Static Game
Let $i \in \{1, 2\}$ denote a player. Each player can execute an action $a_i \in A_i$, where $A_i \subseteq \mathbb R$ denotes the set of feasible actions. Given a pair of actions $a = (a_1, a_2) \in A = A_1 \times A_2$ each player derives a payoff defined by the function $u_i : A \to \mathbb R$. Let $d = (d_1, d_2) \in \mathbb R^2$ denote a given outside option (threat-point). The Nash Bargaining Solution $u(\overline a) = (u_1(\overline a), u_2(\overline a))$ is defined as the payoff that result from the following program \begin{align} &{\overline a} \in \arg\max_{a \in A}(u_1(a) - d_1) \cdot (u_2(a) - d_2)\\ \text{s.t.} \quad & u(a) \geq d \end{align}
Dynamic Game
Let $t \in T = \mathbb R_+$ denote time. Now, each player chooses an action path $a_i : T \to A_i$. Let $s : T \to S$ denote the state variable, where $S \subseteq \mathbb R$ denotes the state-space. Assume that the state is governed by a stationary differential equation $f : S \times A \to S$ such that \begin{align} {\dot s}(t) = \frac{{\text d} s(t)}{{\text d} t} = f(s(t), a(t)). \end{align} Let $\mathcal A_i = \{\alpha_i : S \to A_i\}$ denote the set of feasible stationary Markovian feedback stratgies and $\mathcal A = \mathcal A_1 \times \mathcal A_2$ the joint strategy space. The payoff function $u_i : S \times A \to \mathbb R$ now also depends on the state. The (stationary) payoff functional $U_i : S \times \mathcal A \to \mathbb R$ is defined as \begin{align} U_i(s, \alpha) = \left[\int_{t}^{\infty}{e^{-\delta(\tau - t)} u_i(s(\tau), \alpha(s(\tau))){\text d} \tau} ~ \Bigg| ~ s = s(t)\right] \end{align} where $\delta > 0$ denotes the time-preference rate. Let $d_i : S \to \mathbb R$ denote a state-dependend outside option.
For $s \in S$, the Dynamic Nash Bargaining Solution $U(s, \overline \alpha)$ is defined as the payoff that results from the following program \begin{align} &\overline \alpha \in \arg\max_{\alpha \in \mathcal A} (U_1(s, \alpha) - d_1(s)) \cdot (U_2(s, \alpha) - d_2(s))\\ \text{s.t.} \quad &\dot s = f(s, \alpha)\\ & U(s, \alpha) \geq d(s) \end{align}
- Can something be said about solvability of this program?
- Is it possible to apply standard optimal control techniques (say, Maximum Principle or Dynamic Programming)?