     

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.      

Bargaining sets and the core in partitioning games

Partitioning games are useful on two counts: first, in modeling situations with restricted cooperative possibilities between the agents; second, as a general framework for many unrestricted cooperative games generated by combinatorial optimization problems.We show that the family of partitioning games defined on a fixed basic collection is closed under the strategic equivalence of games, and also for taking the monotonic cover of games. Based on these properties we establish the coincidence of the Mas-Colell, the classical, the semireactive, and the reactive bargaining setswith the core for interesting balanced subclasses of partitioning games, including assignment games, tree-restricted superadditive games, and simple network games. Prepared during the author’s Bolyai János Research Fellowship. Also supported by OTKA grant T46194.     

多选择NTU对策核心的一个非空条件及公理化

提出了\pi-均衡多选择NTU对策的概念,证明了\pi-均衡多选择NTU对策的核心非空, 定义了多选择NTU对策的非水平性质和缩减对策,给出了相容性和逆相容性等概念. 用个体合理性、单人合理性、相容性和逆相容性对非水平多选择NTU对策的核心进行了公理化.     

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.     

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.     

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.     

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.     

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.     

Solutions for some bargaining games under the Harsanyi-Selten solution theory,part II: Analysis of specific bargaining games

Part II of the paper (for Part I see Harsanyi (1982)) describes the actual solutions the Harsanyi-Selten solution theory provides for some important classes of bargaining games, such as unanimity games; trade between one seller and several potential buyers; and two-person bargaining games with incomplete information on one side or on both sides. It also discusses some concepts and theorems useful in computing the solution; and explains how our concept of risk dominance enables us to analyze game situations in terms of some intuitively very compelling probabilistic (subjective-probability) considerations disallowed by classical game theory.     

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.     

The core and related solution concepts for infinite assignment games

Assignment problems where both sets of agents that have to be matched are countably infinite, the so-called infinite assignment problems, are studied as well as the related cooperative assignment games. Further, several solution concepts for these assignment games are studied. The first one is the utopia payoff for games with an infinite value. In this solution each player receives the maximal amount he can think of with respect to the underlying assignment problem. This solution is contained in the core of the game. Second, we study two solutions for assignment games with a finite value. Our main result is the existence of core-elements of these games, although they are hard to calculate. Therefore another solution, the f-strong ε-core is studied. This particular solution takes into account that due to organisational limitations it seems reasonable that only finite groups of agents will eventually protest against unfair proposals of profit distributions. The f-strong ε-core is shown to be nonempty. These authors’ research is partially supported by the Generalitat Valenciana (Grant number GV-CTIDIA-2002-32) and by the Government of Spain (through a joint research grant Universidad Miguel Hernández — Università degli Studi di Genova HI2002-0032).     

Consistent extensions and subsolutions of the core of multi-choice NTU games

We extend the reduced games introduced by Davis and Maschler (Naval Res Log Q 12:223–259, 1965) and Moulin (J Econ Theory 36:120–148, 1985) to multi-choice non-transferable utility games and define two related properties of consistency. We also show that the core proposed by Hwang and Li (Math Methods Oper Res 61:33–40, 2005) violates these two consistency properties. In order to investigate how seriously it violates these two consistency properties, we provide consistent extensions and consistent subsolutions of the core.     

Some results related to the Harsanyi-Shapley solution

The main result of the paper is to characterize the Harsanyi-Shapley solution on a specific class of NTU games by reduced game property. Final version July 2001     

Axiomatizing the Harsanyi solution,the symmetric egalitarian solution and the consistent solution for NTU-games

The validity of the axiomatization of the Harsanyi solution for NTU-games by Hart (1985) is shown to depend on the regularity conditions imposed on games. Following this observation, we propose two related axiomatic characterizations, one of the symmetric egalitarian solution (Kalai and Samet, 1985) and one of the consistent solution (Maschler and Owen, 1992). The three axiomatic results are studied, evaluated and compared in detail.Revised October 2004We thank an anonymous referee and an associate editor for their helpful comments. Geoffroy de Clippel also thanks Professors Sergiu Hart, Jean-François Mertens and Enrico Minelli. Horst Zank thanks the Dutch Science Foundation NWO and the British Council for support under the UK-Netherlands Partnership Programme in Science (PPS 706). The usual disclaimer applies.     

On the core of network synthesis games

We use polynomial formulations to show that several rational and discrete network synthesis games, including the minimum cost spanning tree game, satisfy the assumptions of Owen's linear production game model. We also discuss computational issues related to finding and recognizing core points for these classes of games.     

Egalitarian solutions in the core

In this paper we define the Lorenz stable set, a subset of the core consisting of the allocations that are not Lorenz dominated by any other allocation of the core. We introduce the leximin stable allocation, which is derived from the application of the Rawlsian criterion on the core. We also define and axiomatize the egalitarian core, a set of core allocations for which no transfer from a rich player to a poor player is possible without violating the core restrictions. We find an inclusive relation of the leximin stable allocation and of the Lorenz stable set into the egalitarian core. Received: October 1999/Final version: July 2001     

The core and the Weber set of games on augmenting systems

This paper deals with cooperative games in which only certain coalitions are allowed to form. There have been previous models developed to confront the problem of unallowable coalitions. Games restricted by a communication graph were introduced by Myerson and Owen. In their model, the feasible coalitions are those that induce connected subgraphs. Another type of model is introduced in Gilles, Owen and van den Brink. In their model, the possibilities of coalition formation are determined by the positions of the players in a so-called permission structure. Faigle proposed a general model for cooperative games defined on lattice structures. In this paper, the restrictions to the cooperation are given by a combinatorial structure called augmenting system which generalizes antimatroid structure and the system of connected subgraphs of a graph. In this framework, the core and the Weber set of games on augmenting systems are introduced and it is proved that monotone convex games have a non-empty core. Moreover, we obtain a characterization of the convexity of these games in terms of the core of the game and the Weber set of the extended game.