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This paper deals with a constrained egalitarian solution for convex multi-choice games named the d value. It is proved that the d value of a convex multi-choice game belongs to the precore, Lorenz dominates each other element of the precore and possesses a population monotonicity property regarding players’ participation levels. Furthermore, an axiomatic characterization is given where a specific consistency property plays an important role.

Uncertainty accompanies almost every situation in real world and it influences our decisions. In sequencing situations it may affect parameters used to determine an optimal order in the queue, and consequently the decision of whether (or not) to rearrange the queue by sharing the realized cost savings. This paper extends the analysis of one-machine sequencing situations and their related cooperative games to a setting with interval data, i.e. when the agents’ costs per unit of time and/or processing time in the system lie in intervals of real numbers obtained by forecasting their values. The question addressed here is: How to determine an optimal order (if the initial order in the queue is not so) and which approach should be used to motivate the agents to adopt the optimal order? This question is an important one that deserves attention both in theory and practice.

This paper introduces and studies a class of cooperative interval games suitable to model market situations with two corners where players face interval uncertainty regarding the outcome of cooperation. In one corner there is a powerful player called the big boss; the other corner contains players that need the big boss to benefit from cooperation. Various characterizations of big boss interval games are given. The interval core of a big boss interval game is explicitly described, bi-monotonic allocation schemes using interval core elements are introduced, and it is shown that each element of the interval core of a big boss interval game is extendable to such a scheme. Two value-type interval solution concepts are defined on the class of big boss interval games which generate for each such game the same interval core allocation which is extendable to a bi-monotonic interval allocation scheme.

Uncertainty accompanies almost every situation in our lives and it influences our decisions. On many occasions uncertainty is so severe that we can only predict some upper and lower bounds for the outcome of our (collaborative) actions, i.e., payoffs lie in some intervals. Cooperative interval games have been proved useful for solving reward/cost sharing problems in situations with interval data in a cooperative environment. In this paper we propose two procedures for cooperative interval games. Both transform an interval allocation, i.e., a payoff vector whose components are compact intervals of real numbers, into a payoff vector (whose components are real numbers) when the value of the grand coalition becomes known (at once or in multiple stages). The research question addressed here is: How to determine for each player his/her/its payoff generated by cooperation within the grand coalition – in the promised range of payoffs to establish such cooperation – after the uncertainty on the payoff for the grand coalition is resolved? This question is an important one that deserves attention both in the literature and in game practice.

Uncertainty is a daily presence in the real world. It affects our decision making and may have influence on cooperation. Often uncertainty is so severe that we can only predict some upper and lower bounds for the outcome of our actions, i.e., payoffs lie in some intervals. A suitable game theoretic model to support decision making in collaborative situations with interval data is that of cooperative interval games. Solution concepts that associate with each cooperative interval game sets of interval allocations with appealing properties provide a natural way to capture the uncertainty of coalition values into the players' payoffs. This paper extends interval-type core solutions for cooperative interval games by discussing the set of undominated core solutions which consists of the interval nondominated core, the square interval dominance core, and the interval dominance core. The interval nondominated core is introduced and it is shown that it coincides with the interval core. A straightforward consequence of this result is the convexity of the interval nondominated core of any cooperative interval game. A necessary and sufficient condition for the convexity of the square interval dominance core of a cooperative interval game is also provided.

We give a non-cooperative and an axiomatic characterization of the Average Lexicographic value (AL-value) on the class of balanced games. The AL-value is a single-valued solution for balanced TU-games. It is defined as the average of lexicographic maximum of the core of the game with respect to all orders on the player set, and it can be seen as a core selection based on the priority orders on the players. In both of our characterizations of the AL-value, a consistency property which we call average consistency plays an important role, and the property is obtained by the consistency property à la Davis and Maschler of the lexicographic vectors with respect to any order of players.

This paper focuses on new characterizations of convex multi-choice games using the notions of exactness and superadditivity. Furthermore, level-increase monotonic allocation schemes (limas) on the class of convex multi-choice games are introduced and studied. It turns out that each element of the Weber set of such a game is extendable to a limas, and the (total) Shapley value for multi-choice games generates a limas for each convex multi-choice game.

The Shapley value, one of the most widespread concepts in operations Research applications of cooperative game theory, was defined and axiomatically characterized in different game-theoretic models. Recently much research work has been done in order to extend OR models and methods, in particular cooperative game theory, for situations with interval data. This paper focuses on the Shapley value for cooperative games where the set of players is finite and the coalition values are compact intervals of real numbers. The interval Shapley value is characterized with the aid of the properties of additivity, efficiency, symmetry and dummy player, which are straightforward generalizations of the corresponding properties in the classical cooperative game theory.

This paper deals with cooperative fuzzy games arising from two economic situations: a linear production situation and a labor-time availability situation. In both economic situations we derive cooperative fuzzy games and study conditions for nonemptiness of Aubin's core. Also, relations between reasona-ble (stable) payoff schemes for the economic situations and core elements of the related cooperative fuzzy games are studied

In this paper, the egalitarian solution for convex cooperative fuzzy games is introduced. The classical Dutta–Ray algorithm for finding the constrained egalitarian solution for convex crisp games is adjusted to provide the egalitarian solution of a convex fuzzy game. For arbitrary fuzzy games, the equal division core is introduced. It turns out that both the equal division core and the egalitarian solution of a convex fuzzy game coincide with the corresponding equal division core and the constrained egalitarian solution, respectively, of the related crisp game.

This chapter examines several monotonicity properties of value-type interval solutions on the class of convex interval games and focuses on the Dutta–Ray (DR) solution for such games. Well-known properties for the classical DR solution are extended to the interval setting. In particular, it is proved that the interval DR solution of a convex interval game belongs to the interval core of that game and Lorenz dominates each other interval core element. Consistency properties of the interval DR solution in the sense of Davis–Maschler and of Hart–Mas-Colell are verified. An axiomatic characterization of the interval DR solution on the class of convex interval games with the help of bilateral Hart–Mas-Colell consistency and the constrained egalitarianism for two-person interval games is given.

This paper examines several monotonicity properties of value-type interval solutions on the class of convex interval games and focuses on the Dutta-Ray (DR) solution for such games. Well known properties for the classical DR solution are extended to the interval setting. In particular, it is proved that the interval DR solution of a convex interval game belongs to the interval core of that game and Lorenz dominates each other interval core element. Consistency properties of the interval DR solution in the sense of Davis-Maschler and of Hart-Mas-Colell are verified. An axiomatic characterization of the interval DR solution on the class of convex interval games with the help of bilateral Hart-Mas-Colell consistency and the constrained egalitarianism for two-person interval games is given.

In this paper, convex interval games are introduced and some characterizations are given. Some economic situations leading to convex interval games are discussed. The Weber set and the Shapley value are defined for a suitable class of interval games and their relations with the interval core for convex interval games are established. A square operator is introduced which allows us to obtain interval solutions starting from classical cooperative game theory solutions. It turns out that on the class of convex interval games the square Weber set coincides with the interval core.

This paper focuses on new characterizations of convex interval games using the notions of exactness and superadditivity. We also relate big boss interval games with concave interval games and obtain characterizations of big boss interval games in terms of exactness and subadditivity.

Convex interval games are introduced and characterizations are given. Some economic situations leading to convex interval games are discussed. The Weber set and the Shapley value are defined for a suitable class of interval games and their relations with the interval core for convex interval games are established. The notion of population monotonic interval allocation scheme (pmias) in the interval setting is introduced and it is proved that each element of the Weber set of a convex interval game is extendable to such a pmias. A square operator is introduced which allows us to obtain interval solutions starting from the corresponding classical cooperative game theory solutions. It turns out that on the class of convex interval games the square Weber set coincides with the interval core.

A new way is presented to define for minimum cost spanning tree (mcst) games the irreducible core, which is introduced by Bird in 1976. The Bird core correspondence turns out to have interesting monotonicity and additivity properties and each stable cost monotonic allocation rule for mcst-problems is a selection of the Bird core correspondence. Using the additivity property an axiomatic characterization of the Bird core correspondence is obtained.

In this paper we provide characterizations of convex games and total clan games by using properties of their corresponding marginal games. We show that a dualize and restrict procedure transforms total clan games with zero worth for the clan into monotonic convex games. Furthermore, each monotonic convex game generates a total clan game with zero worth for the clan by a dualize and extend procedure. These procedures are also useful for relating core elements and elements of the Weber set of the corresponding games.

A new way is presented to define for minimum cost spanning tree (mcst-) games the irreducible core, which is introduced by Bird in 1976. The Bird core correspondence turns out to have interesting monotonicity and additivity properties and each stable cost monotonic allocation rule for mcst-problems is a selection of the Bird core correspondence. Using the additivity property an axiomatic characterization of the Bird core correspondence is obtained.

We introduce the class of Obligation Rules for Minimum Cost Spanning Tree Situations. The main result of this paper is that such rules are cost monotonic and induce also population monotonic allocation schemes. Another characteristic of Obligation Rules is that they assign to a minimum cost spanning tree situation a vector of cost contributions which can be obtained as product of a double stochastic matrix with the cost vector of edges in the optimal tree provided by the Kruskal algorithm. It turns out that the Potters value (P-value) is an element of this class.

In this paper the equal split-off set is introduced as a new solution concept for cooperative games. This solution is based on egalitarian considerations and it turns out that for superadditive games the equal split-off set is a subset of the equal division core. Moreover, the proposed solution is single valued on the class of convex games and it coincides with the Dutta-Ray constrained egalitarian solution.

In this paper we consider information collecting (IC) situations where an action taker in an uncertain situation can improve his action choices by gathering information from some players who are more informed about the situation. Then the problem of sharing the gains when cooperating with informants is tackled by constructing an appropriate game, the IC?game corresponding to the IC?situation. It turns out that the cone of IC?games, given a fixed set of players, coincides with the cone of 0?normalized monotonic games with a veto player. Also special classes of convex IC?games and big boss IC?games are considered, for which more is known about the solution concepts.

In this paper we study convex games with an infinite countable set of agents and provide characterizations of this class of games. To do so, and in order to overcome some shortcomings related to the difficulty of dealing with infinite orderings, we need to use a continuity property. Infinite sequencing situations where the number of jobs is infinite countable can be related to convex cooperative TU games. It is shown that some allocations turn out to be extreme points of the core of an infinite sequencing game.