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my question is about general representation of a dynamic Stackelberg game which is played in continuous time. We have maximization problems of two agents who play this game. Agents are 'Leader' and 'Follower'.

Leader has the payoff (benefit) as a function of his decision $y$, Follower's decision $x$ and current state $s$ which is $U_{L}(x,y,s)$. Follower's payoff is the function $U_{F}(x,y,s)$. At each instant in time Leader decides his strategy $y$, Follower observes this decision and makes his decision $x$. These decisions affect the current state $s$ with the transition function $\dot{s} = g(x,y,s)$. In the end, after the decisions made and state changed, the game will be replayed at new state with the same rules.

The game is played with the feedback (Markovian) strategies, hence players make their decisions by observing the current state $s$. This will result as considering opponents strategy in your maximization programme as a function of state, such that Leader will consider $x(s)$ and Follower will consider $y(s)$.

The HJB equation for follower agent is:
$ \rho V_{F}(s) = \max _{x}\{U_{F}(x,y(s),s) +V'_{F}(s)g(x,y(s),s) \} $
where $V_{F}$ is the value function of the follower.

This programme will give us the best strategy of Follower as some function $x^* =\phi_{F} (s,y(s))$.

Leader will know that Follower will observe his decision and play with the rule defined above. Hence Leader's HJB equation will be as follows:
$ \rho V_{L}(s) = \max _{y}\{U_{L}(\phi_{F} (s,y(s)),y,s) +V'_{L}(s)g(\phi_{F} (s,y(s)),y,s) \} $
where $V_{L}$ is the value function of the Leader.

The existence of $y(s)$ in this programme makes it a non-standard optimal control problem. Usually a form for $y(s)$ is assumed (such as linear in s) and hence $\phi_{F} (s,y(s))$ is transformed to $\phi'_{F} (s)$ which makes solution could be obtained and we can define Nash equilibrium.

This question is stressed in the book by Engelbert J. Dockner and others but they do not go far from above assumption. Are there another possible approaches or potential use of different techniques for this type of problem, especially on this kind of general characterization? I will appreciate your opinions and suggestions. Thanks.

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