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Construction of Compromise Values for Cooperative Games

Robert P. Gilles, René van den Brink

TL;DR

The paper develops a unifying framework for compromise values in cooperative TU-games by introducing bound pairs $(\mu,\eta)$ that simultaneously serve as lower and upper payoff bounds. It defines bound-balanced game classes and shows how a unique compromise value $\gamma(\cdot;\mu,\eta)$ emerges on these classes, characterised axiomatically by Minimal rights and Restricted proportionality. The authors generalise the classic $\tau$-value and exhibit multiple constructions: LBC values derived from regular lower bounds (yielding Egalitarian and CIS values) and UBC values derived from translation-covariant upper bounds (yielding the $\tau$-value, $\chi$-value, and CIS-value in new guises). They also introduce the Kikuta-Milnor KM-value, defined on all TU-games via $(\underline{M},\overline{M})$, which is self-dual and reduces to $\tau$ on convex games. The EANSC value is presented as a dual construct that can be obtained both from an upper-bound and a lower-bound perspective, thereby providing a comprehensive, interconnected foundation for a wide family of compromise values with broad applicability in cooperative game analysis.

Abstract

We explore a broad class of values for cooperative games in characteristic function form, known as \emph{compromise values\/}. These values efficiently allocate payoffs by linearly combining well-specified upper and lower bounds on payoffs. We identify subclasses of games that admit non-trivial efficient allocations within the considered bounds, which we call \emph{bound-balanced games}. Subsequently, we define the associated compromise value. We also provide an axiomatisation of this class of compromise values using variants of the minimal rights property and restricted proportionality. We introduce two construction methods for properly devised compromise values. Under mild conditions, one can use either a lower or an upper bound to construct a well-defined compromise value. We construct and axiomatise various well-known and new compromise values based on these methods, including the $τ$-, the $χ$-, the Gately, the CIS-, the PANSC-, the EANSC-, and the new KM-values. We conclude that this approach establishes a common foundation for a wide range of different values.

Construction of Compromise Values for Cooperative Games

TL;DR

The paper develops a unifying framework for compromise values in cooperative TU-games by introducing bound pairs that simultaneously serve as lower and upper payoff bounds. It defines bound-balanced game classes and shows how a unique compromise value emerges on these classes, characterised axiomatically by Minimal rights and Restricted proportionality. The authors generalise the classic -value and exhibit multiple constructions: LBC values derived from regular lower bounds (yielding Egalitarian and CIS values) and UBC values derived from translation-covariant upper bounds (yielding the -value, -value, and CIS-value in new guises). They also introduce the Kikuta-Milnor KM-value, defined on all TU-games via , which is self-dual and reduces to on convex games. The EANSC value is presented as a dual construct that can be obtained both from an upper-bound and a lower-bound perspective, thereby providing a comprehensive, interconnected foundation for a wide family of compromise values with broad applicability in cooperative game analysis.

Abstract

We explore a broad class of values for cooperative games in characteristic function form, known as \emph{compromise values\/}. These values efficiently allocate payoffs by linearly combining well-specified upper and lower bounds on payoffs. We identify subclasses of games that admit non-trivial efficient allocations within the considered bounds, which we call \emph{bound-balanced games}. Subsequently, we define the associated compromise value. We also provide an axiomatisation of this class of compromise values using variants of the minimal rights property and restricted proportionality. We introduce two construction methods for properly devised compromise values. Under mild conditions, one can use either a lower or an upper bound to construct a well-defined compromise value. We construct and axiomatise various well-known and new compromise values based on these methods, including the -, the -, the Gately, the CIS-, the PANSC-, the EANSC-, and the new KM-values. We conclude that this approach establishes a common foundation for a wide range of different values.

Paper Structure

This paper contains 33 sections, 13 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Constructing bound pairs and their associated compromise values.
  3. Axiomatisations
  4. Structure of the paper.
  5. Preliminaries: Cooperative games and values
  6. Game properties and values
  7. The $\tau$-value
  8. Semi-balanced games
  9. Compromise Values
  10. Bound pairs
  11. Linear bound pairs
  12. Bound balanced games
  13. An axiomatic characterisation of compromise values
  14. Three illustrations of compromise values
  15. The PANSC value
  16. ...and 18 more sections

Key Result

Proposition 3.6

Every linear bound pair $(\mu , \eta )$ is a bound pair on $\mathbb V ( \mu , \eta ) = \{ v \in \mathbb V^N \mid \mu (v) \leqslant \eta (v) \} \neq \varnothing$, which is a non-empty closed convex cone in $\mathbb V^N$.

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Definition 3.5
  • Proposition 3.6
  • Proposition 3.7
  • Definition 3.8
  • Proposition 3.9
  • ...and 12 more