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\begin{center}
\textbf{\Large Homework 4}
\textbf{due March 16, 2017}
\end{center}
\bigskip\noindent\textbf{Problem 1} A seller has a painting for sale that is
either good or bad. A good painting is worth $1$ to the seller. A bad painting
is worth 0 to the seller. The seller knows the painting's quality. The buyer
does not know whether the painting is good or bad, only that it is good with
probability $\frac12$ and bad with probability $\frac12$. A good painting is
worth $v$ to the buyer. A bad painting is worth 0 to the buyer.
The buyer makes a one-time offer to the seller, which the seller can accept or
reject. To keep the problem simple, assume that the seller accepts offers
where she is indifferent.
\medskip\noindent\textbf{a. } Suppose $v=1$. What offer should the buyer make? What is his expected profit?
\medskip\noindent\textbf{b. } Suppose $v=1.5$. What offer should the buyer make? What is his expected profit?
\medskip\noindent\textbf{c. } Suppose $v=5$. What offer should the buyer make? What is his expected profit?
\medskip\noindent\textbf{d. } What is the lowest value of $v$ such that both types of the painting are traded in equilibrium?
\medskip\noindent\textbf{e. } Discuss the efficiency of the outcome in a., b. and c. What is the source of the inefficiency, if any?
\bigskip\noindent\textbf{Problem 2} Consider a two-player Bayesian
game where both players are not sure whether they are playing game X or game Y,
and they both think that the two games are equally likely. This game has a
unique Bayesian Nash equilibrium, which involves only pure strategies. What is
it? (Hint: start by looking for Player 2's best response to each of Player 1's
actions.)
\begin{center}
$\begin{game}{2}{3}[Player 1][Player 2]
& L & M &R \\
T & 1,.2 & 1,0 & 1,.3 \\
B & 2,2 & 0,0 & 0,3
\end{game}$
Game X
\end{center}
\begin{center}
$\begin{game}{2}{3}[Player 1][Player 2]
& L & M &R \\
T & 1,.2 & 1,.3 & 1,0 \\
B & 2,2 & 0,3 & 0,0
\end{game}$
Game Y
\end{center}
\bigskip\noindent\textbf{Problem 3} Now consider a variant of this game (from Problem 2) in which Player 2 knows
which game is being played (but Player 1 still does not). This game
also has a unique Bayesian Nash equilibrium. What is it? (Hint:
Player 2's strategy must specify what she chooses in the case that
the game is X and in the case that it is Y.) Compare Player 2's
payoff in the games from Problems 2 and 3. What seems strange about
this?
\bigskip\noindent\textbf{Problem 4} Firm 1 is considering taking over Firm 2. It does not know Firm 2's current value, but
believes that is equally likely to be any dollar amount from 0 to
100. If Firm 1 takes over firm 2, it will be worth 50\% more than
its current value, which Firm 2 knows to be $x$. Firm 1 can bid any
amount $y$ to take over Firm 2 and Firm 2 can accept or reject this
offer. If 2 accepts 1's offer, 1's payoff is $\frac32 x-y$, and 2's
payoff is $y$. If 2 rejects 1's offer, 1's payoff is 0 and 2's
payoff is $x$.
\medskip\noindent\textbf{a. } Find the unique Bayesian Nash equilibrium of this game.
\medskip\noindent\textbf{b. } Can you explain why the result you obtained in part a is sometimes called ``adverse selection''? Give two other examples of markets that may exhibit adverse selection.
\bigskip\noindent\textbf{Problem 5} Two bidders are bidding on a
bottle of Scotch in a first-price, sealed bid auction. Bidder 1 values the
bottle at $v_1$, and bidder 2 values the bottle at $v_2$. Neither bidder knows
the other's valuation, but each knows that $v_i\thicksim U[0,1]$, and that
$v_1$ and $v_2$ are independent (note that this setting is identical to the
first example studied in class). Bidders simultaneously submit hidden bids; the
highest bidder gets the bottle for the price he paid.
\medskip\noindent\textbf{a. } Show that there is a Nash equilibrium in bidding
strategies in which player $i$ bids $b_i=\frac{v_i}{2}$.
\medskip\noindent\textbf{b. } Suppose that bidder 2 is irrational, and will bid $b_2=v_2$.
Demonstrate that $b_1=\frac{v_1}{2}$ remains the best response for player 1.
\medskip\noindent\textbf{c. } Suppose a third bidder arrives to bid on the bottle of Scotch.
Like bidders 1 and 2, bidder 3's valuation is private information, but is
distributed $v_3\thicksim U[0,1]$. $v_1$, $v_2$, and $v_3$ are independent.
Show that there is a Nash equilibrium in which player $i$ bids $b_i=\frac{2
v_1}{3}$.
To answer part c., you will need to use the fact that if $X_1$ and $X_2$ are
independent $U[0,1]$ random variables, $P(\max\{X_1,X_2\}