How to approach such question of two periods and asymmetric information game?

There are two players 1 and 2, playing a two-period public good contribution game. The benefits from the public good per-period to each player is $$b_1$$ and $$b_2$$ respectively, which are common knowledge. For the provision of the public good only one player needs to contribute. Once a player contributes and the good is provided then both can derive the benefits. The public good lasts for only one period and contributions are required in each period to renew it. The per-period cost of contribution for player 1 is private information which can take two values, $$\underline c$$ (low type with probability $$p$$) and $$\overline c$$ (high type with probability $$1 − p$$) for which following condition holds $$\underline c < b_1 < \overline c$$. However, the cost of contribution for player 2 takes the value $$c_2$$ which is common knowledge and is lower than its benefit i.e. $$c_2 < b_2$$. The game is played as per the following sequence: First period: Only player 1 moves. He can choose to ‘contribute’($$C$$) or ‘Not to Contribute’($$N$$) towards the first period contribution of public good. If player 1 chooses $$N$$ then both gets 0 otherwise if player 1 chooses $$C$$ then both players get their benefits but only player 1 pays the cost. Note that the player 2 observes the player 1’s action of either $$C$$ or $$N$$ but not its type (i.e. the private information regarding costs).

Second period: The game is played sequentially where player 2 moves first and chooses $$C$$ or $$N$$. Player 1 observes it and decides on $$C$$ or $$N$$. If at least one of them contributes then both players get their benefits bi and those who contribute pay their respective costs. If no one contributes each gets 0. Players discount their payoffs by $$\delta$$, which is common to both. In answering the questions below, you need to write strategies and beliefs precisely.

(a) Find an equilibrium where different types of player 1 choose different actions in first period (separating equilibrium).

(b) Find an equilibrium where different types player 1 choose the same action in first period (pooling equilibrium).

(c) Find a hybrid equilibrium where low type of player 1 in first period chooses $$C$$ with probability $$\alpha$$ and $$N$$ with probability $$(1 − \alpha)$$, and player 2 in second period randomizes between $$C$$ with probability $$\beta$$ and $$N$$ with probability $$(1 − \beta)$$.

I have attached my workings. • Why don't you show us what you have done and we can lead you the way? – Bayesian Apr 30 at 12:17