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\blfootnote{Omer Tamuz. Email: tamuz@caltech.edu.}
\section*{PS/Ec 172, Set 5\\Due Friday, May 19\textsuperscript{th} at
11:59pm \\ Resubmission due Friday, June 7\textsuperscript{th} at
11:59pm}
Collaboration on homework is encouraged, but individually written
solutions are required. Also, please name all collaborators and
sources of information on each assignment; any such named source may
be used.
\mbox{}
\begin{myenumerate}
\item {\em Reserve prices}. Michael and Thierno would both like to
buy an item owned by Nishka. Michael and Thierno's valuations are
chosen independently from the uniform distribution on
$[0,1]$, and each is known only to himself.
\begin{myenumerate}
\item {\em 20 points.} What is Nishka's expected revenue from a
second price auction?
\item {\em 20 points.} Nishka now introduces a {\em reserve price}
$b_r \in [0,1]$: if the maximum bid is under $b_r$ then the
auction is canceled, no one gets the item and no one
pays. Otherwise, the winner pays the maximum of $b_r$ and the
loser's bid. What is her expected revenue, as a function of $b_r$?
\item {\em 10 points.} What is the maximal expected revenue she can
get by choosing $b_r$ optimally?
\end{myenumerate}
\item {\em Bundling}. Moya walks into a store with the intention of
buying a loaf of bread and a stick of butter. Her valuations for the
two items are chosen independently from the uniform distribution on
$[0,1]$. Lilly, the store owner, has to set the prices. We assume
that Moya will buy for any price that is lower than her valuation.
\begin{myenumerate}
\item {\em 20 points.} Assume first that Lilly sets a price $b_l$
for the loaf and $b_s$ for the stick. What is her expected
revenue, as a function of $b_l$ and $b_s$?
\item {\em 5 points.} What is the maximal expected revenue she can
get?
\item {\em 20 points.} Lilly now decides to {\em bundle}: she sets a
price $b_b$ for buying both items together, and does not offer
each one of them separately. That is, she offers Moya to either buy
both for $b_b$, or else get neither. What is her expected revenue,
as a function of $b_b$?
\item {\em 5 points.} What is the maximal expected revenue she can
get now?
% \item {\em Bonus question (1 point).} Assume now that Lilly sets
% three different prices: $b_l$ for the loaf, $b_s$ for the stick,
% and $b_b$ for both, so that Moya can choose if to buy just the
% loaf (for $b_l$), just the stick (for $b_s$), or both (for
% $b_b$). Assume that he will choose to buy whichever items maximize
% his utility, which is his value for the bought items minus the
% price paid. What is the maximal expected revenue she can get now?
\end{myenumerate}
\item {\em Bonus: a riddle with both prisoners and hats
(Gabay-O'Connor game).} There are $n$ prisoners standing in a
line. The first can observe all the rest. The second can observe all
except the first, etc. Each is given either a red or a blue hat
which he cannot see. Now, starting with the first prisoner, each in
turn has to guess the color of his hat, a guess which the rest can
hear.
\begin{myenumerate}
\item {\em 1 point.} The prisoners are allowed to decide on a
strategy ahead of time. Find one in which they all guess the
color correctly, except maybe the first prisoner.
\item {\em 1 point.} Do the same, but for an infinite line of
prisoners.
\item {\em 1 point.} For an infinite line of deaf prisoners, find
a strategy in which at most finitely many of them guess
incorrectly.
% \item {\em 1 point.} For an infinite line of deaf prisoners,
% assume that each is assigned a hat independently and uniformly
% at random. Show that regardless of the strategy the prisoners
% agree on, each has a probability of $1/2$ of guessing his hat
% color correctly. Explain why this means that with probability
% one infinitely many prisoners will guess incorrectly. Resolve
% the apparent conflict with your answer from the previous
% question.
\end{myenumerate}
\end{myenumerate}
\end{document}