This paper considers an infinite-horizon differential game played by two direct marketers. Each player controls the number of emails sent to potential customers at each moment in time. There is a cost associated to the messages sent, as well as a potential reward. The latter is assumed to depend on the state variable defined as the level of the representative consumer’s attention.

Assistant Professor Olivier Rubel and co-authors Guiomar Martin-Herran from the University of Valladolid and Georges Zaccour from HEC Montreal include two features in the model: marginal decreasing returns and bounded rationality. By the latter, they mean that the representative consumer has a limited capacity for processing the information received. The evolution of this capacity depends on its level, as well as on the emails sent by both players.

This provides environmental flavour where, usually, one player’s pollution emissions (here emails) also affect the payoff of the other player by damaging the common environment (here, the stock of consumer attention).