Roy is coming to plant flowers in Zoe’s garden. Zoe loves flowers, her utility for a garden with $x$ flowers is

$z(x) = x.$

Roy plants a unit mass of seeds and the fraction of these that will bloom into flowers depends on how attentive Roy is as a gardener. Roy’s attentiveness is his type $\theta$. In particular when Roy’s type is $\theta$, absent any sabotage by Zoe, there will be $\theta$ flowers in Zoe’s garden in Spring. Roy’s attentiveness is unknown to everyone and it is believed by all to be uniformly distributed on the unit interval.

Jane, Zoe’s neighbor, is looking for a gardener for the following Spring. Jane has high standards, she will hire Roy if and only if he is sufficiently attentive. In particular, Jane’s utility for hiring Roy when his true type is $\theta$ is given by

$j(\theta) = \theta - 2/3.$

(Her utility is zero if she does not hire Roy.)

Roy tends to one and only one garden per year. Therefore Roy will continue to plant flowers in Zoe’s garden for a second year if and only if Jane does not hire him away from her.

Consequently, Zoe is contemplating sabotaging Roy’s flowers this year. If Zoe destroys a fraction $1 - \alpha$ of Roy’s seeds then the total number of flowers in Zoe’s garden when Spring arrives will be $x = \alpha\theta$. Of course sabotage is costly for Zoe because she loves flowers.

There will be no sabotage in the second year because after two years of gardening Roy goes into retirement. Therefore, if Zoe destroys $1-\alpha$ in the first year and Roy continues to work for Zoe in the second year, Zoe’s total payoff will be

$z(\alpha\theta) + z(\theta)$

whereas if Roy is hired away by Jane, then Zoe’s total payoff is just $z(\alpha\theta)$.

This is a two-player (Zoe and Jane) extensive-form game with incomplete information. The timing is as follows. First, Roy’s type is realized. Nobody observes Roy’s type. Zoe moves first and chooses $\alpha \in [0,1]$. Then Spring arrives and the flowers bloom.  Jane does not observe $\alpha$ but does observe the number of flowers in Zoe’s garden. Then Jane chooses whether or not to hire Roy away from Zoe. Then the game ends.

Describe the set of all Perfect Bayesian Equilibria.