SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof. To show that  $\rho_{*}$  is a steady state we can follow exactly the same argument as in the proof of Corollary 12, just replacing  $-\frac{1}{\alpha} \int f_1(z, x) \mu_*(x)$  with  $+\frac{1}{\alpha} \int f_1(z, b(\rho_*))$ . It remains to show that  $\mathrm{supp}(\rho_*) = \mathrm{supp}(\tilde{\rho})$ . As  $\rho^*$  is a maximizer, it is in particular a critical point, and therefore satisfies that  $\delta_{\rho} G_b[\rho_*)(z)$  is constant on all connected components of  $\mathrm{supp}(\rho_*)$ . Thanks to Lemma 18, this means there exists a constant  $c[\rho_*]$  (which may be different on different components of  $\mathrm{supp}(\rho_*)$ ) such that
$$
f _ {1} (z, b (\rho_ {*})) - \alpha \log \left(\frac {\rho_ {*} (z)}{\tilde {\rho} (z)}\right) - W * \rho_ {*} (z) = c [ \rho_ {*} ] \quad \text {o n}
$$
Rearranging, we obtain (for a possible different constant  $c[\rho_{*}] \neq 0$ )
$$
\rho_ {*} (z) = c \left[ \rho_ {*} \right] \tilde {\rho} (z) \exp \left[ \frac {1}{\alpha} \left(f _ {1} \left(z, b \left(\rho_ {*}\right)\right) - W * \rho_ {*} (z)\right) \right] \quad \text {o n} \operatorname {s u p p} \left(\rho_ {*}\right). \tag {32}
$$
Firstly, note that  $\mathrm{supp}(\rho_{*})\subset \mathrm{supp}(\tilde{\rho})$  since  $\rho_{*}$  is absolutely continuous with respect to  $\tilde{\rho}$ . Secondly, note that  $\exp \frac{1}{\alpha} f_1(z,b(\rho_*))\geq 1$  for all  $z\in \mathbb{R}^d$  since  $f_{1}\geq 0$ . Finally, we claim that  $\exp \left(-\frac{1}{\alpha} W*\rho_{*}(z)\right) > 0$  for all  $z\in \mathbb{R}^d$ . In other words, we claim that  $W*\rho_{*}(z) < \infty$  for all  $z\in \mathbb{R}^d$ . This follows by exactly the same argument as in Corollary 12, see equation (19). We conclude that  $\mathrm{supp}(\rho_{*}) = \mathrm{supp}(\tilde{\rho})$ .
Remark 11. If we have in addition that  $\tilde{\rho} \in L^{\infty}(\mathbb{R}^{d})$  and  $f_{1}(\cdot, x) \in L^{\infty}(\mathbb{R}^{d})$  for all  $x \in \mathbb{R}^{d}$ , then the maximizer  $\rho_{*}$  of  $G_{b}$  is in  $L^{\infty}(\mathbb{R}^{d})$  as well. This follows directly by bounding the right-hand side of (32).
With the above preliminary results, we can now show the HWI inequality, which implies again a Talagrand-type inequality and a generalized logarithmic Sobolev inequality.
Proposition 24 (HWI inequalities). Define the dissipation functional
$$
D _ {b} (\gamma) := \iint | \nabla_ {z} \delta_ {\rho} G _ {b} [ \rho ] (z) | ^ {2} \mathrm {d} \rho (z).
$$
Assume  $\alpha, \beta > 0$  such that  $\alpha \tilde{\lambda} > \Lambda_1 + \sigma^2$ , and let  $\lambda_b$  as defined in (27). Denote by  $\rho_*$  the unique maximizer of  $G_b$ .
$(\mathbf{HW})$  Let  $\rho_0, \rho_1 \in \mathcal{P}_2(\mathbb{R}^d)$  such that  $G_b(\rho_0), G_b(\rho_1), D_b(\rho_0) < \infty$ . Then
$$
G _ {b} \left(\rho_ {0}\right) - G _ {b} \left(\rho_ {1}\right) \leq \bar {W} \left(\rho_ {0}, \rho_ {1}\right) \sqrt {D _ {b} \left(\rho_ {0}\right)} - \frac {\lambda_ {b}}{2} W _ {2} \left(\rho_ {0}, \rho_ {1}\right) ^ {2} \tag {33}
$$
(logSobolev) Any  $\rho \in \mathcal{P}_2(\mathbb{R}^d)$  such that  $G(\rho), D_b(\rho) < \infty$  satisfies
$$
D _ {b} (\rho) \geq 2 \lambda_ {b} G _ {a} (\rho | \rho_ {*}). \tag {34}
$$
(Talagrand) For any  $\rho \in \mathcal{P}_2(\mathbb{R}^d)$  such that  $G_{b}(\rho) < \infty$ , we have
$$
W _ {2} \left(\rho , \rho_ {*}\right) ^ {2} \leq \frac {2}{\lambda_ {b}} G _ {b} \left(\rho \mid \rho_ {*}\right). \tag {35}
$$
Proof. The proof for this result follows analogously to the arguments presented in the proofs of Proposition 13, Corollary 14 and Corollary 15, using the preliminary results established in Proposition 21 and Proposition 22.  $\square$
Proof of Theorem 3. Following the same approach as in the proof of Theorem 2, the results in Theorem 3 immediately follow by combining Proposition 22, Corollary 23 and Proposition 24 applied to solutions of the PDE (5).
The proof for this theorem uses similar strategies as that of Theorem 3, but considers the evolution of an ODE rather than a PDE. Recall that for any  $x \in \mathbb{R}^d$  the best response  $r(x)(\cdot) \in \mathcal{P}(\mathbb{R}^d)$  in (6) is defined as
$$
r(x):= \operatorname *{argmax}_{\hat{\rho}\in \mathcal{P}}G_{c}(\hat{\rho},x)  ,
$$
where the energy  $G_{c}(\rho ,x):\mathcal{P}(\mathbb{R}^{d})\times \mathbb{R}^{d}\mapsto [-\infty ,\infty ]$  is given by
$$
G _ {c} (\rho , x) = \int f _ {1} (z, x) \mathrm {d} \rho (z) + \int f _ {2} (z, x) \mathrm {d} \bar {\rho} (z) + \frac {\beta}{2} \| x - x _ {0} \| ^ {2} - \alpha K L (\rho | \tilde {\rho}) - \frac {1}{2} \int \rho W * \rho .
$$
Lemma 25. Let Assumptions 2-4 hold and assume  $\alpha \tilde{\lambda} > \Lambda_1$ . Then for each  $x \in \mathbb{R}^d$  there exists a unique maximizer  $\rho_* := r(x)$  solving  $\operatorname{argmax}_{\hat{\rho} \in \mathcal{P}_2} G_c(\hat{\rho}, x)$ . Further,  $r(x) \in L^1(\mathbb{R}^d)$ ,  $\operatorname{supp}(r(x)) = \operatorname{supp}(\tilde{\rho})$ , and there exists a function  $c: \mathbb{R}^d \mapsto \mathbb{R}$  such that the best response  $\rho_*(z) = r(x)(z)$  solves the Euler-Lagrange equation
$$
\delta_ {\rho} G _ {c} \left[ \rho_ {*}, x \right] (z) := \alpha \log \rho_ {*} (z) - \left(f _ {1} (z, x) + \alpha \log \tilde {\rho} (z)\right) + \left(W * \rho_ {*}\right) (z) = c (x) \text {f o r a l l} (z, x) \in \operatorname {s u p p} (\tilde {\rho}) \times \mathbb {R} ^ {d}. \tag {36}
$$
Proof. Equivalently, consider the minimization problem for  $F(\rho) = -\int f_1(z,x)\mathrm{d}\rho (z) + \alpha KL(\rho |\tilde{\rho}) + \frac{1}{2}\int \rho W*\rho$  with some fixed  $x$ . Note that we can rewrite  $F(\rho)$  as
$$
F (\rho) = \alpha \int \rho \log \rho d z + \int V (z, x) d \rho (z) + \frac {1}{2} \int \rho W * \rho
$$
where  $V(z,x)\coloneqq -(f_1(z,x) + \alpha \log \tilde{\rho} (z))$  is strictly convex in  $z$  for fixed  $x$  by Assumptions 2 and 4. Together with Assumption 3, we can directly apply the uniqueness and existence result from [CMV03, Theorem 2.1 (i)].
The result on the support of  $r(x)$  and the expression for the Euler-Lagrange equation follows by exactly the same arguments as in Corollary 12 and Corollary 23.
Lemma 26. The density of the best response  $r(x)$  is continuous on  $\mathbb{R}^d$  for any fixed  $x \in \mathbb{R}^d$ .
Proof. Instead of solving the Euler-Lagrange equation (36), we can also obtain the best response  $r(x)$  as the long-time asymptotics for the following gradient flow:
$$
\partial_ {t} \rho = \operatorname {d i v} \left(\rho \nabla \delta_ {\rho} F [ \rho ]\right). \tag {37}
$$
Following Definitions 8 and 9, we can characterize the steady states  $\rho_{\infty}$  of the PDE (37) by requiring that  $\rho_{\infty} \in L_{+}^{1}(\mathbb{R}^{d}) \cap L_{loc}^{\infty}(\mathbb{R}^{d})$  with  $\| \rho_{\infty} \|_{1} = 1$  such that  $\rho_{\infty} \in W_{loc}^{1,2}(\mathbb{R}^{d})$ ,  $\nabla W * \rho_{\infty} \in L_{loc}^{1}(\mathbb{R}^{d})$ ,  $\rho_{\infty}$  is absolutely continuous with respect to  $\tilde{\rho}$ , and  $\rho_{\infty}$  satisfies
$$
\nabla_ {z} \left(- f _ {1} (z, x) + \alpha \log \left(\frac {\rho_ {\infty} (z)}{\tilde {\rho} (z)}\right) + W * \rho_ {\infty} (z)\right) = 0 \quad \forall z \in \mathbb {R} ^ {d}, \tag {38}
$$
in the sense of distributions. Noting that because the energy functional  $F(\rho)$  differs from  $G_{a}(\rho, \mu)$  only in the sign of  $f_{1}(z, x)$  if viewing  $G_{a}(\rho, \mu)$  as a function of  $\rho$  only. Note that  $F(\rho)$  is still uniformly displacement convex in  $\rho$  due to Assumption 4. Then the argument to obtain that  $\rho_{\infty} \in C(\mathbb{R}^{d})$  follows exactly as that of Lemma 8.
Lemma 27. Let  $i\in \{1,\dots,d\}$ . If the energy  $H_{i}:\mathcal{P}(\mathbb{R}^{d})\to \mathbb{R}^{d}$  given by
$$
H _ {i} (\rho , x) := \frac {\alpha}{2} \int \frac {\rho (z) ^ {2}}{r (x) (z)} \mathrm {d} z + \frac {1}{2} \int \rho W * \rho - \int \partial_ {x _ {i}} f _ {1} (z, x) \mathrm {d} \rho (z), \tag {39}
$$
admits a critical point at  $x \in \mathbb{R}^d$ , then the best response  $r(x) \in \mathcal{P}(\mathbb{R}^d)$  is differentiable in the  $i$ th coordinate direction at  $x \in \mathbb{R}^d$ . Further, the critical point of  $H_i$  is in the subdifferential  $\partial_{x_i} r(x)$ .
Proof. First, note that  $DF[r(x)](x)(u) = 0$  for all directions  $u \in C_c^\infty(\mathbb{R}^d)$  and for all  $x \in \mathbb{R}^d$  thanks to optimality of  $r(x)$ . Here,  $DF$  denotes the Fréchet derivative of  $F$ , associating to every  $\rho \in \mathcal{P}(\mathbb{R}^d)$  the bounded linear operator  $DF[\rho]: C_c^\infty \to \mathbb{R}$
$$
D F [ \rho ] (u) := \int \delta_ {\rho} F [ \rho ] (z) u (z) \mathrm {d} z,
$$
and we note that  $F(\rho)$  depends on  $x$  through the potential  $V$ . Fixing an index  $i \in \{1, \dots, d\}$ , and differentiating the optimality condition with respect to  $x_i$  we obtain
$$
\partial_ {x _ {i}} D F [ r (x) ] (x) (u) + D ^ {2} F [ r (x) ] (x) \left(u, \partial_ {x _ {i}} r (x)\right) = 0 \quad \forall u \in C _ {c} ^ {\infty} \left(\mathbb {R} ^ {d}\right). \tag {40}
$$
Both terms can be made more explicit using the expressions for the Fréchet derivative of  $F$ :
$$
\partial_ {x _ {i}} D F [ r (x) ] (x) (u) = - \int \partial_ {x _ {i}} f _ {1} (z, x) u (z) \mathrm {d} z,
$$
and for the second term note that the second Fréchet derivative of  $F$  at  $\rho \in \mathcal{P}(\mathbb{R}^d)$  along directions  $u, v \in C_c^\infty(\mathbb{R}^d)$  such that  $(\operatorname{supp}(u) \cup \operatorname{supp}(v)) \subset \operatorname{supp}(\rho)$  is given by
$$
D ^ {2} F [ \rho ] (x) (u, v) = \alpha \int \frac {u (z) v (z)}{\rho (z)} \mathrm {d} z + \iint W (z - \tilde {z}) u (z) v (\tilde {z}) \mathrm {d} z \mathrm {d} \tilde {z}.
$$
In other words, assuming  $\mathrm{supp}(r(x)) = \mathrm{supp}(\tilde{\rho}) = \mathbb{R}^d$ , relation (40) can be written as
$$
\alpha \int \frac {\partial_ {x _ {i}} r (x)}{r (x) (z)} u (z) d z + \int \left(W * \partial_ {x _ {i}} r (x)\right) (z) u (z) v d z - \int \partial_ {x _ {i}} f _ {1} (z, x) u (z) d z = 0,
$$
For ease of notation, given  $r(x)\in \mathcal{P}(\mathbb{R}^d)$ , we define the function  $g:\mathcal{P}(\mathbb{R}^d)\to L_{loc}^1 (\mathbb{R}^d)$  by
$$
g [ \rho ] (z) := \alpha \frac {\rho (z)}{r (x) (z)} + W * \rho - \partial_ {x _ {i}} f _ {1} (z, x).
$$
The question whether the partial derivative  $\partial_{x_i}r(x)$  exists then reduces to the question whether there exists some  $\rho_{*}\in \mathcal{P}(\mathbb{R}^{d})$  such that  $\rho = \rho_{*}$  solves the equation
$$
g [ \rho ] (z) = c \quad \text {f o r a l m o s t e v e r y} z \in \mathbb {R} ^ {d}.
$$
and for some constant  $c > 0$ . This is precisely the Euler-Lagrange condition for the functional  $H_{i}$  defined in (39), which has a solution thanks to the assumption of Lemma 27.
We observe that the first term in  $H_{i}$  is precisely (up to a constant) the  $\chi^2$ -divergence with respect to  $r(x)$ ,
$$
\int \left(\frac {\rho}{r (x)} - 1\right) ^ {2} r (x) d z = \int \frac {\rho^ {2}}{r (x)} d z - 1.
$$
Depending on the shape of the best response  $r(x)$ , the  $\chi^2$ -divergence may not be displacement convex. Similarly, the last term  $-\int \partial_{x_i}f_1(z,x)\mathrm{d}\rho (z)$  in the energy  $H_{i}$  is in fact displacement concave due to the convexity properties of  $f_{1}$  in  $z$ . The interaction term is displacement convex thanks to Assumption 3. As a result, the overall convexity properties of  $H_{i}$  are not known in general. Proving the existence of a critical for  $H_{i}$  under our assumptions on  $f_{1},f_{2},\tilde{\rho}$  and  $W$  would be an interesting result in its own right, providing a new functional inequality that expands on the literature of related functional inequalities such as the related Hardy-Littlewood-Sobolev inequality [Lie83].
It remains to show that  $H_{i}$  indeed admits a critical point. Next, we provide examples of additional assumptions that would guarantee for Lemma 27 to apply.
Lemma 28. If either  $C \coloneqq \sup_{z \in \mathbb{R}^d} |W(z)| < \infty$ , or
$$
C := \sup  _ {z \in \mathbb {R} ^ {d}} | \alpha \log (r (x) (z) / \tilde {\rho} (z)) + f _ {1} (z, x) + c | <   \infty ,
$$
then for each  $x \in \mathbb{R}^d$  and for large enough  $\alpha > 0$ , the best response  $r(x)$  is differentiable with the gradient coordinate  $\partial_{x_i} r(x)$  given by the unique coordinate-wise solutions of the Euler-Lagrange condition for  $H_i$ .
Proof. We will show this result using the Banach Fixed Point Theorem for the mapping  $T_{i}: L^{1}(\mathbb{R}^{d}) \to L^{1}(\mathbb{R}^{d})$  for each fixed  $i \in \{1, \dots, d\}$  given by
$$
T _ {i} (\rho) = - \frac {r (x) (z)}{\alpha} [ (W * \rho) (z) - \partial_ {x _ {i}} f _ {1} (z, x) + c ],
$$
noting that  $\rho_{*} = T_{i}(\rho_{*})$  is the Euler-Lagrange condition for a critical point of  $H_{i}$ . It remains to show that  $T_{i}$  is a contractive mapping. For the first assumption, note that
$$
\begin{array}{l} \left\| T _ {i} (\rho) - T _ {i} \left(\rho^ {\prime}\right) \right\| _ {1} = \frac {1}{\alpha} \int r (x) | W * (\rho - \rho^ {\prime}) | d z \\ \leq \frac {1}{\alpha} \iint r (x) (z) W (z - \hat {z}) | \rho (\hat {z}) - \rho^ {\prime} (\hat {z}) | \mathrm {d} \hat {z} \mathrm {d} z \\ \leq \frac {\| W \| _ {\infty}}{\alpha} \left(\int r (x) (z) \mathrm {d} z\right) \left(\int | \rho (\hat {z}) - \rho^ {\prime} (\hat {z}) | \mathrm {d} \hat {z}\right) \leq \frac {C}{\alpha} \| \rho - \rho^ {\prime} \| _ {1}. \\ \end{array}
$$
Similarly, for the second assumption we estimate
$$
\begin{array}{l} \left\| T _ {i} (\rho) - T _ {i} \left(\rho^ {\prime}\right) \right\| _ {1} = \frac {1}{\alpha} \int r (x) | W * (\rho - \rho^ {\prime}) | d z \\ \leq \frac {1}{\alpha} \iint r (x) (z) W (z - \hat {z}) | \rho (\hat {z}) - \rho^ {\prime} (\hat {z}) | \mathrm {d} \hat {z} \mathrm {d} z \\ = \frac {1}{\alpha} \int (W * r (x)) (z) | \rho (z) - \rho^ {\prime} (z) | \mathrm {d} z \\ \leq \frac {1}{\alpha} \left\| W * r (x) \right\| _ {\infty} \left\| \rho - \rho^ {\prime} \right\| _ {1} \\ \end{array}
$$
which requires a bound on  $\| W * r(x) \|_{\infty}$ . Using
$$
\left\| W * r (x) \right\| _ {\infty} = \sup  _ {z \in \mathbb {R} ^ {d}} \left| \alpha \log \left(r (x) / \tilde {\rho}\right) + f _ {1} (z, x) + c \right| = C <   \infty ,
$$
we conclude that  $T_{i}$  is a contraction map for large enough  $\alpha$ . In both cases, we can then apply the Banach Fixed-Point Theorem to conclude that  $\nabla_{x}r(x)$  exists and is unique.
Lemma 29. Let  $r(x)$  as defined in (6). If  $r(x)$  is differentiable in  $x$ , then we have  $\nabla_{x}G_{d}(x) = (\nabla_{x}G_{c}(\rho ,x))|_{\rho = r(x)}$ .
Proof. We start by computing  $\nabla_x G_d(x)$ . We have
$$
\begin{array}{l} \nabla_ {x} G _ {d} (x) = \nabla_ {x} \left(G _ {c} (r (x), x)\right) = \int \delta_ {\rho} [ G _ {c} (\rho , x) ] | _ {\rho = r (x)} (z) \nabla_ {x} r (x) (z) d z + \left. (\nabla_ {x} G _ {c} (\rho , x)) \right| _ {\rho = r (x)} \\ = c (x) \nabla_ {x} \int r (x) (z) \mathrm {d} z + (\nabla_ {x} G _ {c} (\rho , x)) | _ {\rho = r (x)} = (\nabla_ {x} G _ {c} (\rho , x)) | _ {\rho = r (x)}, \\ \end{array}
$$
where we used that  $r(x)$  solves the Euler-Lagrange equation (36) and that  $r(x)\in \mathcal{P}(\mathbb{R}^d)$  for any  $x\in \mathbb{R}^d$  so that  $\int r(x)(z)\mathrm{d}z$  is independent of  $x$ .
Lemma 30. Let Assumption 1 hold. Then  $G_{d}:\mathbb{R}^{d}\to \mathbb{R}\cup \{+\infty \}$  is strongly convex with constant  $\lambda_{d}:= \lambda_{1} + \lambda_{2} + \beta >0$ .
Proof. The energy  $G_{c}(\rho, x)$  is strongly convex in  $x$  due to our assumptions on  $f_{1}, f_{2}$ , and the regularizing term  $\| x - x_{0} \|_{2}^{2}$ . This means that for any  $\rho \in \mathcal{P}$ ,
$$
G _ {c} (\rho , x) \geq G _ {c} (\rho , x ^ {\prime}) + \nabla_ {x} G _ {c} (\rho , x ^ {\prime}) ^ {\top} (x - x ^ {\prime}) + \frac {\lambda_ {d}}{2} \| x - x ^ {\prime} \| _ {2} ^ {2}.
$$
Selecting  $\rho = r(x')$ , we have
$$
G _ {c} \left(r \left(x ^ {\prime}\right), x\right) \geq G _ {c} \left(r \left(x ^ {\prime}\right), x ^ {\prime}\right) + \nabla_ {x} G _ {c} \left(r \left(x ^ {\prime}\right), x ^ {\prime}\right) ^ {\top} \left(x - x ^ {\prime}\right) + \frac {\lambda_ {d}}{2} \left\| x - x ^ {\prime} \right\| _ {2} ^ {2}.
$$
Since  $G_{c}(r(x^{\prime}),x) \leq G_{c}(r(x),x)$  by definition of  $r(x)$ , we obtain the required convexity condition:
$$
G _ {d} (x) = G _ {c} \left(r (x), x\right) \geq G _ {c} \left(r \left(x ^ {\prime}\right), x ^ {\prime}\right) + \nabla_ {x} G _ {c} \left(r \left(x ^ {\prime}\right), x ^ {\prime}\right) ^ {\top} \left(x - x ^ {\prime}\right) + \frac {\lambda_ {d}}{2} \left\| x - x ^ {\prime} \right\| _ {2} ^ {2}.
$$
Proof of Theorem 4. For any reference measure  $\rho_0\in \mathcal{P}$ , we have
$$
G _ {d} (x) \geq G _ {c} (\rho_ {0}, x) \geq - \alpha K L (\rho_ {0} \mid \tilde {\rho}) - \frac {1}{2} \int \rho_ {0} W * \rho_ {0} + \frac {\beta}{2} \| x - x _ {0} \| ^ {2}
$$
and therefore,  $G_{d}$  is coercive. Together with the strong convexity provided by Lemma 30, we obtain the existence of a unique minimizer  $x_{\infty} \in \mathbb{R}^{d}$ . Convergence in norm now immediately follows also using Lemma 30: for solutions  $x(t)$  to (6), we have
$$
\frac {1}{2} \frac {\mathrm {d}}{\mathrm {d} t} \| x (t) - x _ {\infty} \| ^ {2} = - \left(G _ {d} (x (t)) - G _ {d} (x _ {\infty})\right) \cdot \left(x (t) - x _ {\infty}\right) \leq - \lambda_ {d} \| x (t) - x _ {\infty} \| ^ {2}.
$$
A similar result holds for convergence in entropy using the Polyak-Lojasiewicz convexity inequality
$$
\frac {1}{2} \left\| \nabla G _ {d} (x) \right\| _ {2} ^ {2} \geq \lambda_ {d} \left(G _ {d} (x) - G _ {d} \left(x _ {\infty}\right)\right),
$$
which is itself a direct consequence of strong convexity provided in Lemma 30. Then
$$
\frac {\mathrm {d}}{\mathrm {d} t} \left(G _ {d} (x (t)) - G _ {d} (x _ {\infty})\right) = \nabla_ {x} G _ {d} (x (t)) \cdot \dot {x} (t) = - \| \nabla_ {x} G _ {d} (x (t)) \| ^ {2} \leq - 2 \lambda_ {d} \left(G _ {d} (x (t)) - G _ {d} (x _ {\infty})\right),
$$
and so the result in Theorem 4 follows.
E Additional Simulation Results
We simulate a number of additional scenarios to illustrate extensions beyond the setting with provable guarantees and in the settings for which we have results but no numerical implementations in the main paper. First, we simulate the aligned objectives setting in one dimension, corresponding to (4). Then we consider two settings which are not covered in our theory: (1) the previously-fixed distribution  $\bar{\rho}$  is also time varying, and (2) the algorithm does not have access to the full distributions of  $\rho$  and  $\bar{\rho}$  and instead samples from them to update. Lastly, we illustrate a classifier with the population attributes in two dimensions, which requires a different finite-volume implementation [CCH15, Section 2.2] than the one dimension version of the PDE due to flux in two dimensions.
E.1 Aligned Objectives
Here we show numerical simulation results for the aligned objectives case, where the population and distribution have the same cost function. In this setting, the dynamics are of the form
$$
\begin{array}{l} \partial_ {t} \rho = \operatorname {d i v} \left(\rho \nabla_ {z} \delta_ {\rho} G _ {a} [ \rho , \mu ]\right) \\ = \operatorname {d i v} \left(\rho \nabla_ {z} \left(\int f _ {1} (z, x) \mathrm {d} \mu (x) + \alpha \log (\rho / \tilde {\rho}) + W * \rho\right)\right) \\ \end{array}
$$
$$
\frac {\mathrm {d}}{\mathrm {d} t} x = - \nabla_ {x} \left(\int f _ {1} (z, x) \mathrm {d} \rho (z) + \int f _ {2} (z, x) \mathrm {d} \bar {\rho} (z) + \frac {\beta}{2} \| x - x _ {0} \| ^ {2}\right)
$$
where  $f_{1}$  and  $f_{2}$  are as defined in section 4.1, and  $W = \frac{1}{20}(1 + z)^{-1}$ , a consensus kernel. Note that  $W$  does not satisfy Assumption 3, but we still observe convergence in the simulation. This is expected; in other works such as [CMV03], the assumptions on  $W$  are relaxed and convergence results proven given sufficient convexity of other terms. The regularizer  $\tilde{\rho}$  is set to  $\rho_0$ , which models a penalty for the effort required of individuals to alter their attributes. The coefficient weights are  $\alpha = 0.1$  and  $\beta = 1$ , with discretization parameters  $dz = 0.1$ ,  $dt = 0.01$ .
Figure 4: The dynamics include a consensus kernel, which draws neighbors in  $z$ -space closer together. We see that the population moves to make the classifier performer better, as the two distributions become more easily separable by the linear classifier.
In Figure 4, we observe the strategic distribution separating itself from the stationary distribution, improving the performance of the classifier and also improving the performance of the population itself. The strategic distribution and classifier appear to be stationary by time  $t = 40$ .
E.2 Multiple Dynamical Populations
We also want to understand the dynamics when both populations are strategic and respond to the classifier. In this example, we numerically simulate this and in future work we hope to prove additional results regarding convergence. This corresponds to modeling the previously-fixed distribution  $\bar{\rho}$  as time-dependent; let this distribution be  $\tau \in \mathcal{P}_2$ . We consider the case where  $\rho$  is competitive with  $x$  and  $\tau$  is aligned with  $x$ , with dynamics given by
$$
\partial_ {t} \rho = - \operatorname {d i v} \left(\rho \nabla_ {z} \left(f _ {1} (z, x) - \alpha \log (\rho / \tilde {\rho}) - W * \rho)\right) \right.
$$
$$
\partial_ {t} \tau = \operatorname {d i v} \left(\tau \nabla_ {z} \left(f _ {2} (z, x) + \alpha \log (\tau / \tilde {\tau}) + W * \tau\right)\right)
$$
$$
\frac {\mathrm {d}}{\mathrm {d} t} x = - \nabla_ {x} \left(\int f _ {1} (z, x) \mathrm {d} \rho (z) + \int f _ {2} (z, x) \mathrm {d} \tau (z) + \frac {\beta}{2} \| x - x _ {0} \| ^ {2}\right).
$$
We use  $W = 0$  and  $f_{1}, f_{2}$  as in section 4.1 and the same discretization parameters as in Section E.1. In Figure 5,
Figure 5: The population  $\rho$  aims to be classified with the  $\tau$  population, while the classifier moves to delineate between the two. We observe that  $\tau$  adjusts to improve the performance of the classifier while  $\rho$  competes against it. The distributions are plotted at time  $t = 0$ , corresponding to  $\tilde{\rho}$  and  $\tilde{\tau}$ , and time  $t = 20$ , corresponding to  $\rho$  and  $\tau$ .
we observe that the  $\tau$  population moves to the right, assisting the classifier in maintaining accurate scoring. In contrast,  $\rho$  also moves to the right, rendering the right tail to be classified incorrectly, which is desirable for individuals in the  $\rho$  population but not desirable for the classifier. While we leave analyzing the long-term behavior mathematically for future work, the distributions and classifier appear to converge by time  $t = 20$ .
E.3 Sampled Gradients
In real-world applications of classifiers, the algorithm may not know the exact distribution of the population, relying on sampling to estimate it. In this section we explore the effects of the classifier updating based on an approximated gradient, which is computed by sampling the true underlying distributions  $\rho$  and  $\bar{\rho}$ . We use the same parameters for the population dynamics as in section 4.1, and for the classifier we use the approximate gradient
$$
\nabla_ {x} L (z, x _ {t}) \approx \frac {1}{n} \sum_ {i = 1} ^ {n} \left(\nabla_ {x} f _ {1} \left(z _ {i}, x _ {t}\right) + \nabla_ {x} f _ {2} \left(\bar {z} _ {i}, x _ {t}\right)\right) + \beta \left(x _ {t} - x _ {0}\right), \quad z _ {i} \sim \rho_ {t}, \quad \bar {z} _ {i} \sim \bar {\rho} _ {t}.
$$
First, we simulate the dynamics with the classifier and the strategic population updating at the same rate, using  $\alpha = 0.05$ ,  $\beta = 1$ , and the same consensus kernel as used previously, with the same discretization parameters as in E.1. In Figure 6, we observe no visual difference between the two results with  $n = 4$  versus  $n = 40$  samples, which suggests that not many samples are needed to estimate the gradient.
Next, we consider the setting where the classifier is best-responding to the strategic population.