Application of Poisson process in economic modelling To understand the emergence of constitution, Myerson (2008) models a scernario that a political leader gathers supports from captains in order to defeat challengers whose arrival is modelled by a Poisson process.

I found using Poisson process seems to be simple and reasonable, but it is the only model I know that utilizes Poisson process.Are there any other well-known models utilizing Poisson process in a similar way? If not, what's drawback of Poisson process?
Most of the literature on "Strategic experimentation" (or Bandits) uses Poisson processes. Here players can use either a risky or safe arm and one of them generates a fixed stream of payoffs (usually the safe arm) the other one generates lump-sum payments whose arrival times are described by a Poisson process.

Some examples are:

Klein, Nicolas, and Sven Rady. "Negatively correlated bandits." The Review of Economic Studies (2011).

Keller, Godfrey, and Sven Rady. "Strategic experimentation with Poisson bandits." Theoretical Economics 5.2 (2010): 275-311.

Keller, Godfrey, Sven Rady, and Martin Cripps. "Strategic experimentation with exponential bandits." Econometrica 73.1 (2005): 39-68.