  The decisions that human beings make to allocate time has significant bearing
on economic output and to the sustenance of social networks. The time
allocation problem motivates our formal analysis of the resource allocation
game, where agents on a social network, who have asymmetric, private
interaction preferences, make decisions on how to allocate time, a bounded
endowment, over their neighbors. Unlike the well-known opinion formation game
on a social network, our game appears not to be a potential game, and the
Best-Response dynamics is non-differentiable making the analysis of
Best-Response dynamics non-trivial.
  In our game, we consider two types of player behavior, namely optimistic or
pessimistic, based on how they use their time endowment over their neighbors.
To analyze Best-Response dynamics, we circumvent the problem of the game not
being a potential game, through the lens of a novel two-level potential
function approach. We show that the Best-Response dynamics converges point-wise
to a Nash Equilibrium when players are all: optimistic; pessimistic; a mix of
both types. Finally, we show that the Nash Equilibrium set is non-convex but
connected, and Price of Anarchy is unbounded while Price of Stability is one.
Extensive simulations over a stylized grid reveals that the distribution of
quality of the convergence points is unimodal-we conjecture that presence of
unimodality is tied to the connectedness of Nash Equilibrium.
