  In this work, we consider a population composed of a continuum of agents that
seek to maximize a payoff function by moving on a network. The nodes in the
network may represent physical locations or abstract choices. The population is
stratified and hence agents opting for the same choice may not get the same
payoff. In particular, we assume payoff functions that model diminishing
returns, that is, agents in "newer" strata of a node receive a smaller payoff
compared to "older" strata. In this first part of two-part work, we model the
population dynamics under three choice revision policies, each having varying
levels of coordination -- i. no coordination and the agents are selfish, ii.
coordination among agents in each node and iii. coordination across the entire
population. To model the case with selfish agents, we generalize the Smith
dynamics to our setting, where we have a stratified population and network
constraints. To model nodal coordination, we allow the fraction of population
in a node, as a whole, to take the `best response' to the state of the
population in the node's neighborhood. For the case of population-wide
coordination, we explore a dynamics where the population evolves according to
centralized gradient ascent of the social utility, though constrained by the
network. In each case, we show that the dynamics has existence and uniqueness
of solutions and also show that the solutions from any initial condition
asymptotically converge to the set of Nash equilibria.
