A game with coin toss

Question

So I have a game between two players, p1 and p2. Someone(nature?) tosses a biased coin with 80% chance on head. p1 observes the outcome of it and write on a piece of paper head/tail(not neccessarily the outcome of the coin toss, but if p1 writes the outcome, p1 gains additional payoff.). p2 has to guess the outcome of the coin toss. p2 gains payoff base on if p2 guesses correctly, while p1 gains payoff based on p2's choices. How would an extensive form(in terms of a tree) of a game like this looks like?

I'm thinking that p1 and p2's actions can be seen as a simultanous game for each outcome of the coin toss. But I'm having problem interpreting if p1's action is based on the outcome of the coin toss or not; In other words: are there subgames in the game?

The payoffs are:

For p1: 1 point for writing down the outcome, 0 otherwise. 2 points if p2 picks head, and 0 if p2 picks tail.

For p2: 1 point for guessing correctly.

Answer 1

• Nature moves first, choosing "head" with 80% and "tail" with 20%.
• P1 observes the outcome, and then chooses whether to write down $H$, $T$, or nothing ($O$).

Suppose, before her move, P2 does not observe what P1 writes, or whether P1 has written anything down. Then the extensive form looks like this:

Suppose P2 observes what P1 has done/written before making a decision. Then the extensive form looks like this:

In either case, there is only one subgame, which is the whole game itself.

---

NB: I don't think you've fully/correctly specified all the payoffs, so I didn't fill in the payoffs at the terminal nodes. Getting the normal form from the above extensive forms should be straight forward.