On a class of solidarity values

We suggest a new one-parameter family of solidarity values for TU-games. The members of this class are distinguished by the type of player whose removal from a game does not a¤ect the remaining players payo¤s. While the Shapley value and the equal division value are the boundary members of this family, the solidarity value is its center. With exception of the Shapley value, all members of this family are asymptotically equivalent to the equal division value in the sense of Radzik (2012, Math Soc Sci, forthcoming). Keywords: solidarity, null player out, desirability, positivity, asymptotic equivalence 2010 MSC: 91A12 1. Introduction A cooperative game with transferable utility (TU-game) consists of a non-empty and …nite set of players, N; and a coalition function, v : 2 ! R; v (;) = 0; which for each coalition S N describes the worth v (S) that can be generated by its members. Assuming that the grand coalition eventually is formed, the question arises how to distribute the grand coalition’s worth, v (N). The most prominent single-point solution concept that answers this question probably is the Shapley value (Shapley, 1953). While almost all modern societies reveal some degree of solidarity, the Shapley value does not allow for solidarity among the players. Unproductive players (null players) are not only assigned zero payo¤s, but may even leave the game without a¤ecting the other players’ payo¤s. Moreover, a player’s payo¤ depends only on his own marginal contributions. In contrast, the equal division value distributes the worth generated by the grand coalition equally among the players. This can be interpreted as an extreme kind of solidarity. Clearly, the payo¤s according to the equal division value are almost insensitive to a player’s own marginal contributions. Several attempts have been made to provide solution concepts that live in between these two extremes. Sprumont (1990) suggests an example of a population monotonic allocation We are grateful to René van den Brink for valuable comments on this paper. Email addresses: [email protected] (André Casajus), [email protected] (Frank Huettner) URL: www.casajus.de (André Casajus), www.frankhuettner.de (Frank Huettner) Preprint submitted to European Journal of Operational Research February 24, 2013 scheme, later on characterized by Nowak and Radzik (1994) as the solidarity value. Nowak and Radzik (1996) study the convex mixtures of the Shapley value and their solidarity value. The class of egalitarian Shapley values, i.e., convex mixtures of the Shapley value and the equal division value is suggested by Joosten (1996). Joosten (1996) and Driessen and Radzik (2002) come up with the discounted Shapley values. The whole class of e¢ cient, linear, and symmetric values is described by Chameni-Nembua (2012), for example. Kamijo and Kongo (2010, 2012) show that the di¤erence between the Shapley value, the equal division value, and the solidarity value that can be pinpointed to just one axiom. The aforementioned values di¤er in an axiom that speci…es the type of player that can be removed from the player set without a¤ecting the remaining players’payo¤s. For the Shapley value, null players can be excluded. Proportional players can be eliminated for the equal division value, where a proportional player is a player whose entrance to a coalition does not change the per capita worth. Closely related is the notion of a quasi-proportional player, which can be removed for the solidarity value. In this paper, we suggest a family of player types— the -players— that contains the null players, the proportional players, and quasi-proportional players. A -player is a player whose marginal contribution to a coalition is times the per capita worth of the coalition he enters. The value of may depend on the size of the coalition entered, i.e., actually is sequence of of real numbers. For example, a null player is a -player if is constantly zero and a proportional player is a -player if is constantly one. The notion of a -player gives rise to the corresponding -player out axiom. As our …rst result, we determine those sequences for which there exists a value that satis…es e¢ ciency and the -player out axiom. Further, we show that for these sequences there exits a unique value that satis…es e¢ ciency, linearity, symmetry, and the -player out axiom. The family of these values, parametrized by ; contains values that are not economically sound. In particular, these values may fail desirability (Maschler and Peleg, 1966) or positivity (Kalai and Samet, 1987). Desirability ensures that a player who is more productive than another one does not end up with a lower payo¤. Positivity requires that all players obtain non-negative payo¤s in monotonic games, i.e., in games, where all marginal contributions are non-negative. As our second result, we identify the sequences for which the values in our family meet desirability and positivity. The values in this subfamily are called the “generalized solidarity values”. Except the Shapley value and the equal division value, all other egalitarian Shapley values are not members of this subfamily. The solidarity value is the center of the family of generalized solidarity values, which indicates that the name of this class is appropriate. Radzik (2012) establishes that the solidarity value and the equal division value are asymptotically equivalent in two respects. Let us explain the weaker one. Consider a sequence of games such that the set of players strictly increases, but the worth generated by any coalition of the these games is bounded. Fix a player in some of these games. Then, the di¤erence of this player’s payo¤ for the solidarity value and for the equal division values converges to zero as the size of the player set goes to in…nity. Our third result says that the generalized solidarity values with exception of the Shapley value, also are asymptotically equivalent to the equal division value in the sense of Radzik (2012). 2 The plan of this paper is as follows: In Sect. 2, basic de…nitions and notation are given. In Sect. 3, we analyze the player out properties and introduce the generalized solidarity values. In Sect. 4, we establish the asymptotic equivalence of the generalized solidarity values and the equal division value. Some remarks conclude the paper. The appendix contains all the proofs. 2. Basic de…nitions and notation Fix a su¢ ciently large in…nite set U, the universe of players, and let N denote the set of non-empty and …nite subsets of U. For R;S; T;N 2 N , r; s; t; and n denote their cardinalities, respectively. A (TU)-game is a pair (N; v) consisting of a set of playersN 2 N and a coalition function v 2 V (N) := f : 2 ! R j f (;) = 0 ; where 2 denotes the power set of N . For notational parsimony, we refer to v 2 V (N) as a game and indicate the underlying player set as superscript, v . Subsets of N are called coalitions, and v (S) is called the worth of coalition S. For v ; w 2 V (N) ; 2 R; the coalition functions (v + w) 2 V (N) and v 2 V (N) are given by (v + w) (S) = v (S) + w (S) and ( v) (S) = v (S) for all S N: For S N and v 2 V (N) ; v jS 2 V (S) denotes the restriction of v to 2: For T N; T 6= ;; the game uT , uT (S) = 1 if T S and uT (S) = 0 otherwise, is called a unanimity game; the game eT , e N T (S) = 1 if T = S and e N T (S) = 0 otherwise, is called a standard game; the game 0 ; 0 (S) = 0 for all S N; is called a null game. Any v 2 V (N) can be uniquely represented by unanimity games,