Null Player Research Articles

Some properties for probabilistic and multinomial (probabilistic) values on cooperative games

We investigate the conditions for the coefficients of probabilistic and multinomial values of cooperative games necessary and/or sufficient in order to satisfy some properties, including marginal contributions, balanced contributions, desirability relation and null player exclusion property. Moreover, a similar analysis is conducted for transfer property of probabilistic power indices on the domain of simple games.

Axiomatic characterizations under players nullification

Many axiomatic characterizations of values for cooperative games invoke axioms which evaluate the consequences of removing an arbitrary player. Balanced contributions (Myerson, 1980) and balanced cycle contributions (Kamijo and Kongo, 2010) are two well-known examples of such axioms. We revisit these characterizations by nullifying a player instead of deleting her/him from a game. The nullification (Béal et al., 2014a) of a player is obtained by transforming a game into a new one in which this player is a null player, i.e. the worth of the coalitions containing this player is now identical to that of the same coalition without this player. The degree with which our results are close to the original results in the literature is connected to the fact that the targeted value satisfies the null player out axiom (Derks and Haller, 1999). We also revisit the potential approach (Hart and Mas-Colell, 1989) similarly.

Modeling values for TU-games using generalized versions of consistency, standardness and the null player property

In the paper we discuss three general properties of values of TU-games: λ-standardness, general probabilistic consistency and some modifications of the null player property. Necessary and sufficient conditions for different families of efficient, linear and symmetric values are given in terms of these properties. It is shown that the results obtained can be used to get new axiomatizations of several classical values of TU-games.

Implementation and axiomatization of discounted Shapley values

In this paper we introduce discounting in the bidding mechanism of Pérez-Castrillo and Wettstein (J Econ Theory 100:274–294, 2001) who implemented the Shapley value for cooperative transferable utility games. This modification of the mechanism yields the corresponding discounted Shapley value as the payoff distribution in every subgame perfect equilibrium. The class of discounted Shapley values contains the Shapley value and equal division solution as its extreme cases. Interestingly, we obtain axiomatizations of each solution in this class by generalizing the null player property (of the Shapley value) and nullifying player property (of the equal division solution) to the so-called \(\delta \)-reducing player property.

Two-step CS-value without null player and null coalition axioms

Two-step CS-value without null player and null coalition axioms

The SD-prenucleolus for TU games

We introduce and characterize a new solution concept for TU games: The Surplus Distributor Prenucleolus. The new solution is a lexicographic value although it is not a weighted prenucleolus. The SD-prenucleolus satisfies core stability, strong aggregate monotonicity, null player out property in the class of balanced games and coalitional monotonicity in the class of monotonic games with veto players. We characterize the solution in terms of balanced collections of sets and we provide a simple formula for computing it in the class of monotonic games with veto players.

Solidarity within a fixed community

We study the consequences of a solidarity property that specifies how a value for cooperative games should respond if some player forfeits his productivity, i.e., becomes a null player. Nullified solidarity states that in this case either all players weakly gain together or all players weakly lose together. Combined with efficiency, the null game property, and a weak fairness property, we obtain a new characterization of the equal division value.

Allocation Rules for Games with Optimistic Aspirations

A game with optimistic aspirations specifies two values for each coalition of players: the first value is the worth that the players in the coalition can guarantee for themselves in the event that they coordinate their actions, and the second value is the amount that the players in the coalition aspire to get under reasonable but very optimistic assumptions about the demands of the players who are not included in the coalition. In this paper, in addition to presenting this model and justifying its relevance, we introduce allocation rules and extend the properties of efficiency, additivity, symmetry, and null player property to this setting. We demonstrate that these four properties are insufficient to find a unique allocation rule and define three properties involving null players and nullifying players that allow the identification of unique allocation rules. The allocation rules we identify are the Midpoint Shapley Value and the Equal Division Rule.

Characterizations of power indices based on null player free winning coalitions

In this paper, we characterize two power indices introduced in [1] using two different modifications of the monotonicity property first stated by [2]. The sets of properties are easily comparable among them and with previous characterizations of other power indices.

Null players, solidarity, and the egalitarian Shapley values

The Shapley value probably is the most eminent single-point solution concept for TU-games. In its standard characterization, the null player property indicates the absence of solidarity among the players. First, we replace the null player property by a new axiom that guarantees null players non-negative payoffs whenever the grand coalition’s worth is non-negative. Second, the equal treatment property is strengthened into desirability. This way, we obtain a new characterization of the class of egalitarian Shapley values, i.e., of convex combinations of the Shapley value and the equal division solution. Within this characterization, additivity and desirability can be replaced by strong differential monotonicity, which translates higher productivity differentials into higher payoff differentials.

A CONSISTENT IMPUTATION GENERATION METHOD FOR LINGUISTIC COOPERATIVE GAMES AND ITS APPLICATION TO RISK AVERSION

Cooperative game theory is very useful to risk aversion problems in economics and management systems. The existing methods only focus on the situation payoffs take the form of numerical values, ones take the form of linguistic labels are seldom discussed. The aim of this study is to propose the consistent imputation for cooperative games under a linguistic environment. To support risk aversion, a 2-tuple linguistic representation is employed to obtain the valid results and avoid the loss of linguistic information. This paper firstly defines some concepts for linguistic cooperative games, such as linguistic imputation, carrier, core and null player. A set of their desirable properties are also discussed. The linguistic Shapley value is then presented based on three axioms. Moreover, the existence and uniqueness of the linguistic Shapley value are discussed in detail. To adjust the linguistic imputation in accordance with the cardinality of a given original linguistic label set, an adjustment algorithm for generating consistent imputation is proposed. Finally, we give the application of linguistic imputation in solving risk aversion problems to illustrate the validity of the consistent imputation generation (CIG) method.

Efficiency and collusion neutrality in cooperative games and networks

In the literature various axiomatizations of the Shapley and Banzhaf value can be found where the only difference is the Shapley value satisfying efficiency and the Banzhaf value satisfying collusion neutrality. Both properties seem to be desirable. In this paper, we show that there is no solution satisfying efficiency, collusion neutrality and the null player property. After that we show that this impossibility is resolved when cooperation among players is restricted by a cycle-free communication network such that only connected coalitions are feasible. In particular, all hierarchical outcomes introduced by Demange (2004) and their convex combinations satisfy these three properties.

OUTSIDE OPTIONS IN PROBABILISTIC COALITION SITUATIONS

In this paper, we introduce an extension of (TU) games with a coalition structure. Taking a situation where all coalitions are already established is not reasonable in order to forecast the reality; there is not only one possible coalition, there are several. We consider situations where coalitions are not established yet and take into account the likelihood of each possible coalition. This leads to a generalized, probabilistic setting for coalition structures. Probabilistic versions of known axioms as efficiency, symmetry or the null player axiom are introduced as well as new probabilistic axioms, the probabilistic influence axioms. The focus is on a generalization of the outside-option-sensitive χ-value in the new setting and an axiomatic characterization of it. The problematic of the impossibility of a direct axiomatization via deterministic pendants is discussed: As an example for a direct characterization we introduce and characterize a probabilistic version of the outside-option-insensitive pendant of the χ-value, the component restricted Shapley value. As another example for an outside-option-sensitive value without direct characterization we introduce a probabilistic version of the Owen value and show that a direct characterization is not possible; we link this to the problem of component decomposability.

The Consensus Value for Games in Partition Function Form

This paper studies a generalization of the consensus value (cf.Ju, Borm and Ruys (2004)) to the class of partition function form games. The concepts and axioms, related to the consensus value, are extended. This value is characterized as the unique function that satisfies efficiency, complete symmetry, the quasi null player property and additivity. By means of the transfer property, a second characterization is provided. Moreover, it is shown that this value satisfies the individual rationality under a certain condition, and well balances the trade-off between coalition effects and externality effects. By modifying the stand-alone reduced game, a recursive formula for the value is established. A further generalization of the consensus value is discussed. Finally, two applications of the consensus value are given: one is for oligopoly games in partition function form and the other is about participation incentives in free-rider situations.

Linear efficient and symmetric values for TU-games: Sharing the joint gain of cooperation

Recently, Hermandez-Lamoneda et al. (2008) and independently Chameni and Andjiga (2008) gave an analytic formulation for all valued solutions to the n-person TU-games that satisfy linearity, efficiency and symmetry axioms. Our main purpose in this paper is to recast the proposed formulation to a more potentially interpretational one. We are focused on an interpretation based on the idea of marginal contribution, a concept already familiar in the Shapley value and the Solidarity value. A general null player axiom is introduced, and it turns out that any valued solution satisfying the three properties is characterized by a null player model.

Two new power indices based on winning coalitions

Deegan and Packel (1979) and Holler (1982) proposed two power indices for simple games: the Deegan–Packel index and the Public Good Index. In the definition of these indices, only minimal winning coalitions are taken into account. Using similar arguments, we define two new power indices. These new indices are defined taking into account only those winning coalitions that do not contain null players. The results obtained with the different power indices are compared by means of two real-world examples taken from the political field.

Fair‐ranking properties of a core selection and the Shapley value

The fair‐ranking principle states that if all the underlying outside data of some coalition are the same in two games, then the relative positions of the players in the coalition should be the same in the two games. We investigate the implications of the principle in the context of transferable utility games. Our first axiom, coalitional fair‐ranking, requires that if the worth of a given coalition changes while the worths of other coalitions remain fixed, then the relative positions of all players in the given coalition should not be affected. We show that for games with at least three players, there is no core selection satisfying coalitional fair‐ranking. If the change is limited to the worth of the grand coalition, then there is a core selection satisfying the requirement. In addition, we present a new axiomatic characterization of the Shapley value on the basis of strong coalitional fair‐ranking: it requires that if a game is added to another game with two symmetric players, then the relative positions of the symmetric players should not be affected. The Shapley value is the only value satisfying strong coalitional fair‐ranking together with efficiency, the null player property and strategic equivalence.

Axiomatization and Implementation of Discounted Shapley Values

We generalize the null player property (satisfied by the Shapley value) and nullifying player property (satisfied by the equal division solution) to the so-called delta-reducing player property, stating that a delta-reducing player (being a player such that any coalition containing this player earns a fraction delta in [0,1] of the worth of that coalition without that player) earns a zero payoff. This property yields the null player property for delta = 1 and the nullifying player property for delta = 0. We show that efficiency, symmetry, linearity and this delta-reducing player property characterizes the corresponding delta-discounted Shapley value. Moreover, we provide a strategic implementation of these solutions where delta is a discount factor that determines the decrease in value to be distributed in the next round after the proposal is rejected and the remaining players (without the proposer) play a new round of bidding.

The Shapley value for arbitrary families of coalitions

We address the problem of finding a suitable definition of a value similar to that of Shapley’s, when the games are defined on a subfamily of coalitions with no structure. We present two frameworks: one based on the familiar efficiency, linearity and null player axioms, and the other on linearity and the behavior on unanimity games. We give several properties and examples in each case, and give necessary and sufficient conditions on the family of coalitions for the approaches to coincide.

Differential marginality, van den Brink fairness, and the Shapley value

We revisit the characterization of the Shapley value by van den Brink (Int J Game Theory, 2001, 30:309–319) via efficiency, the Null player axiom, and some fairness axiom. In particular, we show that this characterization also works within certain classes of TU games, including the classes of superadditive and of convex games. Further, we advocate some differential version of the marginality axiom (Young, Int J Game Theory, 1985, 14: 65–72), which turns out to be equivalent to the van den Brink fairness axiom on large classes of games.