Suppose that Bernie and Beto are considering to run together on a ticket with Bernie as President and Beto as Vice-Presi

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Suppose that Bernie and Beto are considering to run together on
a ticket with Bernie as President and Beto as Vice-President.
Beto’s quality as candidate (in terms of attracting campaign
resources and votes) is unknown to Bernie at the time at which they
must decide whether to run together, but Bernie knows that, with
probability p, Beto is a highquality candidate (H) and, with
probability 1-p Beto is a low-quality candidate (L). Beto knows his
own quality. If these two candidates do not run together, then each
receive a payoff of 0. If they run together, then their payoffs are
as follows: If Beto’s type is H, then each candidate receives 10.
If Beto turns out to be L, then Beto receives 5 and Bernie receives
-4 (which has to do with Bernie having to work very hard for at
least 4 more years to run again for office).
(a) Consider the game in which Bernie chooses between running
together (R) or not running together (NR). Draw the extensive form
of this game (using a move of Nature at the beginning to select
Beto’s type). Note that only one candidate has a move in this game.
What is the Bayesian Nash equilibrium of the game? (Hint: It
depends on p.)
(b) Now suppose that before Bernie decides whether to invite
Beto to run together, Beto has the opportunity to publicly endorse
Bernie’s candidacy. Bernie observes Beto’s choice between E and NE
(Endorse or Non-Endorse) before selecting R or NR. Endorsing
entails a cost of g units of utility for Beto; this cost is
subtracted from Beto’s payoff designated above. On the other hand,
endorsing adds w to Bernie’s utility. If Beto does not campaign,
then it costs him nothing. Assume that g and w are positive
numbers. Draw the extensive form of this new game.
(c) Under what conditions (values of g and w) does a
(separating) perfect Bayesian equilibrium exist in which the
low-quality type of Beto does not endorse, the high-quality type of
Beto endorses, and Bernie invite Beto to run together only if the
endorsement is provided? Completely specify such an
equilibrium.
(d) Is there a pooling equilibrium in this game? Fully describe
it and note how it depends on p.