<p>Suppose there is a leader with net profit function $EC(P_{1},Y,X)$ where $P_{1},Y$ are decision variables of leader. The follower's problem is given as follows:
$$\max_{X}ENP(P_{1},Y,X)=\max_{X}P_{1}X-c_{1}(X^2+\sigma^2)-p_{1}\int_{LB}^{Y-X}(Y-X-\epsilon)f(\epsilon)d\epsilon$$ where $c_{1},p_{1}$ are parameters of the model and $X$ is follower's decision variable. Here $\epsilon$ is random variable with support in $[LB,UB]$ ,probability density function $f(.)$ and cumulative distribution function $F(.)$. The follower's problem is solved. $ENP$ is concave function of follower's decision variable $X$ with optimal $X$ coming from first order condition given by:$P_{1}-2c_{1}X^{*}+p_{1}F(Y-X^{*})=0$. From this expression it can be seen that $X^{*}$ is an implicit function of model parameters and customer's decision variables. Then how do we substitute $X^{*}$ in leader's problem and solve the leader's problem.</p>

CC BY-SA 3.0
2014-12-19T08:04:23.847Z
What have you tried? People don't like to answer homework questions, unless there has been show some effort. Your question is not also very clear.
Thorst
CC BY-SA 3.0
2014-12-19T09:06:32.177Z
my idea is to optimize the leader's problem with the first order condition expression as equality constraint.
Pradipta
CC BY-SA 3.0
2014-12-19T09:56:00.293Z
@Pradipta Why don't you try your idea and post it here as an answer (you can answer your own question) ? It sounds promising.
Alecos Papadopoulos
<game-theory>
How to solve leader's problem in stackelberg game when follower's optimal decision is implicit function of model parameters?
Pradipta
# How to solve leader's problem in stackelberg game when follower's optimal decision is implicit function of model parameters?

(Asked by: username_0 on 2014-12-19)

Suppose there is a leader with net profit function $EC(P\_{1},Y,X)$ where $P\_{1},Y$ are decision variables of leader. The follower's problem is given as follows:
$$\max\_{X}ENP(P\_{1},Y,X)=\max\_{X}P\_{1}X-c\_{1}(X^2+\sigma^2)-p\_{1}\int\_{LB}^{Y-X}(Y-X-\epsilon)f(\epsilon)d\epsilon$$ where $c\_{1},p\_{1}$ are parameters of the model and $X$ is follower's decision variable. Here $\epsilon$ is random variable with support in $[LB,UB]$ ,probability density function $f(.)$ and cumulative distribution function $F(.)$. The follower's problem is solved. $ENP$ is concave function of follower's decision variable $X$ with optimal $X$ coming from first order condition given by:$P\_{1}-2c\_{1}X^{\*}+p\_{1}F(Y-X^{\*})=0$. From this expression it can be seen that $X^{\*}$ is an implicit function of model parameters and customer's decision variables. Then how do we substitute $X^{\*}$ in leader's problem and solve the leader's problem.

**Tags:** game-theory
> **Comments:**
> **username_1**: What have you tried? People don't like to answer homework questions, unless there has been show some effort. Your question is not also very clear.  
> **username_0**: my idea is to optimize the leader's problem with the first order condition expression as equality constraint.  
> **username_2**: @Pradipta Why don't you try your idea and post it here as an answer (you can answer your own question) ? It sounds promising.  

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## 0 answers

