The bounded core for games with precedence constraints

Michel Grabisch,Peter Sudhölter

doi:10.1007/s10479-012-1228-9

Abstract

An element of the possibly unbounded core of a cooperative game with precedence constraints belongs to its bounded core if any transfer to a player from any of her subordinates results in payoffs outside the core. The bounded core is the union of all bounded faces of the core, it is nonempty if the core is nonempty, and it is a continuous correspondence on games with coinciding precedence constraints. If the precedence constraints generate a connected hierarchy, then the core is always nonempty. It is shown that the bounded core is axiomatized similarly to the core for classical cooperative games, namely by boundedness (BOUND), nonemptiness for zero-inessential two-person games (ZIG), anonymity, covariance under strategic equivalence (COV), and certain variants of the reduced game property (RGP), the converse reduced game property (CRGP), and the reconfirmation property. The core is the maximum solution that satisfies a suitably weakened version of BOUND together with the remaining axioms. For games with connected hierarchies, the bounded core is axiomatized by BOUND, ZIG, COV, and some variants of RGP and CRGP, whereas the core is the maximum solution that satisfies the weakened version of BOUND, COV, and the variants of RGP and CRGP.

Highlights

In the classical theory of cooperative games one assumes that all players may cooperate, i.e., any coalition may form
Theorem 4.4 The bounded core is the unique solution on Γ that satisfies zero-inessential two-person games (ZIG), AN, covariance under strategic equivalence (COV), reduced game property (RGP), RCPcg, converse reduced game property (CRGP), and BOUND
Lemma 4.5 Any solution σ that satisfies ZIG, AN, COV, RGP, RCPcg, CRGP, and BOUNDs is a subsolution of the core

Summary

Introduction

In the classical theory of cooperative games one assumes that all players may cooperate, i.e., any coalition may form. The well-known fact that the core of a TU game with precedence constraints may be unbounded seems counterintuitive and has created several attempts to define a meaningful subset of the core that is bounded (see, e.g., Grabisch (2011)) The core of such a game is a convex polyhedral set that contains no lines, but, in contrast to the core of a classical TU game, it may have unbounded faces. The bounded core of (N, , v), denoted by Cb(N, , v), is the set of all elements x ∈ C(N, , v) that satisfy the following condition for any i, j ∈ N with i ≺· j: There is no ε > 0 such that x + ε χ{j} − χ{i} ∈ C(N, , v).

Properties of the core

Axiomatization of the bounded core

The bounded core for games with connected hierarchies