  A two-type version of the frog model on $\mathbb{Z}^d$ is formulated, where
active type $i$ particles move according to lazy random walks with probability
$p_i$ of jumping in each time step ($i=1,2$). Each site is independently
assigned a random number of particles. At time 0, the particles at the origin
are activated and assigned type 1 and the particles at one other site are
activated and assigned type 2, while all other particles are sleeping. When an
active type $i$ particle moves to a new site, any sleeping particles there are
activated and assigned type $i$, with an arbitrary tie-breaker deciding the
type if the site is hit by particles of both types in the same time step. We
show that the event $G_i$ that type $i$ activates infinitely many particles has
positive probability for all $p_1,p_2\in(0,1]$ ($i=1,2$). Furthermore, if
$p_1=p_2$, then the types can coexist in the sense that $\mathbb{P}(G_1\cap
G_2)>0$. We also formulate several open problems. For instance, we conjecture
that, when the initial number of particles per site has a heavy tail, the types
can coexist also when $p_1\neq p_2$.
