Game-theoretical approach for task allocation problems with constraints

TL;DR

This paper introduces a game-theoretical framework for constrained task allocation, proving the uniqueness and optimality of solutions, and validating results through algorithms and simulations.

Contribution

It develops a potential game model for constrained task allocation, deriving conditions for optimal solutions and proposing algorithms validated by simulations.

Findings

Nash equilibrium can be the unique globally optimal solution under certain conditions.

Analytical solutions are derived for exponential and quadratic cost functions.

Algorithms and Monte Carlo simulations confirm theoretical results.

Abstract

The distributed task allocation problem, as one of the most interesting distributed optimization challenges, has received considerable research attention recently. Previous works mainly focused on the task allocation problem in a population of individuals, where there are no constraints for affording task amounts. The latter condition, however, cannot always be hold. In this paper, we study the task allocation problem with constraints of task allocation in a game-theoretical framework. We assume that each individual can afford different amounts of task and the cost function is convex. To investigate the problem in the framework of population games, we construct a potential game and calculate the fitness function for each individual. We prove that when the Nash equilibrium point in the potential game is in the feasible solutions for the limited task allocation problem, the Nash equilibrium point is the unique globally optimal solution. Otherwise, we also derive analytically the unique globally optimal solution. In addition, in order to confirm our theoretical results, we consider the exponential and quadratic forms of cost function for each agent. Two algorithms with the mentioned representative cost functions are proposed to numerically seek the optimal solution to the limited task problems. We further perform Monte Carlo simulations which provide agreeing results with our analytical calculations.