Player $A$ and Player $B$ contribute hours towards building a house. If player $A$ contributes $x ≥ 0$ hours and player $B$ contributes $y ≥ 0$ hours, then the value of the house will be $x + y + \frac{xy}{2}$. The contribution of each player comes at a cost:
- Contributing $x$ hours costs player $A$: $\frac{x^2}{2}$.
- Contributing $y$ hours costs player $B$: $ay^2$, where $ a = \frac{1}{2}$ or $a = 1$.
Player $B$ knows the value of $a$, but player $A$ does not:
Player $A$ believes that $a = \begin{cases}\frac{1}{2}\text{ with probability } p\\1\text{ with probability } 1-p\end{cases}$.
(i) What is the maximisation problem that player B needs to solve to optimise their own effort? Show also that
$$y_{opt} = \frac{2 + x}{4a} .$$
(ii) Find the Bayes-Nash equilibrium as a function of $p$ for $x$ and $y$ at the two cost values.
I'm not sure how to solve this. What I have so far is:
Player A's strategy is $(x)$ hours.
Player B's strategy is $(y)$ hours.
There are 2 types. B either chooses $ a = \frac{1}{2}$ or $a = 1$.
If $ a = \frac{1}{2}$ then the payoff will be
$$ \begin{array}{c|lcr} & \text{y}\\ \hline x & p (x + y + \frac{xy}{2}) \end{array} $$
If $ a = 1$ then the payoff will be
$$ \begin{array}{c|lcr} & \text{y}\\ \hline x & (1-p) (x + y + \frac{xy}{2}) \end{array} $$