Define and characterize equilibria of the following game Consider the following game between a monopolist firm and a consumer. Consumer's income is $1$, and he needs to allocate it between period 1 and period 2 consumption to maximize his utility $u(c_1,c_2)=c_1+c_2$, where $c_t$ is the consumer's period $t$ consumption. Period 2 prices are given exogenously by $P$. Assume that $P$ is random and takes value in $\{\bar{p},\underline{p}\}$ with equal probability where $\bar{p}&gt;\underline{p}&gt;0$. It is common knowledge that the firm knows $P$, and the consumer knows $P$ with probability $\alpha$. The timing of the game is as follow. The firm chooses the first period price $p$ to maximize its profits in period 1. After observing $p$, the consumer decides how to allocate his income between $c_1$ and $c_2$. 

1) Define an equilibrium for this game 

2) Characterize the equilibria.

I am having difficulties in formally setting up dynamic games and I would therefore appreciate if someone could check out the solution I provided below, and most importantly whether the definition of equilibrium I provide (strategy sets, strategies, optimality conditions, etc) is correct, and if not why?
Thank You
SOLUTION: 

1) DEFINITION OF EQUILIBRIUM

\begin{equation} \begin{split}
&amp;p \in \mathbb{R}_+\text{ is the price chosen by the firm in period 1}\\
&amp;a_U \in [0,1] \text{ is the share of income the Uninformed consumer allocates in period 1}\\
&amp;a_I \in [0,1] \text{ is the share of income the Informed consumer allocates in period 1}\\
&amp;\Theta = \{I,U\} \text{ is the type of consumer: Informed, Uninformed}\\
&amp; S_I = \{ a : \{\underline{p},\bar{p}\} \times \mathbb{R}_+ \to [0,1] \} \text{ is the strategy set of the I. consumer}\\
&amp; S_U = \{ a : \mathbb{R}_+ \to [0,1] \} \text{ is the strategy set of the U. consumer}\\
&amp; S_f = \{ p : \{\bar{p},\underline{p}\} \to \mathbb{R}_+ \} \text{ is the strategy set of the firm}\\
&amp;\mu_U(p) = Pr(P =\bar{p}|p) \text{ : consumer of type U belief that the second period price is } \bar{p}, \text{ given } p\\
&amp;\mu_I(P) = Pr(P =\bar{p}|p) \text{ : consumer of type I belief that the second period price is } \bar{p}, \text{ given } p\\
\end{split} \end{equation}

An equilibrium of the game is a triple of strategies, $s^*=(a^*_I, a^*_U,p^*)$ for each type of consumer and the monopolist, together with a system of belief $\mu$ such that: for each player $i$, $s_i^*$ is sequentially rational, and the beliefs are determined by the strategy profile and Bayes rule whenever it applies. \ 

In particular for each price $p$, the consumer of type $i \in \{U.I\}$ solves:
\begin{equation} \begin{split}
&amp;a_i^*(p) = \text{ argmax } \sum_p \big[ ( \frac{a}{p} + \frac{1-a}{\bar{p}} )\mu_I(P) + ( \frac{a}{p} + \frac{1-a}{\underline{p}} )(1-\mu_I(P)) \big]
\end{split} \end{equation}

Also, in the case of the Informed consumer, we know that 
\begin{equation} \begin{split}
 &amp;\mu_I(P) = Pr(P =\bar{p}|p) = 
\begin{cases}
1 &amp;\text{ if } P =\bar{p}\\
0  &amp;\text{ if } P = \underline{p}\\
\end{cases} \\
\end{split} \end{equation}

given the consumers optimal strategies and $P$, $p^*(P)$ maximizes the monopolist first period profits: 

\begin{equation} \begin{split}
&amp;\pi(P)^* = \text{ argmax } \alpha [a^*_I(p) p] + (1-\alpha) [a^*_U(p) p]\\
\end{split} \end{equation}

2) CHARACTERIZATION OF EQUILIBRIA

for any realization of $P$ in period 2, we have that 
\begin{equation} \begin{split}
a_I(p)^* = 
\begin{cases}
1 &amp;\text{ if } p &lt;  P\\
0  &amp;\text{ if } p &gt;P\\
\in [0,1] &amp;\text{ if } p =  P \\
\end{cases} \\
\end{split} \end{equation}

\begin{equation} \begin{split}
a_U(p)^* = 
\begin{cases}
1 &amp;\text{ if } p &lt;  \mu_U(p) \bar{p} + (1-\mu_U(p))\underline{p}\\
0  &amp;\text{ if } p &gt; \mu_U(p) \bar{p} + (1-\mu_U(p))\underline{p}\\
\in [0,1] &amp;\text{ if } p = \mu_U(p) \bar{p} + (1-\mu_U(p))\underline{p}
\end{cases} \\
\end{split} \end{equation}

To define the optimal strategy of the monopolist consider the following cases, where $\pi$ denotes the monopolist first period profits: 

a)Suppose $p=\bar{p}, \forall P$, then By Bayes rule $\mu_U(p) = 1/2$; also suppose that off the equilibrium path $\mu_U(p) = 1/2, \forall p \neq\bar{p}$ ; then the monopolist profits are given by: 

\begin{equation} \begin{split}
\pi(P) = 
\begin{cases}
\alpha \bar{p} a_I^*(\bar{p}) &amp;\text{ if } P=\bar{p} \\
0  &amp;\text{ if } P =  \underline{p} \\
\end{cases} \\
\end{split} \end{equation}
This is not an equilibrium strategy because the monopolist can deviate and increase its profits when $P=\underline{p}$ by setting, for example, $\hat{p} = 1/2\bar{p} + 1/2\underline{p}$; this will give him a profit of $\alpha a_U^*(\hat{p})  &gt;0$. 
Similarly it is possible to prove that no-matter what the off the equilibrium path are, there is always a profitable deviation for the monopolist. \

b)Suppose $p=\underline{p}, \forall P$, then $\mu_U(p) = 1/2$ and also suppose that off the equilibrium path $\mu(p) = 1/2, \forall p \neq\underline{p}$ ; then the monopolist profits are given by: 

\begin{equation} \begin{split}
\pi(P) = 
\begin{cases}
\underline{p} &amp;\text{ if } P=\bar{p} \\
\underline{p} (\alpha a_I^*(\underline{p}) +(1-\alpha) ) &amp;\text{ if } P =  \underline{p}\\
\end{cases} \\
\end{split} \end{equation}
this is not an equilibrium strategy because $\exists \epsilon &gt; 0 $ such that monopolist will deviate and increase its profits when $P=\bar{p}$; for example by setting  $\hat{p} = 1/2\bar{p} + 1/2\underline{p} -\epsilon&gt;\underline{p}$ which will give him a profit of $\hat{p}&gt; \underline{p}$ . 

c)Suppose $p=\underline{p}, \text{ if } P= \underline{p}$, $p=\bar{p}, \text{ if } P= \bar{p}$ and $\mu_U(p) = 1, \forall p &gt; \underline{p}$ then

\begin{equation} \begin{split}
\pi_1^*(P) = 
\begin{cases}
\bar{p}(\alpha a_I^*(\bar{p}) +(1-\alpha) a_U^*(\bar{p})) &amp;\text{ if } P=\bar{p} \\
\underline{p} (\alpha a_I^*(\underline{p}) +(1-\alpha) a_U^*(\underline{p})) &amp;\text{ if } P =  \underline{p}\\
\end{cases} \\
\end{split} \end{equation}
this represents an equilibrium as long as  $a_I^*(\bar{p}) = a_U^*(\bar{p}) = 1$; otherwise the monopolist can set a price $\hat{p}&lt; \bar{p}$ and can achieve a profit of $\bar{p} - \epsilon &gt;\alpha a_I^*(\bar{p}) +(1-\alpha) a_U^*(\bar{p})$

d) Suppose $p = 1/2\bar{p} + 1/2\underline{p}, \forall P$ then $\mu_U(p)=1/2$ and and also suppose that off the equilibrium path $\mu(p) = 1/2, \forall  p \neq ( 1/2\bar{p} + 1/2\underline{p}) $ 
\begin{equation} \begin{split}
\pi_2^*(P) = 
\begin{cases}
(1/2\bar{p} + 1/2\underline{p})  (\alpha  +(1-\alpha) a_U^*(p)) &amp;\text{ if } P=\bar{p} \\
( 1/2\bar{p} + 1/2\underline{p} ) ( 0 + (1-\alpha)a_U^*(p)) &amp;\text{ if } P =  \underline{p}\\
\end{cases} \\
\end{split} \end{equation}
This is an equilibrium strategy as long as there are no profitable deviation; notice that the monopolist can set $p=\bar{p}$ when $P=\bar{p}$ and given $\mu_U = 1/2$ its profits would equal $\alpha \bar{P}a_I^*(\bar{p})$. Also, when $P=\underline{p}$ the monopolist could set $p=\underline{p}$ and get a profit of $\underline{p}$. So we need to insure that neither deviation is possible.

e) ?

We can therefore characterize the following (Perfect Bayesian) Equlibria: 

1ST EQUILIBRIUM) 
\begin{equation} \begin{split}
&amp;a_I(p)^*  = 1, a_U(p)^* = 1 \\  
&amp;\pi_1^*(P) \text{ defined as above } \\
&amp;\mu_U(p) = 1 \text{ both "on and off" the equilibrium path } \\
&amp;\text{(that is, if the monopolist deviates } \\
&amp;\text{from the equilibrium strategy } \mu_U(p)=1 \text{ still holds} \\
\end{split} \end{equation}

SET OF EQUILIBRIA) 
\begin{equation} \begin{split}
&amp;a_I(p)^* =1, a_U(p)^* \in[0,1] \\  
&amp;\pi_2^*(P) \text{ defined as above } \\
&amp;\mu_U(p) = 1/2 \text{ both "on and off" the equilibrium path } \\
&amp;\text{(that is, if the monopolist deviates } \\
&amp;\text{from the equilibrium strategy } \mu_U(p)=1/2 \text{ still holds} \\
&amp; \text{Also, The following two conditions below hold:}\\
&amp;\alpha \bar{P}a_I^*(\bar{p}) &lt; (1/2\bar{p} + 1/2\underline{p})  (\alpha a_U^*(\bar{p}) +(1-\alpha) ) \\
&amp;\underline{p} &lt; ( 1/2\bar{p} + 1/2\underline{p} ) (\alpha a_U^*(\bar{p}) +0)\\
\end{split} \end{equation}