Carles Rafels

The assignment game: the τ-value

Abstract. We provide some formulae for the τ-value in the case of the assignment game and prove that it coincides with the midpoint between the buyers-optimal and the sellers-optimal core allocations. As a consequence, the τ-value of an assignment game always lies in the core. Some comparative statics of this solution is analyzed: the pairwise monotonicity and the effect of new entrants.

Average Monotonic Cooperative Games

Abstract A subclass of monotonic transferable utility (T.U.) games is studied: average monotonic games. These games are totally balanced. We prove that the core coincides with both the bargaining set a la Davis and Maschler and the bargaining set a la Mas-Colell. To obtain this result a technique based on reduced games is used. Journal of Economic Literature Classification Number: C71

Characterization of the extreme core allocations of the assignment game

Although assignment games are hardly ever convex, in this paper a characterization of their set or extreme points of the core is provided, which is also valid for the class of convex games. For each ordering in the player set, a payoff vector is defined where each player receives his marginal contribution to a certain reduced game played by his predecessors. We prove that the whole set of reduced marginal worth vectors, which for convex games coincide with the usual marginal worth vectors, is the set of extreme points of the core of the assignment game.

On extreme points of the core and reduced games

Given a balanced cooperative game, we prove that the extreme points of the core have the reduced game property with respect to the Davis and Maschler reduced game. One particular case of this reduction gives when we name marginal games. These games allow us to define the reduced marginal worth vectors, where every player gets his marginal contribution to a successive marginal game. This set of vectors is proved to be the set of extreme points of the core of those balanced games which are almost convex, that is, those balanced games such that all proper subgames are convex.

On the dimension of the core of the assignment game

The set of optimal matchings in the assignment matrix allows to define a reflexive and symmetric binary relation on each side of the market, the equal-partner binary relation. The number of equivalence classes of the transitive closure of the equal-partner binary relation determines the dimension of the core of the assignment game. This result provides an easy procedure to determine the dimension of the core directly from the entries of the assignment matrix and shows that the dimension of the core is not as much determined by the number of optimal matchings as by their relative position.

The Böhm–Bawerk horse market: a cooperative analysis

Single–valued solutions for the case of two-sided market games without product differentiation, also known as Böhm–Bawerk horse market games, are analyzed. The nucleolus is proved to coincide with the τ value, and is thus the midpoint of the core. The Shapley value is in the core only if the game is a square glove market, and in this case also coincides with the two aforementioned solutions.

The aggregate-monotonic core

We introduce the aggregate-monotonic core as the set of allocations of a transferable utility cooperative game attainable by single-valued solutions that satisfy core-selection and aggregate-monotonicity. We provide a necessary and sufficient condition for the coincidence of the core and the aggregate-monotonic core. Finally, we introduce upper and lower aggregate-monotonicity for set-valued solutions, and characterize the aggregate-monotonic core using core-selection and upper and lower aggregate-monotonicity.

Even and odd marginal worth vectors, Owen's multilinear extension and convex games

In this paper we characterize convex games by means of Owens multilinear extension and the marginal worth vectors associated with even or odd permutations.Therefore we have obtained a refinement of the classic theorem; Shapley (1971), Ichiishi (1981) in order to characterize the convexity of a game by its marginal worth vectors.We also give new expressions for the marginal worth vectors in relation to unanimity coordinates and the first partial derivatives of Owens multilinear extension. A sufficient condition for the convexity is given and also one application to the integer part of a convex game.

Symmetrically multilateral-bargained allocations in multi-sided assignment markets

We extend the notion of symmetrically pairwise-bargained (SPB) allocations (Rochford, J Econ Theory, 34:262–281, 1984) to balanced assignment games with more than two sides. A symmetrically multilateral-bargained (SMB) allocation is a core allocation such that any agent’s payoff remains invariant after a negotiation process between all agents based on what they could receive—and use as a threat—in their preferred alternative matching to any given optimal matching. We prove that, for balanced multi-sided assignment games, the set of SMB is always nonempty and that, unlike the two-sided case, it does not coincide in general with the kernel (Davis and Maschler, Naval Res Logist Q 12:223–259, 1965). We also give an answer to an open question formulated by Rochford by introducing a kernel-based set whose intersection with the core coincides with the set of SMB.

A Simple Procedure to Obtain the Extreme Core Allocations of an Assignment Market

Given an assignment market, we introduce a set of vectors, one for each possible ordering on the player set, which we name the max-payoff vectors. Each one of these vectors is obtained recursively only making use of the assignment matrix. Those max-payoff vectors that are efficient turn out to give the extreme core allocations of the market. When the assignment game has a large core, all the max-payoff vectors are extreme core allocations.