Assignment games with population monotonic allocation schemes

TL;DR

This paper characterizes assignment games that admit population monotonic allocation schemes (PMAS) using structural properties of their defining matrices, linking game features to specific matrix substructures.

Contribution

It provides a complete characterization of PMAS-admissible assignment games based on matrix properties and identifies the conditions under which core allocations extend to PMAS.

Findings

PMAS-admissible games contain a veto player or a dominant veto pair.

All core allocations in PMAS-admissible games can be extended to a PMAS.

In PMAS-admissible games, the nucleolus and tau-value coincide.

Abstract

We characterize the assignment games which admit a population monotonic allocation scheme (PMAS) in terms of efficiently verifiable structural properties of the nonnegative matrix that induces the game. We prove that an assignment game is PMAS-admissible if and only if the positive elements of the underlying nonnegative matrix form orthogonal submatrices of three special types. In game theoretic terms it means that an assignment game is PMAS-admissible if and only if it contains a veto player or a dominant veto mixed pair or is composed of from these two types of special assignment games. We also show that in a PMAS-admissible assignment game all core allocations can be extended to a PMAS, and the nucleolus coincides with the tau-value.