Abstract
In this paper, we analyze linear-quadratic stochastic differential games with a continuum of players interacting through graphon aggregates, each state being subject to idiosyncratic Brownian shocks. The major technical issue is the joint measurability of the player state trajectories with respect to samples and player labels, which is required to compute for example costs involving the graphon aggregate. To resolve this issue we set the game in a Fubini extension of a product probability space. We provide conditions under which the graphon aggregates are deterministic and the linear state equation is uniquely solvable for all players in the continuum. The Pontryagin maximum principle yields equilibrium conditions for the graphon game in the form of a forward-backward stochastic differential equation, for which we establish existence and uniqueness. We then study how graphon games approximate games with finitely many players over graphs with random weights. We illustrate some of the results with a numerical example.
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Acknowledgements
The authors would like to thank Yeneng Sun, François Delarue, and Dan Lacker for helpful discussions.
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This work was partially supported by NSF DMS-1716673; ARO W911NF-17-1-0578 and AFSOR FA9550-19-1-0291.
A Proofs
A Proofs
1.1 A.1 Proof of Proposition 1
We drop the superscript \({\underline{\alpha }}\) since the strategy profile does not change throughout the proof. By Lemma 3, \({\mathbb {X}}^z\in L^2_\boxtimes (\varOmega \times I; E)\) so by Lemma 1, the mapping U is well defined. We turn to the contraction property.
To prove that U is a strict contraction, let \(z, {\tilde{z}} \in L^2_{\boxtimes }(\varOmega \times I; E)\). By Gronwall’s inequality and the boundedness of the graphon,
where \(C>0\) is a finite constant depending only on T, \(\Vert W\Vert _2\), and the coefficient bound. Iterating the inequality (23) and making use of the fact that
we have
Hence, for some \(N\in {\mathbb {N}}\), \((U^N)\) is a contraction mapping on from \(L^2_{\boxtimes }(\varOmega \times I; E)\) to itself. The existence of a unique fixed point to U in the Banach space \(L^2_{\boxtimes }(\varOmega \times I; E)\) then follows by the Banach fixed-point theorem for iterated mappings, see e.g. [5]. \(\square \)
1.2 A.2 Proof of Theorem 3
The coefficient matrices in (16) are given in terms of \(a,b,c,C_f\) and \(C_h\) as follows:
Step 0: Uniqueness in the Fubini extension
Assume that (X, p, q) and \(({\widetilde{X}}, {\widetilde{p}}, \widetilde{q})\) are solutions to the FBSDE (16) in the sense that (16) is satisfied \({\mathbb {P}}\boxtimes \lambda \)-a.s. and \({\mathbb {E}}^\boxtimes \big [ \Vert X\Vert _E^2 + \Vert p\Vert _E^2 + \int _0^T|q_t|^2dt\big ] < \infty \). Uniqueness, i.e., that
can be proven along standard lines of proof.
Step 1: An ansatz for \(p^x\)
We will look for a solution defined with the following ansatz: for each \(x\in I\) there exists differentiable and deterministic mappings \(t\mapsto \eta ^x_t\) and \(t\mapsto \zeta ^x_t\) such that
Plugging (24) into (16) and matching terms we obtain \(\eta ^x = q^x\) and the following system for \(({\hat{X}}^x,\zeta ^x,\eta ^x; x\in I)\):
The Riccati equation for \(\eta ^x\) in (25) does not depend on the other variables and can be solved independently. Furthermore, under Assumption 1 and 2 it has a unique solution \((\eta ^x_t)_{t\in [0,T]}\) for all \(x\in I\) and \(\sup _{(t,x)\in [0,T]\times I}|\eta ^x_t| < \infty \), see for example [9, Sec. 2.4.1]. Thus, to prove existence of a solution to (25) it is sufficient to study the forward-backward system for \(({\hat{X}}, \zeta )\), which is the subject matter of the next steps.
Step 2: Unique solvability of (25) for short time horizons
If we fix a collection of aggregates \({\hat{Z}}^x\in E\), \(x\in I\), then \(\zeta ^x\) and subsequently \({\hat{X}}^x\) can be solved explicitly for all \(x\in I\). This "decoupling" property of the aggregate provides us with a simple proof of short time existence and uniqueness. By a fixed-point argument there exists a unique solution \(({\hat{X}}^x,\zeta ^x)\) to (25) in \(L^2_\boxtimes (\varOmega \times I; E)\times L^2(I;E)\) when T is small enough.
Step 3: Setting the stage for the induction approach
Inspired by the induction approach, described in detail in [9, Sec. 4.1.2.], we now extend existence and uniqueness from the previous step to any finite time horizon.
For any \(\tau \in [T_0,T]\), where \(T_0 := T-c_0\) and \(c_0>0\), let \(\xi _\tau \) be such that \((\xi ^x_\tau )_{x\in I}\) are e.p.i. and \(\xi ^x_\tau \) is \({\mathcal {F}}^x_\tau \)-measurable for all \(x\in I\). Assume that \(c_0>0\) is small enough so that
has a unique solution as found in Step 2. Denote the solution \(({\hat{X}}^{0:x,\tau ,\xi _\tau ^\cdot }_t,\zeta ^{0:x,\tau ,\xi _\tau ^\cdot }_t; t\in [\tau ,T])\).
Assume now that the forward-backward system (25) has a solution over the full time horizon: \(({\hat{X}}^x_t,\zeta ^x_t; t\in [0,T])\). It is also a solution to (26) on the subinterval \([T_0,T]\) with \({\hat{X}}_{T_0}^x\) as initial condition at \(T_0 = T-c_0\). By the unique solvability of (26),
Now consider for some \(\tau \in [0,T_0]\) the forward-backward system
with \(\xi ^x_\tau \) satisfying the same assumptions as above but with the new \(\tau \in [0, T_0]\) replacing the old \(\tau \in [T_0,T]\). System (27) differs from (25) in the terminal condition for the backward equation. In the next steps we deduce small-time existence and uniqueness of solutions of (27), we patch the solution with \(({\hat{X}}^{0:x,T_0,{\hat{X}}^\cdot _{T_0}}_t, \zeta ^{0:x,T_0,{\hat{X}}^\cdot _{T_0}}_t; t\in [T_0,T])\), then we repeat and show that after a finite number of patching rounds we are left with the unique solution to (25).
Step 4: Unique solvability of (27) for short time horizons
Let \(\tau \in [0, T_0]\) and \(E_{[\tau ,T_0]} := C([\tau ,T_0])\). Let \(V_{[\tau ,T_0]}\) and \(V_{\boxtimes ,[\tau ,T_0]}\) denote \(L^2(I; E_{[\tau ,T_0]})\) and \(L^2_\boxtimes (\varOmega \times I; E_{[\tau ,T_0]})\), respectively. A fixed-point argument can be made to prove the existence of a \(c_1>0\) such that if \(T_0-\tau \le c_1\), then there is a unique solution to (27) in \(V_{\boxtimes ,T_0}\times V_{T_0}\). Denote the solution by \(({\hat{X}}^{1:x,\tau ,\xi ^\cdot _{\tau }}_t, \zeta ^{1:x,\tau ,\xi ^\cdot _{\tau }}_t; t\in [\tau ,T_0])\).
Let \(T_1 := T_0 - c_1\). Most importantly (since we aim to use the induction approach) \(c_1\) can take any value smaller than a constant \({{\bar{C}}}\) depending only on the time horizon T and the coefficient function bound (from Assumption 1). The fixed-point calculations are tedious and omitted here. The main difficulty comes from that the terminal condition of the backward part of the equation depends on the solution of (26) initiated at the solution of (27). This can however be overcome by using the fixed-point argument for (26) from Step 2.
Step 5: Patching the solutions over \([T_1, T_0]\) and \([T_0, T]\)
We now patch the two solutions. Let \(\tau \in [T_1,T_0]\) and \(\xi _\tau \) be the initial condition vector in (27). For any \(s\in [\tau , T]\),
Then, \({\mathbb {P}}\boxtimes \lambda \)-a.s.,
so \(({\hat{X}}^x_s, \zeta ^x_s; \tau \le s\le T)\) is \({\mathbb {P}}\boxtimes \lambda \)-a.s. continuous and the unique solution to the forward-backward system
Step 6: The induction approach
Consider the following forward-backward system: for some \(\tau \in [0, T_1]\),
Repeating the analysis that was done for the interval \([T_1,T_0]\) proves unique solvability for (29) whenever \(T_1 - \tau < c_1\), with \(c_1\) being the same constant that was found above in Step 4. Hence, we have a unique solution to (29) on the time interval \([T-(2c_1+c_0), T-(c_1+c_0)] =: [T_2,T_1]\). This solution can be patched with the solution on \([T_1,T]\) as was done in Step 5. After a finite number N of iterations (an explicit lower bound on N can be found, depending only on T and \({{\bar{C}}}\), the constant from Step 4), the whole interval [0, T] has been covered and the patching has yielded a unique solution to (25) over the interval [0, T] for any finite \(T>0\). \(\square \)
1.3 A.3 Proof of Proposition 3
The proof relies heavily on a bound for the following random variable: for a fixed \(x^\infty \in I^\infty \) and \(N\in {\mathbb {N}}\), we define
We know \(\zeta ^{x^\infty }_N\) is well-defined for all \(x^\infty \in I^\infty \) since the graphon game state \(X^x\) is defined for all \(x\in I\). The proofs of the lemmas below are found in the end of this section.
Lemma 4
For all \(x^\infty \in I^\infty \) and \(N\in {\mathbb {N}}\) there exists a constant C, independent of \(x^\infty \) and N, such that
Before attending to the first claim of the proposition, we establish a useful estimate. It follows from the Burkholder-Davis-Gundy inequality, Gronwall’s lemma, and the uniform integrability of the initial conditions that
for some finite \(C(T,w) > 0\) that depends only on T and w. Adding and subtracting \(\frac{1}{N}\sum _{k=1}^N w(x,x_k){\mathbb {E}}[X^{x_k}_t]\) to \(\zeta ^{x^\infty }_N(t,x)\) results in
By Lemma 1, \((w(x, x_k)(X^{x_k}_t - {\mathbb {E}}[X^{x_k}_t]))_{k=1}^\infty \) are mutually independent \({\bar{\lambda }}^\infty \)-a.s. Thus \({\bar{\lambda }}^\infty \text {-a.s.}\)
The summands on the right hand side of (32) can be bounded by a constant independent of (t, x), using the estimate derived above. We get
for some constant \(C(T,w)>0\) depending only on T and w. We move on to the second term of the right hand side of (31). By the strong law of large numbers, it tends to zero almost surely as \(N\rightarrow \infty \), proving the first claim of the proposition. To prove the second claim, we prove tightness of the term and then apply the Law of Iterated Logarithms.
Let \(m(t,x) := \int _I w(x,y){\mathbb {E}}[X^y_t]\lambda (dy)\) be the mean of \(w(x,x_k){\mathbb {E}}[X^{x_k}_t]\) when \(x_k\) is \(\lambda \)-distributed. Let furthermore \(\theta _k(x^\infty ) : (t,x) \mapsto w(x,x_k){\mathbb {E}}[X^{x_k}_t]-m(t,x)\). \(\theta _k\) is a random variable on \((I^\infty ,{\mathcal {I}}^\infty , \lambda ^\infty )\) into \(C([0,T]\times I)\).
Lemma 5
The collection \((\varTheta _N)_{N\in {\mathbb {N}}}\), where \(\varTheta _N := \frac{1}{\sqrt{N}}\sum _{k=1}^N\theta _k\), is tight.
By Prokhorov’s theorem (see e.g. [26, Theorem 16.3]), this yields relative compactness in distribution of \((\varTheta _N)_N\). Moreover, the finite-dimensional distributions converge. Indeed, for any \(n \in {\mathbb {N}}\) and any \(r_1,\dots ,r_n \in [0,T] \times I\), we have that the sequence \((\varTheta _N(r_1), \dots , \varTheta _N(r_n))_{N=1,2,\dots }\) converges in distribution by the standard central limit theorem. Hence, by [26, Lemma 16.2], \((\varTheta _N)_N\) converges in distribution. By definition, this means that \(\theta _1(x^\infty ):(t,x) \mapsto w(x,x_1){\mathbb {E}}[X^{x_1}_t]-m(t,x)\) satisfies CLT (see [31, Section 10.1]). Note that \(\theta _k,k=2,3,\dots ,\) are independent copies of \(\theta _1\). Thus, \((\varTheta _N)_N\) satisfies a Law of the Iterated Logarithm (see e.g. [31, Theorem 10.12]). More precisely, we obtain the following.
Lemma 6
There exists a constant C such that
In other words, Lemma 6 says that, for \(\lambda ^\infty \)-a.e. \(x^\infty \in I^\infty \):
and we have for \(\lambda ^\infty \)-a.e. \(x^\infty \in I^\infty \), for all \(N \ge N_\varepsilon (x^\infty )\)
The last statement also holds a.s. in \((I^\infty , \bar{{\mathcal {I}}}^\infty ,{\bar{\lambda }}^\infty )\), see [21, Section 6]. \(\square \)
1.3.1 A.3.1 Proof of Lemma 4
From standard estimates for linear SDEs and BSDEs we get
Next, we will the estimate for the right hand side term containing off-diagonal adjoint state variables. Consider the following auxiliary BSDE system: for \(k=1,\dots , N\), \(p^{kk,N}_t = 0\) and \(h = 1,\dots , N\), \(h\ne k\),
The difference \(p^{kh,N}_t - {\widetilde{p}}^{kh,N}_t\), \(1\le h\ne k \le N\), satisfies the BSDE
Standard BSDE estimates (relying on the integrability of \(X^{k,N}, Z^{k,N}\), and \(p^{kk,N}\)) yield
with C independent of k, h, and N. The unique solution to (33) is \({\widetilde{p}}^{k h, N}_t = {\widetilde{q}}^{k h \ell }_t = 0\) for \(t\in [0,T]\), \(\ell = 1,\dots , N\), \(1\le k\ne h\le N\). \(\square \)
1.3.2 A.3.2 Proof of Lemma 5
We will apply the tightness criterion provided by [26, Corollary 16.9], which stems from the Kolmogorov-Chentsov criterion. We first note that
where the random variables are i.i.d. (and with mean 0). So the sequence \((\varTheta _N(0,0))_N\) is tight. Moreover, we prove that there exists a constant \(C>0\) and there exists a positive integer \(N_0\in {\mathbb {N}}\) such that for every \((t,x), (t',x') \in [0,T]\times I\), \(N \ge N_0\),
As a consequence of [26, Corollary 16.9], we will obtain that the sequence of \((\varTheta _N)_N\) is tight and the limiting processes are \(\lambda ^\infty \)-a.s. locally Hölder continuous with exponent 1/2.
We now prove the claim. Let \((t,x), (t',x') \in [0,T]\times I\), \(N\in {\mathbb {N}}\). Letting \({\mathbf {T}}_k := \theta _k(i^\infty )(t,x) - \theta _k(x^\infty )(t',x')\), we note that
The first term can be made arbitrarily small by taking N large enough. The third term is zero. To bound the second term from above, we observe that:
Furthermore, for the last term, we have by definition of m,
Hence:
\(\square \)
1.3.3 A.3.3 Proof of Lemma 6
Using (30), we have that
where \(C(w,T) > 0\) is a finite constant depending only on the graphon w and T. Let \(Lt := \max (1,\log t)\) for \(t\ge 0\). Using the uniform bound derived above, we see that
The claim now follows from the Law of Iterated Logarithms in Banach spaces, see, e.g., [31, Theorem 10.12]. \(\square \)
1.4 A.4 Proof of Proposition 4
We first prove the convergence claim. Recall that the two equilibria are linear functions of state, costate, and aggregate, so the propagation of chaos from Proposition 3 immediately yields
Moving on to the approximation claim, let \(\hat{\alpha }^{-k,N} := (\hat{\alpha }^{1,N},\dots , \hat{\alpha }^{k-1,N},\hat{\alpha }^{k+1,N},\dots , \hat{\alpha }^{N,N})\). Since \((\hat{\alpha }^{1,N},\dots , {\hat{\alpha }}^{N,N})\) is a Nash equilibrium for the N-player game
Let \(\beta \) be an admissible control for player 1 in the N-player game. Let \(({{\bar{X}}}^{k,N,\beta })_{k=1}^N\) be the player states when player 1 is using \(\beta \) and the others are using the graphon game equilibrium control
Let \(({{\hat{X}}}^{k,N,\beta })_{k=1}^N\) be the player states when player 1 is using \(\beta \) and the others are using the N-player game equilibrium control
We have for any admissible \(\beta \) and \({\bar{\lambda }}^\infty \)-a.e. \(x^\infty \in I^\infty \) that
In the same way we can perturb the other players’ actions and that concludes the proof. \(\square \)
1.5 A.5 Conditions for Well-Posedness of the Time-Varying Riccati Equation (20)
Let
Set \({\mathbb {D}}^k := ({\mathbb {B}}^k_t)^2 + {\mathbb {C}}^k {\mathbb {A}}_t - \dot{{\mathbb {B}}}^k_t\). Note that \({\mathbb {D}}^k\) does not depend on t. Indeed, using the ODE satisfied by \(\eta \) (i.e., (25) but with constant coefficients) and the identity \(\varGamma _{11} = -\varGamma _{22}\), we have
Last, we let:
Proposition 5
Assume \({\mathbb {D}}^k \ge 0\) and, for all \(t \in [0,T]\), \({\mathbb {F}}^k e^{\sqrt{{\mathbb {D}}^k}t} - e^{-\sqrt{{\mathbb {D}}^k}t} \ne 0\). Then the Riccati equation (20) has a unique solution.
Remark 4
Note that we can rewrite \({\mathbb {D}}^k\) as:
So, to ensure \({\mathbb {D}}^k \ge 0\) independently of the value of \(\lambda _k\), a sufficient condition is:
Proof
Existence: We first consider the ODE with the mixed initial condition
A solution is given by \(\nu ^k_t = {\mathbb {F}}^k e^{\sqrt{{\mathbb {D}}^k}t} - e^{-\sqrt{{\mathbb {D}}^k}t}\). Let \(\theta ^k_t = \nu ^k_{T-t}\), for \(t \in [0,T]\). It solves the ODE with the mixed terminal condition:
To conclude, let
Then it can be checked that \(\pi ^k_t\) solves
which is equivalent to (20).
Uniqueness: Let us consider \(\pi ^k\) and \({\tilde{\pi }}^k\) solving (20). Reverting the above change of variables yields solutions \(\nu ^k\) and \({\tilde{\nu }}^k\) to (34) such that \({\tilde{\nu }}^k_t \ne 0\) and \(\nu ^k_t \ne 0\) for all \(t \in [0,T]\). For any such solutions, there exist constants \(C_1, C_2, {{\tilde{C}}}_1, {{\tilde{C}}}_2\) such that
Due to (34), we have:
As a consequence, we necessarily have \(C_1/C_2 = {{\tilde{C}}}_1 / \tilde{C}_2 = - {\mathbb {F}}^k\). We deduce that \(\pi ^k_t = {\tilde{\pi }}^k_t\) for all \(t \in [0,T]\). \(\square \)
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Aurell, A., Carmona, R. & Laurière, M. Stochastic Graphon Games: II. The Linear-Quadratic Case. Appl Math Optim 85, 39 (2022). https://doi.org/10.1007/s00245-022-09839-2
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DOI: https://doi.org/10.1007/s00245-022-09839-2
Keywords
- Stochastic differential games
- Continuum of players
- Graphons
- Fubini extensions
- Exact law of large numbers