Differential Games with Asymmetric and Correlated Information

Abstract

Differential games with asymmetric information were introduced by Cardaliaguet (SIAM J Control Optim 46:816–838, 2007). As in repeated games with lack of information on both sides (Aumann and Maschler in Repeated games with incomplete information, with the collaboration of R. Stearns, 1995), each player receives a private signal (his type) before the game starts and has a prior belief about his opponent’s type. Then, a differential game is played in which the dynamic and the payoff functions depend on both types: each player is thus partially informed about the differential game that is played. The existence of the value function and some characterizations have been obtained under the assumption that the signals are drawn independently. In this paper, we drop this assumption and extend these results to the general case of correlated types. As an application, we provide a new characterization of the asymptotic value of repeated games with incomplete information on both sides, as the unique dual solution of a Hamilton–Jacobi equation.

Appendix

Let us describe precisely the standard transformation from a Bolza to a Mayer problem, which allows to assume without loss of generality that there is no running payoff.

The past controls being commonly observed, both players can compute the $K\times L$ potential integral payoffs and positions induced by the pair $(\mathbf {u},\mathbf {v})\in \mathcal {U}(t_0)\times \mathcal {V}(t_0)$ at time $t\in [t_0,1]$:

$$\begin{aligned} \int _{t_0}^t \gamma ^{k\ell }(s,\mathbf {x}^{k\ell }[t_0,x^{k\ell }_0,\mathbf {u},\mathbf {v}](s),\mathbf {u}(s),\mathbf {v}(s))\mathrm{d}s, \quad (k,\ell )\in K\times L \end{aligned}$$

$$\begin{aligned} \mathbf {x}^{k\ell }[t_0,x^{k\ell }_0,\mathbf {u},\mathbf {v}](t), \quad (k,\ell )\in K\times L. \end{aligned}$$

Define a new state variable in $(\mathbb {R}\times \mathbb {R}^n)^{K\times L }$ which contains this information. Let the dynamic be given by

$$\begin{aligned} F:[0,1]\times (\mathbb {R}\times \mathbb {R}^n)^{K\times L}\times U\times V&\rightarrow (\mathbb {R}\times \mathbb {R}^n)^{K\times L},\\ F^{k\ell }\left( t,(y^{k\ell },x^{k\ell })_{(k,\ell )},u,v\right)&= \left( \gamma ^{k\ell } (t,x^{k\ell },u,v ), f^{k\ell }(t,x^{k\ell },u,v)\right) . \end{aligned}$$

Define new terminal payoff functions by setting, for each $(k,\ell )\in K\times L$,

$$\begin{aligned} G^{k\ell }:(\mathbb {R}\times \mathbb {R}^n)^{K\times L}&\rightarrow \mathbb {R},\\ G^{k\ell }((y^{k,\ell },x^{k\ell })_{(k,\ell )})&= y^{k\ell }+g^{k\ell }(x^{k\ell }). \end{aligned}$$

Let $N=(1+n)|K| |L|$ and let $X_0:=(0,x_0^{k\ell })_{(k,\ell )\in K\times L}\in \mathbb {R}^N$. The regularity of $f$, $\gamma$ and $g$ ensures that $F$ and $G^{k\ell }$ satisfy Assumption 2.1, for each $(k,\ell )$. Define an auxiliary differential game with asymmetric information as follows: before the game starts, $(k,\ell )\in K\times L$ is drawn according to $\pi$; $k$ (resp. $\ell$) is told to player $1$ (resp. $2$). Then, the standard differential game

$$\begin{aligned} (X_0,F,0,(G^{k\ell })_{k,\ell }) \end{aligned}$$

is played. This game is equivalent to $\mathcal {G}(t_0,\pi )$ since any couple of controls $(\mathbf {u},\mathbf {v})\in \mathcal {U}(t_0)\times \mathcal {V}(t_0)$ induces the same payoff in both games.