On a filtered probability space
(
Ω
,
F
,
P
,
F
=
(
F
t
)
0
≤
t
≤
T
)
(\Omega ,\mathcal {F},P,\mathbb {F}=(\mathcal {F}_t)_{0\leq t\leq T})
, we consider stopper-stopper games
C
¯
:=
inf
ρ
sup
τ
∈
T
E
[
U
(
ρ
(
τ
)
,
τ
)
]
\overline C:=\inf _{\boldsymbol {\rho }}\sup _{\tau \in \mathcal {T}} \mathbb {E}[U(\boldsymbol {\rho }(\tau ),\tau )]
and
C
_
:=
\underline C:=
sup
τ
inf
ρ
∈
T
E
[
U
(
ρ
,
τ
(
ρ
)
)
]
\sup _{\boldsymbol {\tau }} \inf _{\rho \in \mathcal {T}}\mathbb {E}[U(\rho ,\boldsymbol {\tau } (\rho ))]
in continuous time, where
U
(
s
,
t
)
U(s,t)
is
F
s
∨
t
\mathcal {F}_{s\vee t}
-measurable (this is the new feature of our stopping game),
T
\mathcal {T}
is the set of stopping times, and
ρ
,
τ
:
T
↦
T
\boldsymbol {\rho },\boldsymbol {\tau }:\mathcal {T} \mapsto \mathcal {T}
satisfy certain non-anticipativity conditions. We show that
C
¯
=
C
_
\overline C=\underline C
, by converting these problems into a corresponding Dynkin game.