$\displaystyle\int_{\mathbb{R}^{n}\times\mathbb{R}^{m}}\pi(x_{1},x_{2},y_{1},y_{2})d(x_{2},y_{2})=\int_{\mathbb{R}^{n}}\pi_{1}(x_{1},x_{2})dx_{2}\int_{\mathbb{R}^{m}}\pi_{2}(y_{1},y_{2})dy_{2}=\mu_{1}(x_{1})\mu_{2}(y_{1}).$

Therefore, we get

$\displaystyle d_{W}(\mu,\nu)$
$\displaystyle\leq\int_{\mathbb{R}^{n+m}\times\mathbb{R}^{n+m}}d_{\mathbb{R}^{n+m}}(x_{1},y_{1},x_{2},y_{2})d\pi(x_{1},x_{2},y_{1},y_{2})$
$\displaystyle\leq\int_{\mathbb{R}^{n+m}\times\mathbb{R}^{n+m}}d_{\mathbb{R}^{n}}(x_{1},y_{1})+d_{\mathbb{R}^{m}}(x_{2},y_{2})d\pi(x_{1},x_{2},y_{1},y_{2})$
$\displaystyle=\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}d_{\mathbb{R}^{n}}(x_{1},y_{1})d\pi_{1}(x_{1},y_{1})+\int_{\mathbb{R}^{m}\times\mathbb{R}^{m}}d_{\mathbb{R}^{m}}(x_{2},y_{2})d\pi_{2}(x_{2},y_{2})$
$\displaystyle=d_{W}(\mu_{1},\nu_{1})+d_{W}(\mu_{2},\nu_{2})$

$\square$

###### Lemma 11.7

Let $\beta_{it},\beta_{it}^{\prime}\in\Sigma_{it}^{\text{p}}$,
$\beta_{-it}\in\Sigma_{-it}$ and $\beta_{<t}\in\Sigma_{<t}$. Under Assumption
7.1, it holds that

$\displaystyle
d_{W}\left(P_{<t+1}\left(\,\cdot\,\,|\left(\beta_{i<t},\beta_{it}\right),\left(\beta_{-i<t},\beta_{-it}\right)\right),P_{<t+1}\left(\,\cdot\,\,|\left(\beta_{i<t},\beta_{it}^{\prime}\right),\left(\beta_{-i<t},\beta_{-it}\right)\right)\right)\leq||\beta_{it}-\beta_{it}^{\prime}||_{\infty}.$

##### Proof:

We start with showing the following step first

$\displaystyle
d_{W}\left(P_{t}\left(\,\cdot\,\,|\,a_{<t},\beta_{it},\beta_{-it}\right),P_{t}\left(\,\cdot\,\,|\,a_{<t},\beta_{it}^{\prime},\beta_{-it}\right)\right)\leq||\beta_{it}-\beta_{it}^{\prime}||_{\infty}\text{
for all }a_{<t}\in\mathcal{A}_{<t}.$ (35)

Let $a_{<t}\in\mathcal{A}_{<t}$ be arbitrary. Then note first that as
$\beta_{it},\beta_{it}^{\prime}$ are pure strategies,
$\beta_{it}(\,\cdot\,\,|\,\sigma_{it}(a_{<t}))$ and
$\beta_{it}^{\prime}(\,\cdot\,\,|\,\sigma_{it}(a_{<t}))$ are Dirac-measures.
Therefore, we can use the well-known fact that the Wasserstein distance
between these is simply the distance between the points with positive measure
(see Example 6.3 in (Villani 2009, p.94)), i.e.,

$\displaystyle
d_{W}\left(\beta_{it}(\,\cdot\,\,|\,\sigma_{it}(a_{<t})),\beta_{it}^{\prime}(\,\cdot\,\,|\,\sigma_{it}(a_{<t}))\right)=\left|\left|\beta_{it}(\sigma_{it}(a_{<t}))-\beta_{it}^{\prime}(\sigma_{it}(a_{<t}))\right|\right|_{\infty}\leq\left|\left|\beta_{it}-\beta_{it}^{\prime}\right|\right|_{\infty},$

where we abused notation and treated $\beta_{it},\beta_{it}^{\prime}$ once as
mapping to Dirac-measures and once as mapping to elements in
$\mathcal{A}_{it}$. By Assumption 7.1, $P_{t}$ is a product measure on
$\mathbb{R}^{m}$ for some $m\in\mathbb{N}$. Therefore, one can use Lemma 11.6
and get

$\displaystyle
d_{W}\left(P_{t}\left(\,\cdot\,\,|\,a_{<t},\beta_{it},\beta_{-it}\right),P_{t}\left(\,\cdot\,\,|\,a_{<t},\beta_{it}^{\prime},\beta_{-it}\right)\right)$
$\displaystyle\leq
d_{W}\left(\beta_{it}(\,\cdot\,\,|\,\sigma_{it}(a_{<t})),\beta_{it}^{\prime}(\,\cdot\,\,|\,\sigma_{it}(a_{<t}))\right)\leq\left|\left|\beta_{it}-\beta_{it}^{\prime}\right|\right|_{\infty},$

which shows Equation 35. Consequently, we get

$\displaystyle
d_{W}\left(P_{<t+1}\left(\,\cdot\,\,|\left(\beta_{i<t},\beta_{it}\right),\left(\beta_{-i<t},\beta_{-it}\right)\right),P_{<t+1}\left(\,\cdot\,\,|\left(\beta_{i<t},\beta_{it}^{\prime}\right),\left(\beta_{-i<t},\beta_{-it}\right)\right)\right)$
$\displaystyle=\sup_{||f||_{\text{Lip}}\leq
1}\int_{\mathcal{A}_{<t}}\int_{\mathcal{A}_{\cdot t}}f\left(a_{<t},a_{\cdot
t}\right)dP_{t}\left(a_{\cdot t}\,|\,a_{<t},\beta_{it},\beta_{-it}\right)$
$\displaystyle-\int_{\mathcal{A}_{\cdot t}}f\left(a_{<t},a_{\cdot
t}\right)dP_{t}\left(a_{\cdot
t}\,|\,a_{<t},\beta_{it}^{\prime},\beta_{-it}\right)dP_{<t}\left(a_{<t}\,|\beta_{<t}\right)$
$\displaystyle\overset{(\ref{thm:kantorovich-rubinstein-
duality})}{\leq}\int_{\mathcal{A}_{<t}}d_{W}\left(P_{t}\left(\,\cdot\,\,|\,a_{<t},\beta_{it},\beta_{-it}\right),P_{t}\left(\,\cdot\,\,|\,a_{<t},\beta_{it}^{\prime},\beta_{-it}\right)\right)dP_{<t}\left(a_{<t}\,|\beta_{<t}\right)$
$\displaystyle\overset{Equ.(\ref{equ:step-t-bound-on-wasserstein-distance-by-
norm-
difference})}{\leq}\int_{\mathcal{A}_{<t}}||\beta_{it}-\beta_{it}^{\prime}||_{\infty}dP_{<t}\left(a_{<t}\,|\beta_{<t}\right)=||\beta_{it}-\beta_{it}^{\prime}||_{\infty}.$

$\square$

###### Lemma 11.8

Let $(X,d_{X}),(Y,d_{Y})$ be metric Polish spaces, $f:X\times
Y\rightarrow\mathbb{R}$ be a $K_{f}$-Lipschitz continuous function and
$\mu(\,\cdot\,\,|\,x)$ be a measure on $Y$ for every $x\in X$. Furthermore,
the mapping $x\mapsto\mu(\,\cdot\,\,|\,x)$ is $K_{\mu}$-Lipschitz continuous
with respect to the Wasserstein distance $d_{W}$. Then it holds that

$\displaystyle
g_{f}:X\rightarrow\mathbb{R},x\mapsto\int_{Y}f(x,y)d\mu(y|x)\text{ is
}(K_{f}+K_{f}K_{\mu})-\text{Lipschitz}.$

##### Proof:

Let $x,x^{\prime}\in X$, then

$\displaystyle|g_{f}(x)-g_{f}(x^{\prime})|=\left|\int_{Y}f(x,y)d\mu(y|x)-\int_{Y}f(x^{\prime},y)d\mu(y|x^{\prime})\right|$
$\displaystyle\leq\left|\int_{Y}f(x,y)-f(x^{\prime},y)d\mu(y|x)\right|+\left|\int_{Y}f(x^{\prime},y)d\mu(y|x)-\int_{Y}f(x^{\prime},y)d\mu(y|x^{\prime})\right|$
$\displaystyle\leq\int_{Y}K_{f}\cdot d_{X\times
Y}\left(\left(x,y\right)^{T},\left(x^{\prime},y\right)^{T}\right)d\mu(y|x)+\sup_{||g||_{\text{Lip}}\leq
K_{f}}\left|\int_{Y}g(y)d\mu(y|x)-\int_{Y}g(y)d\mu(y|x^{\prime})\right|$
$\displaystyle\overset{(\ref{thm:kantorovich-rubinstein-
duality})}{=}K_{f}\int_{Y}d_{X}(x,x^{\prime})d\mu(y|x)+K_{f}\cdot
d_{W}\left(\mu\left(\,\cdot\,\,|\,x\right),\mu\left(\,\cdot\,\,|\,x^{\prime}\right)\right)$
$\displaystyle=K_{f}\left(d_{X}(x,x^{\prime})+d_{W}\left(\mu\left(\,\cdot\,\,|\,x\right),\mu\left(\,\cdot\,\,|\,x^{\prime}\right)\right)\right)$
$\displaystyle\leq
K_{f}\left(d_{X}(x,x^{\prime})+L_{\mu}d_{X}(x,x^{\prime})\right)=\left(K_{f}+K_{f}K_{\mu}\right)d_{X}(x,x^{\prime}).$

$\square$

###### Lemma 11.9

Let Assumptions 7.1 and 7.2 hold, and let
$\beta_{<t},\beta_{<t}^{\prime}\in\Sigma_{<t}$ and $\beta_{\cdot
t}\in\Sigma_{\cdot t}^{\text{Lip}}$. Then, there exists $K>0$ such that

$\displaystyle d_{W}\left(P_{<t+1}\left(\,\cdot\,\,|\,\beta_{<t},\beta_{\cdot
t}\right),P_{<t+1}\left(\,\cdot\,\,|\,\beta_{<t}^{\prime},\beta_{\cdot
t}\right)\right)\leq K\cdot
d_{W}\left(P_{<t}\left(\,\cdot\,\,|\,\beta_{<t}\right),P_{<t}\left(\,\cdot\,\,|\,\beta_{<t}^{\prime}\right)\right)$

##### Proof:

By Assumption 7.2, there exist constants $K_{\sigma_{it}}>0$ for $it\in L$,
such that $\sigma_{it}$ is $K_{\sigma_{it}}$-Lipschitz continuous in
$\mathcal{A}_{<t}$. Also, denote with $K_{0t}$ nature’s Lipschitz constant
with respect to $d_{W}$ in stage $t$. Similarly, as $\beta_{\cdot
t}\in\Sigma_{\cdot t}^{\text{Lip}}$, there exist constants $K_{\beta_{it}}>0$
such that $\beta_{it}\left(\,\cdot\,\,|\,s_{it}\right)$ is
$K_{\beta_{it}}$-Lipschitz with respect to the Wasserstein distance. Overall,
we get for $it\in L$, $a_{<t},a_{<t}^{\prime}\in\mathcal{A}_{<t}$

$\displaystyle
d_{W}\left(\beta_{it}\left(\,\cdot\,\,|\,\sigma_{it}(a_{<t})\right),\beta_{it}\left(\,\cdot\,\,|\,\sigma_{it}(a_{<t}^{\prime})\right)\right)\leq
K_{\beta_{it}}K_{\sigma_{it}}\cdot\left|\left|a_{<t}-a_{<t}^{\prime}\right|\right|.$

Denote $K_{t}:=K_{0t}+\sum_{i\in\mathcal{N}}K_{\beta_{it}}K_{\sigma_{it}}$.
Then we get that the mapping $a_{<t}\mapsto
P_{t}\left(\,\cdot\,\,|\,a_{<t},\beta_{\,\cdot\,t}\right)$ is
$K_{t}$-Lipschitz continuous with respect to $d_{W}$, which can be seen by

$\displaystyle
d_{W}\left(P_{t}\left(\,\cdot\,\,|\,a_{<t},\beta_{\,\cdot\,t}\right),P_{t}\left(\,\cdot\,\,|\,a_{<t}^{\prime},\beta_{\,\cdot\,t}\right)\right)$
$\displaystyle\overset{(\ref{thm:wasserstein-distance-of-product-measures-
bounded-by-sum-of-
distances})}{\leq}d_{W}\left(p_{0t}\left(\,\cdot\,\,|\,a_{<t}\right),p_{0t}\left(\,\cdot\,\,|\,a_{<t}^{\prime}\right)\right)+\sum_{i\in\mathcal{N}}d_{W}\left(\beta_{it}\left(\,\cdot\,\,|\,\sigma_{it}(a_{<t})\right),\beta_{it}\left(\,\cdot\,\,|\,\sigma_{it}(a_{<t}^{\prime})\right)\right)$
$\displaystyle\leq\left(K_{0t}+\sum_{i\in\mathcal{N}}K_{\beta_{it}}K_{\sigma_{it}}\right)\cdot\left|\left|a_{<t}-a_{<t}^{\prime}\right|\right|_{\infty},$

for all $a_{<t},a_{<t}^{\prime}\in\mathcal{A}_{<t}$. Let
$f:\mathcal{A}_{<t+1}\rightarrow\mathbb{R}$ be $1$-Lipschitz continuous. Then,
we get by Lemma 11.8, that the function
$g_{f}(a_{<t}):=\int_{\mathcal{A}_{<t}}f(a_{<t},a_{\,\cdot\,t})dP_{t}\left(a_{\,\cdot\,t}\,|\,a_{<t},\beta_{\,\cdot\,t}\right)$
is $(1+K_{t})$-Lipschitz continuous. With this, we get

$\displaystyle d_{W}\left(P_{<t+1}\left(\,\cdot\,\,|\,\beta_{<t},\beta_{\cdot
t}\right),P_{<t+1}\left(\,\cdot\,\,|\,\beta_{<t}^{\prime},\beta_{\cdot
t}\right)\right)$ $\displaystyle\overset{(\ref{thm:kantorovich-rubinstein-
duality})}{=}\sup_{||f||_{\text{Lip}}\leq
1}\int_{\mathcal{A}_{<t+1}}f(a_{<t+1})P_{<t+1}\left(a_{<t+1}\,|\,\beta_{<t},\beta_{\cdot
t}\right)-\int_{\mathcal{A}_{<t+1}}f(a_{<t+1})P_{<t+1}\left(a_{<t+1}\,|\,\beta_{<t}^{\prime},\beta_{\cdot
t}\right)$ $\displaystyle=\sup_{||f||_{\text{Lip}}\leq
1}\int_{\mathcal{A}_{<t}}g_{f}(a_{<t})P_{<t}\left(a_{<t}\,|\,\beta_{<t}\right)-\int_{\mathcal{A}_{<t}}g_{f}(a_{<t})P_{<t}\left(a_{<t}\,|\,\beta_{<t}^{\prime}\right)$
$\displaystyle\overset{(\ref{thm:lipschitz-function-conditional-on-lipschitz-
measure-is-again-lipschitz})}{\leq}\sup_{||g||_{\text{Lip}}\leq
1+K_{t}}\int_{\mathcal{A}_{<t}}g(a_{<t})P_{<t}\left(a_{<t}\,|\,\beta_{<t}\right)-\int_{\mathcal{A}_{<t}}g(a_{<t})P_{<t}\left(a_{<t}\,|\,\beta_{<t}^{\prime}\right)$
$\displaystyle\overset{(\ref{thm:kantorovich-rubinstein-
duality})}{=}\left(1+K_{t}\right)\cdot
d_{W}\left(P_{<t}\left(\,\cdot\,\,|\,\beta_{<t}\right),P_{<t}\left(\,\cdot\,\,|\,\beta_{<t}^{\prime}\right)\right),$

which shows the statement. $\square$

###### Lemma 11.10

Let $\Gamma=\left(\mathcal{N},T,S,\mathcal{A},p,\sigma,u\right)$ be a multi-
stage game, where Assumptions 7.1 and 7.2 hold. For strategies
$\beta_{-i}\in\Sigma_{-i}^{\text{Lip}}$, $\beta_{i}\in\Sigma_{i}^{\text{Lip,
p}}$ and $\epsilon>0$, there exists a $\delta>0$ such that for all
$\beta_{i}^{\prime}\in\Sigma_{i}^{\text{p}}$ with
$\left|\left|\beta_{i}-\beta_{i}^{\prime}\right|\right|_{\infty}<\delta$, it
holds that

$\displaystyle
d_{W}\left(P\left(\,\cdot\,\,|\,\beta_{i},\beta_{-i}\right),P\left(\,\cdot\,\,|\,\beta_{i}^{\prime},\beta_{-i}\right)\right)<\epsilon.$

##### Proof:

Let $\epsilon>0$, $\beta_{-i}\in\Sigma_{-i}^{\text{Lip}}$,
$\beta_{i}\in\Sigma_{i}^{\text{Lip, p}}$, and
$\beta_{i}^{\prime}\in\Sigma_{i}^{\text{p}}$. Then we get

$\displaystyle
d_{W}\left(P\left(\,\cdot\,\,|\,\beta_{i},\beta_{-i}\right),P\left(\,\cdot\,\,|\,\beta_{i}^{\prime},\beta_{-i}\right)\right)$
$\displaystyle\overset{(\triangle\text{-inequ.})}{\leq}\sum_{t=1}^{T}d_{W}\left(P\left(\,\cdot\,\,|\,\left(\beta_{i<t}^{\prime},\beta_{it},\beta_{i>t}\right),\beta_{-i}\right),P\left(\,\cdot\,\,|\,\left(\beta_{i<t}^{\prime},\beta_{it}^{\prime},\beta_{i>t}\right),\beta_{-i}\right)\right)$
$\displaystyle\overset{(\ref{thm:backward-bound-for-wasserstein-with-
lipschitz-continuous-
strategies})}{\leq}\sum_{t=1}^{T}\left(\prod_{v>t}K_{v}\right)\cdot
d_{W}\left(P_{<t+1}\left(\,\cdot\,\,|\,\left(\beta_{i<t}^{\prime},\beta_{it}\right),\beta_{-i}\right),P_{<t+1}\left(\,\cdot\,\,|\,\left(\beta_{i<t}^{\prime},\beta_{it}^{\prime}\right),\beta_{-i}\right)\right)$
$\displaystyle\overset{(\ref{thm:beta-beta-prime-lipschitz-condition-in-
p-t})}{\leq}\sum_{t=1}^{T}K_{>t}\left|\left|\beta_{it}-\beta_{it}^{\prime}\right|\right|_{\infty},$

where $K_{>t}=\prod_{v=t+1}^{T}K_{v}$ with
$K_{t}:=K_{0t}+\sum_{i\in\mathcal{N}}K_{\beta_{it}}K_{\sigma_{it}}$ (see proof
of Lemma 11.9) for all $1\leq t\leq T$. By choosing $\delta<\max_{t\in
T}\frac{\epsilon}{K_{>t}\cdot T}$, we get the statement. $\square$