On a class of vertices of the core
Introduction
In the seminal paper of Shapley (1971) was undertaken probably the first study of the geometric properties of the core of TU-games. In particular, it was established that the set of marginal vectors coincides with the set of vertices of the core when the game is convex. This paper was the starting point of numerous publications on the core and its variants, studying its geometric structure (vertices, facets, since it is a closed convex polytope) for various classes of games (in particular, the assignment games of Shapley and Shubik, 1972).
In a parallel way, it was found that the classical view of TU-games, defined as set functions on the power set of the set of players, was too narrow, and the idea of restricted cooperation (i.e., not any coalition can form) germinated in several papers, most notably Aumann and Drèze (1974); Myerson (1977); Owen (1977) and Faigle (1989), who coined the term “restricted cooperation” and precisely studied the core of such games. Many algebraic structures were proposed for the set of feasible coalitions, e.g., (distributive) lattices, antimatroids, convex geometries, etc. It turned out that the structure of the core became much more complex to study, in particular due to the fact that the core on such games may become unbounded (see a survey in Grabisch, 2013). However, as shown by Derks and Gilles (1995), the main result established in Shapley (1971) remains true for games on distributive lattices: for supermodular games, the set of marginal vectors still coincides with the set of extreme points of the core.
The question addressed in this paper arises naturally from the last result: What if the game is not supermodular? Is it possible to know all of its vertices in an analytical form? The question has puzzled many researchers, and so far only partial answers have been obtained, and only in the case of classical TU-games, i.e., without restriction on cooperation. Significant contributions have been done in particular by Núñez and Rafels (1998), and Tijs (2005). In the former work, a family of vertices is obtained, which is shown to cover all vertices of the core when the game is almost convex (i.e., satisfying the supermodularity condition except when the grand coalition is involved). Later, Núñez and Rafels (2003) have shown that this family of vertices is also exhaustive for assignment games, while Trudeau and Vidal-Puga (2017) have shown that the same result holds for minimum cost spanning tree games. In the work of Tijs, another family of vertices is proposed, called leximals, which is leading to the concept of lexicore and the Alexia value.
The present paper lies in the continuity of these works, showing that the two previous families have close links, proposing a wider class of vertices (unfortunately, still not exhaustive in all cases), and most importantly, establishing results in the general context of games on distributive lattices. Such a class of games is of considerable interest, because it has a very simple interpretation: the set of feasible coalitions is induced by a hierarchy (partial order) on the set of players, and feasible coalitions correspond to subsets of players where every subordinate of a member must be present. In the absence of hierarchy, the classical case is recovered.
We summarize the main achievements of the paper. We first give a tight upper bound of the number of vertices of the core, using an argument of Derks and Kuipers (2002). Then we introduce the family of min–max vertices, obtained by minimizing or maximizing in a given order the coordinates of a core element. Minimization (respectively, maximization) is performed if the considered coordinate (player) is a minimal element (respectively, a maximal element) in the sub-hierarchy formed by the remaining players. We prove that these are indeed vertices of the core (Theorem 2), and that in the case of supermodular games, we recover all marginal vectors (Corollary 1). The case of connected hierarchies reveals to be particularly simple, because min–max vertices take a simple form and can be computed directly without solving an optimization problem (Theorem 5). In the general case, a similar computation can be done provided some conditions are satisfied (Formula (15)). Two different orders may yield the same min–max vertex for every game. We show in Theorem 7 that this arises if and only if one of the orders can be obtained from the other one by a sequence of switches exchanging minimal and maximal elements. Lastly, we investigate the limits of the min–max approach to find vertices, and show that there exist balanced games whose core has vertices which are not min–max vertices if and only if (Theorem 8).
The paper is organized as follows. Section 2 introduces the necessary material for games on distributive lattices and cores of such games, which are unbounded in general. We show that the structure of the convex hull of the vertices of the core of games on distributive lattices is more complex than the structure of the core of ordinary games (Proposition 1). Section 3 gives an upper bound of the number of vertices of the core. Section 4 is the main section of the paper, introducing and studying min–max vertices. Section 5 investigates under which condition orders yield identical min–max vertices. Examples illustrating the main results and concepts are given in Section 6, together with a practical summary of how to proceed. The limits of the min–max approach are investigated in Section 7, and the paper finishes with Section 8 detailing the past literature on the topic.
Section snippets
Notation, definitions and preliminaries
A partially ordered set or poset is a set P endowed with a partial order ⪯, i.e., a reflexive, antisymmetric and transitive binary relation. A poset is a lattice if every two elements have a supremum and an infimum, denoted respectively by . The lattice is distributive if obey distributivity. As usual, means and . We say that x covers y, denoted by , if and there is no such that . A chain in is a sequence such that , and
An upper bound for the number of vertices
Let , denote , and introduce , , and, for any , . Then the interior of the intersection of any two of the is empty, their union is the unit cube, and the volumes of all of them are identical so that we conclude that the volume of is . Moreover, we claim that
Indeed, it is a well-known fact that each is an n-dimensional simplex, and its vertices are the characteristic vectors
Min–max vertices
This section presents the construction of our proposed family of vertices. We begin by presenting informally the main idea underlying this family.
Equivalent consistent pairs of permutations and decisions
We begin by a simple observation. If is a consistent pair w.r.t. and differs from d only inasmuch as , then is also consistent, and for any game . Indeed, is the unique player of so that is both maximal and minimal in . Moreover, by definition so that by Pareto optimality of the core. For this reason, we call this operation an irrelevant switch.
We say that two consistent
Examples and summary of the results
We begin by illustrating the computation of the min–max vertices when the hierarchy is connected. Example 2 We consider and the connected hierarchy given in Fig. 1 (the “N” example). Let us consider a strictly supermodular game . Every order is admissible and the simple orders are 1234, 1243, 1423, 1432, 4321, 4312, 4123 and 4132. The linear extensions (which yield all extreme points of the core) are 1324, 1342, 3124, 3142, and 3412. Taking order 1234 and using strict supermodularity, we
Limits of the min–max approach
The following example shows that there exists a game v that possesses a vertex x of its core with the following property: for any , there exist core elements with . It shows that it is not possible in general to find all vertices of the core by taking arbitrary orders and maximizing or minimizing within the core the payoff of the first player, then of the second, etc.
Example 5 Let , , and let be a poset that such that , e.g., the
Related literature
The basic idea of min–max vertices as well as the form of the induced vector (10) have their roots in the past literature, although limited to the classical case . Perhaps the first occurrence of the idea of systematically taking the minimum or maximum over single coordinates of core elements goes back to the paper of Derks and Kuipers (2002), while the induced vector (10) in the particular case where is always considered as a maximal element is due to Núñez and Rafels (1998).
Acknowledgements
The first author thanks the Agence Nationale de la Recherche for financial support under contract ANR-13-BSHS1-0010 (DynaMITE). The second author was supported by the Spanish Ministerio de Economía y Competitividad under project ECO2015-66803-P and by the Danish Council for Independent Research/Social Sciences under the FINQ project (Grant ID: DFF-1327-00097).
References (27)
- et al.
On the restricted cores and the bounded core of games on distributive lattices
Eur. J. Oper. Res.
(2014) - et al.
Characterization of the extreme core allocations of the assignment game
Games Econ. Behav.
(2003) - et al.
An average lexicographic value for cooperative games
Eur. J. Oper. Res.
(2011) - et al.
On the set of extreme core allocations for minimal cost spanning tree problem
J. Econ. Theory
(2017) - et al.
Cooperative games with coalition structures
Int. J. Game Theory
(1974) On the combination of subalgebras
Proc. Camb. Philos. Soc.
(1933)- et al.
The kernel of a cooperative game
Nav. Res. Logist. Q.
(1965) - et al.
Hierarchical organization structures and constraints on coalition formation
Int. J. Game Theory
(1995) - et al.
On the number of extreme points of the core of a transferable utility game
Graph Theory
(2005)
Cores of games with restricted cooperation
ZOR, Z. Oper.-Res.
The Shapley value for cooperative games under precedence constraints
Int. J. Game Theory
A note on submodular functions on distributive lattices
J. Oper. Res. Soc. Jpn.
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