Projection onto the core: An optimal reallocation to correct market failure

TL;DR

This paper introduces algorithms for projecting onto the core of a game to correct market failures by computing optimal side payments, ensuring the state moves closer to a core allocation and evaluating market efficiency.

Contribution

It develops new formulae and algorithms for projection onto the core of a game, addressing computational challenges and applying them to market games to improve economic stability.

Findings

Algorithms effectively compute the closest core allocation from any preimputation.

Optimal side payments can correct market states to align with the core.

Proposed solution minimizes market failure to reach core allocations.

Abstract

This paper provides formulae and algorithms to compute the projection onto the core of a preimputation outside it. The core of a game is described using an exponential number of linear constraints, and we cannot know beforehand which are redundant or defining the polytope. We apply these new results to market games, a class of games in which every game has a nonempty core. Given an initial state of the game represented by a preimputation, it is not guaranteed that the state of the game evolves toward the core following the dynamics induced by the domination relations. Our results identify and compute the most efficient side payment that acts on a given state of the game and yields its closest core allocation. Using this side payment, we propose a way to evaluate the failure of a market to reach a state of the economy belonging to the core, and we propose a new solution concept consisting of preimputations that minimizes this failure.