  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.
