  This paper provides a novel solution to a task allocation problem, by which a
group of agents decides on the assignment of a discrete set of tasks in a
distributed manner. In this setting, heterogeneous agents have individual
preferences and associated rewards for doing each task; however, these rewards
are only known asymptotically. We start by formulating the assignment problem
by means of a combinatorial partition game for known rewards, with no
constraints on number of tasks per agent. We relax this into a weight game,
which together with the former, are shown to contain the optimal task
allocation in the corresponding set of Nash Equilibria (NE). We then propose a
projected, best-response, ascending gradient dynamics (PBRAG) that converges to
a NE in finite time. This forms the basis of a distributed online version that
can deal with a converging sequence of rewards by means of an agreement
sub-routine. We present simulations that support our results
