Question
Three houses share exclusive access to a beach, but it is dirty due to trash washed ashore. A beach clean-up exercise costs $100$, but has a value of $200$ to each household. A clean-up company offers to take on the exercise and suggests that contributions be made sequentially. First, Household 1 will contribute some amount that is $x_1$. Then, after observing $x_1$, Household 2 will contribute some amount that is $x_2$. Finally, after observing $x_1$ and $x_2$, Household 3 will contribute some amount that is $x_3$. If $x_1 + x_2 + x_3 \geq 100$, then the company will go ahead with the clean-up and keep any proceeds. If $x_1 + x_2 + x_3 \leq 100$, then the company keeps all contributions and the clean-up is not done.
Find the subgame perfect Nash equilibrium.
My answer
Consider Household 1. Observe that it is always in Household 1's best interest to have the beach cleaned, since $200 > 100$, so he should offer $100$. Now, Household 2 sees this and knows that enough contribution has been made for the clean-up to happen, since $100 \geq 100$, so he will offer $0$. A similar argument can be made for Household 3. Thus, the equilibrium outcome is $\{x_1 = 100, x_2 = 0, x_3 = 0\}$.
Note
I know that the question asked for the subgame perfect Nash equilibrium, but my professor has specifically stated that, for the purposes of the module we are taking, being able to come up with the equilibrium outcome is sufficient (i.e. We do not know how to solve for the actual subgame perfect Nash equilibrium).
I have two questions.
- Is my equilibrium outcome correct?
- May I know if my reasoning is sufficient/complete/logical to arrive at the outcome I had reached?
We have just covered game theory, so I am still trying to get used to answering such questions. Any help/thoughts on my answer will be greatly appreciated :)