We prove that every bounded Lipschitz function
F
F
on a subset
Y
Y
of a length space
X
X
admits a tautest extension to
X
X
, i.e., a unique Lipschitz extension
u
:
X
→
R
u:X \rightarrow \mathbb {R}
for which
Lip
U
u
=
Lip
∂
U
u
\operatorname {Lip}_U u =\operatorname {Lip}_{\partial U} u
for all open
U
⊂
X
∖
Y
U \subset X\smallsetminus Y
. This was previously known only for bounded domains in
R
n
\mathbb {R}^n
, in which case
u
u
is infinity harmonic; that is, a viscosity solution to
Δ
∞
u
=
0
\Delta _\infty u = 0
, where
\[
Δ
∞
u
=
|
∇
u
|
−
2
∑
i
,
j
u
x
i
u
x
i
x
j
u
x
j
.
\Delta _\infty u = |\nabla u|^{-2} \sum _{i,j} u_{x_i} u_{x_ix_j} u_{x_j}.
\]
We also prove the first general uniqueness results for
Δ
∞
u
=
g
\Delta _{\infty } u = g
on bounded subsets of
R
n
\mathbb {R}^n
(when
g
g
is uniformly continuous and bounded away from
0
0
) and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of
u
u
. Let
u
ε
(
x
)
u^\varepsilon (x)
be the value of the following two-player zero-sum game, called tug-of-war: fix
x
0
=
x
∈
X
∖
Y
x_0=x\in X \smallsetminus Y
. At the
k
t
h
k^{\mathrm {th}}
turn, the players toss a coin and the winner chooses an
x
k
x_k
with
d
(
x
k
,
x
k
−
1
)
>
ε
d(x_k, x_{k-1})> \varepsilon
. The game ends when
x
k
∈
Y
x_k \in Y
, and player I’s payoff is
F
(
x
k
)
−
ε
2
2
∑
i
=
0
k
−
1
g
(
x
i
)
F(x_k) - \frac {\varepsilon ^2}{2}\sum _{i=0}^{k-1} g(x_i)
. We show that
‖
u
ε
−
u
‖
∞
→
0
\|u^\varepsilon - u\|_{\infty } \to 0
. Even for bounded domains in
R
n
\mathbb {R}^n
, the game theoretic description of infinity harmonic functions yields new intuition and estimates; for instance, we prove power law bounds for infinity harmonic functions in the unit disk with boundary values supported in a
δ
\delta
-neighborhood of a Cantor set on the unit circle.