Consider the following two$-$player team production game. The players simultaneously chooses eort levels $($a$1$ for player $1$ and a$2$ for player $2),$ and revenue is given by $2$k$($a$1+$ a$2).$ Suppose that player i$\text{'}$s cost of eort is a$2$i in monetary terms. Revenue is split equally between the players, so$,$ player i$\text{'}$s payo is k$($a$1+$ a$2)$ a$2$i $.$

Assume that the value of k is privately observed by player $1$ prior to her selection of a$1.$ Player $2$ does not know the value of k before choosing a$2.$ Player $2$ knows only that k follows uniform distribution between $[4,6].$

$($a$)$ Find the Bayesian Nash Equilibrium $($eort levels$)$ for this game.

$($b$)$ Consider a variation of the game in which, after observing k and the prior to eort selection, player $1$ can provide evidence of the value of k to player $2.$ That is$,$ after either realization of k$,$ player $1$ chooses either E or N$.$ If she selects E$,$ then k is revealed to player $2$; otherwise, player $2$ observes nothing other than that player $1$ selected N $.$ Represent this game in extensive form and calculate the perfect Bayesian equilibria. Are there any perfect Bayesian equilibria in which, at the time player $2$ selects a$2,$ she is uncertain of k$?$

$($c$)$ Would player $1$ prefer to have the option of revealing k$,$ as in part $($b$)$ relative to part $($a$)?$ Explain.