Equal Surplus Division Research Articles

On egalitarian values for cooperative games with level structures

In this paper we extend the equal division and the equal surplus division values for transferable utility cooperative games to the more general setup of transferable utility cooperative games with level structures. In the case of the equal surplus division value we propose three possible extensions, one of which has already been described in the literature. We provide axiomatic characterizations of the values considered, apply them to a particular cost sharing problem and compare them in the framework of such an application.

The Equal Surplus Division Based the Bargaining Power of Priori Coalition in the Coalition Structure Game

The Equal Surplus Division Based the Bargaining Power of Priori Coalition in the Coalition Structure Game

New results on egalitarian values for games with a priori unions

Several extensions of the equal division value and the equal surplus division value to the family of games with a priori unions have been proposed by Alonso-Meijide et al. [On egalitarian values for cooperative games with a priori unions. TOP. 2020;28:672–688]. In this study, we provide new axiomatic characterizations of these values. Furthermore, using the equal surplus division value in two steps, we propose a new coalitional value. The balanced contributions and quotient game properties generate a different modification of the equal surplus division value.

Compromising between the proportional and equal division values

We introduce a family of values for TU-games that offers a compromise between the proportional and equal division values. Each value, called an α-mollified value, is obtained in two steps. First, linear functions are defined that associate a real number to every TU-game. Second, the weight assigned by this function is used to weigh proportionality and equality principles in allocating the worth of the grand coalition. We provide an axiomatic characterization of this family, and show that this family contains the affine combinations of the equal division value and the equal surplus division value as the only linear values. Further, we identify the proportional division value and the affine combinations of the equal division value and the equal surplus division value as those members of this family, that satisfy projection consistency.

Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency

A solution on a set of transferable utility (TU) games satisfies strong aggregate monotonicity (SAM) if every player can improve when the grand coalition becomes richer. It satisfies equal surplus division (ESD) if the solution allows the players to improve equally. We show that the set of weight systems generating weighted prenucleoli that satisfy SAM is open, which implies that for weight systems close enough to any regular system, the weighted prenucleolus satisfies SAM. We also provide a necessary condition for SAM for symmetrically weighted nucleoli. Moreover, we show that the per capita nucleolus on balanced games is characterized by single-valuedness (SIVA), translation covariance (TCOV) and scale covariance (SCOV), and equal adjusted surplus division (EASD), a property that is comparable to but stronger than ESD. These properties together with ESD characterize the per capita prenucleolus on larger sets of TU games. EASD and ESD can be transformed to independence of (adjusted) proportional shifting, and these properties may be generalized for arbitrary weight systems p to I(A)Sp. We show that the p-weighted prenucleolus on the set of balanced TU games is characterized by SIVA, TCOV, SCOV, and IASp and on larger sets by additionally requiring ISp.

Equal loss under separatorization and egalitarian values

We characterize the equal division value, the equal surplus division value, and the class of their affine combinations for TU-games involving equal loss under separatorization. This axiom requires that, if a player becomes a dummifying player (Casajus and Huettner, 2014), then any two other players are equally affected.

The Equal Surplus Division Value for Cooperative Games with a Level Structure

In this paper, we investigate the equal surplus division value for cooperative games with a level structure, which is a sequence of coalition structures becoming coarser and coarser. We propose three axiomatizations of the value. Among them, the first two use different variations of the recent population solidarity axiom, and the third one invokes a special reduced game consistency axiom. Due to the existence of a level structure, our axioms impose special restrictions on the players we focus. We show that our value can be characterized with these axioms and other variations of well-known axioms, such as efficiency, standardness, and quotient game property. Besides characterizing the value, we also connect it to the recent field of ordered tree cooperative games, wherein we find that the iterative value can be viewed as a special case of our value.

Similarities in axiomatizations: equal surplus division value and first-price auctions

We compare axiomatizations between a value for cooperative games with transferable utilities (TU games), and a rule for auctions. The equal surplus division value on the set of zero-monotonic TU games is characterized by the following: individual rationality, Pareto efficiency, and equal effect of players’ nullification on others. Meanwhile, first-price auctions, on the general preference domain, are characterized by individual rationality, envy-freeness, and weak equal effect of buyers’ nullification on others. Here, envy-freeness implies Pareto efficiency in the model of auctions. Given the agents’ general preferences in the auction model, the characteristic of a weak equal effect of buyers’ nullification on others weakens the requirement of equal effect of players’ nullification on others. Although the two models are different, the corresponding axioms in both models require conditions corresponding to each other. In particular, individual rationality requires voluntary participation of agents, Pareto efficiency (or its stronger axiom of envy-freeness in the model of auctions) requires outcomes with no waste, and (weak) equal effect of players’ (buyers’) nullification on others requires equal treatment of agents when an agent is nullified. Therefore, in terms of axiomatizations, we can similarly interpret the equal surplus division value and first-price auctions.

On egalitarian values for cooperative games with a priori unions

In this paper, we extend the equal division and the equal surplus division values for transferable utility cooperative games to the more general setup of transferable utility cooperative games with a priori unions. In the case of the equal surplus division value we propose three possible extensions. We provide axiomatic characterizations of the new values. Furthermore, we apply the proposed modifications to a particular motivating example and compare the numerical results with those obtained with the original values.

Compromising between the Proportional and Equal Division Values: Axiomatization, Consistency and Implementation

We introduce a family of values for TU-games that offers a compromise between the proportional and equal division values. Each value, called an alpha-mollified value, is obtained in two steps. First, a linear function with respect to the worths of all coalitions is defined which associates a real number to every TU-game. Second, the weight assigned by this function is used to weigh proportionality and equality principles in allocating the worth of the grand coalition. We provide an axiomatic characterization of this family, and show that this family contains the affine combinations of the equal division value and the equal surplus division value as the only linear values. Further, we identify the proportional division value and the affine combinations of the equal division value and the equal surplus division value as those members of this family, that satisfy projection consistency. Besides, we provide a procedural implementation of each single value in this family.

Equal Loss under Separatorization and Egalitarian Values

We characterize the equal division value, the equal surplus division value, and the class of their a ne combinations for TU-games involving equal loss under separatorization. This axiom requires that, if a player becomes a dummifying player (Casajus and Huettner, in Economics Letters 122(2): 167-169, 2014), then any two other players are equally affected.

Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions

Some sub-coalitions can not be formed or fail to satisfy the superadditivity in many realistic cooperative transferable utility (TU) games, which particularly exists in the logistics service industry. To exploit some novel solutions for addressing these TU games, we firstly propose again the equal surplus division value based on the least square method and players’ common excess vector, and then introduce the weighted equal surplus division value. Both of them belong to the family of the least square values. Inspired by the fact that many TU games base the profit distribution strategies not only on the egalitarian principle but also on the utility principle, the equal contribution division value and the weighted equal contribution division value based on the least square method and players’ contribution excess vector are spontaneously generated. An algorithm is described to make the four solutions proposed in this paper satisfy the property of individual rationality. Finally, to show the advantages, the practicability and the rationality of the four solutions, a practical example about the profit distribution strategy of a logistics enterprise coalition is illustrated and the contrastive analysis among them is given.

Coalitional surplus desirability and the equal surplus division value

In this paper, we propose axiomatic results with two variations of the recent coalitional desirability axiom (Beal et al., 2019), respectively named as coalitional surplus desirability and average coalitional surplus desirability. Particularly, we show that while the first variation is incompatible with the well-known efficiency axiom, the second variation is characteristic for the equal surplus division value, together with the well-known efficiency and additivity axioms.

Relationally equal treatment of equals and affine combinations of values for TU games

We axiomatize the set of affine combinations between the Shapley value, the equal surplus division value, and the equal division value in cooperative games with transferable utilities. The set is characterized by efficiency, linearity, the balanced contributions property for equal contributors and outsiders, and the differential null player out property. The balanced contributions property for equal contributors and outsiders requires the balance of contributions between two players who contribute the same amount to the grand coalition and whose singleton coalitions earn the same worth. The differential null player out property requires that an elimination of a null player affects the other players identically. These two relational axioms are obtained by investigating Myerson’s (Int J Game Theory 9:169–182, 1980) balanced contributions property and Derks and Haller’s (Int Game Theory Rev 1:301–314, 1999) null player out property, respectively, from the perspective of a principle of Aristotle’s distributive justice, whereby “equals should be treated equally”.

A new axiomatization of a class of equal surplus division values for TU games

In this paper, we propose a variation of weak covariance named as non-singleton covariance, requiring that changing the worth of a non-singleton coalition in a TU game affects the payoffs of all players equally. We establish that this covariance is characteristic for the convex combinations of the equal division value and the equal surplus division value, together with efficiency and a one-parameterized axiom treating a particular kind of players specially. As special cases, parallel axiomatizations of the two values are also provided.

Effects of Players’ Nullification and Equal (Surplus) Division Values

We provide axiomatic characterizations of the solutions of transferable utility (TU) games on the fixed player set, where at least three players exist. We introduce two axioms on players’ nullification. One axiom requires that the difference between the effect of a player’s nullification on the nullified player and on the others is relatively constant if all but one players are null players. Another axiom requires that a player’s nullification affects equally all of the other players. These two axioms characterize the set of all affine combinations of the equal surplus division and equal division values, together with the two basic axioms of efficiency and null game. By replacing the first axiom on players’ nullification with appropriate monotonicity axioms, we narrow down the solutions to the set of all convex combinations of the two values, or to each of the two values.

Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency

A solution on a set of transferable utility (TU) games satisfies strong aggregate monotonicity (SAM) if every player can improve when the grand coalition becomes richer. It satisfies equal surplus division (ESD) if the solution allows the players to improve equally. We show that the set of weight systems generating weighted prenucleoli that satisfy SAM is open which implies that for weight systems close enough to any regular system the weighted prenucleolus satisfies SAM. We also provide a necessary condition for SAM for symmetrically weighted nucleoli. Moreover, we show that the per capita nucleolus on balanced games is characterized by single-valuedness (SIVA), translation and scale covariance (COV), and equal adjusted surplus division (EASD), a property that is comparable but stronger than ESD. These properties together with ESD characterize the per capita prenucleolus on larger sets of TU games. EASD and ESD can be transformed to independence of (adjusted) proportional shifting and these properties may be generalized for arbitrary weight systems p to I(A)Sp. We show that the p-weighted prenucleolus on the set of balanced TU games is characterized by SIVA, COV, and IASp; and on larger sets by additionally requiring ISp.

A strategic implementation of the sequential equal surplus division rule for digraph cooperative games

We provide a strategic implementation of the sequential equal surplus division rule (Beal et al. in Theory Decis 79:251–283, 2015). Precisely, we design a non-cooperative mechanism of which the unique subgame perfect equilibrium payoffs correspond to the sequential equal surplus division outcome of a superadditive rooted tree TU-game. This mechanism borrowed from the bidding mechanism designed by Perez-Castrillo and Wettstein (J Econ Theory 100:274–294, 2001), but takes into account the direction of the edges connecting any two players in the rooted tree, which reflects some dominance relation between them. Our proofs rely on interesting properties that we provide for a general class of bidding mechanisms.

An Owen-type value for games with two-level communication structure

We introduce an Owen-type value for games with two-level communication structure, which is a structure where the players are partitioned into a coalition structure such that there exists restricted communication between as well as within the a priori unions of the coalition structure. Both types of communication restrictions are modeled by an undirected communication graph. We provide an axiomatic characterization of the new value using an efficiency, two types of fairness (one for each level of the communication structure), and a new type of axiom, called fair distribution of the surplus within unions, which compares the effect of replacing a union in the coalition structure by one of its maximal connected components on the payoffs of these components. The relevance of the new value is illustrated by an example. We also show that for particular two-level communication structures the Owen value and the Aumann–Dreze value for games with coalition structure, the Myerson value for communication graph games, and the equal surplus division solution appear as special cases of this new value.

Axioms of invariance for TU-games

We introduce new axioms for the class of all TU-games with a fixed but arbitrary player set. These axioms require either invariance of an allocation rule or invariance of the payoff assigned by an allocation rule to a specified player in two related TU-games. Combinations of these new axioms are used to characterize the Shapley value, the Equal Division rule, and the Equal Surplus Division rule. The classical axioms of Efficiency, Anonymity, Equal treatment of equals, Additivity and Linearity are not used.