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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}