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\title{
  MATH 157: Mathematics in the world \\
  \Large{Homework 10 (Due April 23th, 2019 at 1:00PM)}
}
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\vspace{-1cm}

\subsection*{Problem 1}
Consider a population of voters uniformly distributed along the ideological spectrum
from left $(x = 0)$ to right $(x = 1)$. Each of the candidates for a single office
simultaneously chooses a campaign platform (i.e., a point on the interval $[0,1]$). 
The voters observe the candidates' choices, and then each
voter votes for the candidate whose platform is closest to the voter's position on
the spectrum. 
If there are two candidates and they choose platforms $x_1 = 0.3$ and
$x_2 = 0.6$, for example, then all voters to the left of $x = 0.45$ vote for candidate $1$, all
those to the right vote for candidate $2$, and candidate 2 wins the election with $55$
percent of the vote. 
Suppose that the candidates care only about being elected - they do not really care about their platforms at all!
\begin{enumerate}
\item What is the strategy space for each candidate?
\item If there are two candidates, what is the pure-strategy Nash equilibrium.\footnote{Assume that any candidates who choose the same platform equally split the votes
cast for that platform, and that ties among the leading vote-getters are resolved
by coin flips.}
\item If there are three candidates, exhibit a pure-strategy Nash equilibrium.\footnote{There are many Nash equilibria, you only need to find one (and prove it is a Nash equilibrium)!}
\end{enumerate}

\subsection*{Problem 2}
Suppose a parent and child play the following game. First, the child takes an action $A\in \mathbb{R}$, that produces income for the child, $I_C(A) = 5-(A-3)^2$, and
income for the parent, $I_P (A) = 5-(A-1)^2$. 
Second, the parent observes the incomes $I_C$ and $I_P$ and then chooses a bequest, $B$, to leave to the child. The
child's payoff is $U(I_C+B)$; the parent?s is $V (I_P-B) + U(I_C+B)$, where the
utility functions $U(x) = \log x$ and $V (x) = \log(4 + x)$.
\begin{enumerate}
\item What are the strategy spaces for child and for parents?
\item Use the backwards-induction to find the (subgame perfect) Nash equilibrium of this game.
\item Prove the `Rotten Kid' Theorem\footnote{See \href{https://en.wikipedia.org/wiki/Rotten_kid_theorem}{Rotten Kid Theorem}}: in the backwards-induction outcome, the child chooses the action that maximizes the family's aggregate income,
$I_C(A) + I_P (A)$, even though only the parent's payoff exhibits altruism.
\item (Extra credit) Now consider general functions $I_C$, $I_P$, $U$ and $V$. 
Assume that all functions
are differentiable and strictly concave, and $U$ and $V$ are strictly increasing.
Assume also that maximizers of the parent's payoff and the child's payoff
exist. Show that the Rotten Kid Theorem holds true.
\end{enumerate}



\subsection*{Problem 3}
Consider the following first-price sealed-bid auction. Suppose there are two bidders,
$i = 1, 2$. The bidders' valuations $v_1$ and $v_2$ for a good are independently and
uniformly distributed on $[0, 1]$. 
The bidders have preferences represented by the utility functions $u_i = x^{\alpha_i}$, where $0<\alpha_i \leq 1$, $i = 1,2$.
Bidders submit their bids $b_1$ and $b_2$ simultaneously. The higher bidder wins the good and pays her bidding
price, so that he/she gains $u_i(v_i-b_i)$; the other bidder gets and pays nothing.
In the case that $b_1 = b_2$, the winner is determined by a flip of a coin. Find a
Bayesian Nash equilibrium $(b_1, b_2)$ in which $b_i$ is a linear function of $v_i$, $i = 1, 2$.

\section*{A Double Auction}
There are two players: a buyer and a seller.
The buyer's valuation for the seller's good is $v_b$, the seller's is $v_s$.
The valuations are private information and are drawn from certain independent distributions on $[0,1]$.
The seller names an asking price, $p_s$, and the buyer simultaneously names an offer price, $p_b$.
If $p_b \geq p_s$, then trade occurs at price $p = (p_b+p_s)/2$; if $p_b < p_s$, then no trade occurs.

The buyer's payoff is
$$
\pi_b(p_s, p_b, v_b) = 
\begin{cases} 
      v_b - (p_b+p_s)/2 & p_b\leq p_s \\
      0 & p_b < p_s
   \end{cases}
$$

The seller's payoff is
$$
\pi_s(p_s, p_b, v_s) = 
\begin{cases} 
      (p_b+p_s)/2 - v_s & p_b\leq p_s \\
      0 & p_b < p_s
   \end{cases}
$$

There are many, many Bayesian Nash equilibria of this game. In the following, we will consider two types.

\subsection*{Problem 4 - One-price equilibria}
For any value $x\in [0,1]$, which is given exogenously and is known by both players, the {\em one-price strategies at $x$} are as follows:\\
The buyer offers $x$ if $v_b \geq x$ and $0$ otherwise;\\
The seller demands $x$ if $v_s \leq x$ and $1$ otherwise.
\begin{enumerate}
\item Show that if buyer and seller both play the one-price strategies at $x$ is a Bayesian Nash equilibrium.
\item Compute the region where the trade occurs. 
\item A trading mechanism is called \href{https://en.wikipedia.org/wiki/Pareto_efficiency}{Pareto efficient} if the item is finally given to the agent who values it the most.
Use your computation in the previous part to show that the one-price strategies at $x$ is not Pareto efficient. \footnote{This is a special case of \href{https://en.wikipedia.org/wiki/Myerson?Satterthwaite_theorem}{Myerson-Satterthwaite theorem}}
\item We define the expected total gains from trade to be the sum of expected gains of the buyer and the seller. Compute the expected total gains from trade for one-price equilibrium at $x$.
\end{enumerate}

\subsection*{Problem 5 - Linear equilibria}
Now we look for {\em linear strategies}
$$
p_i(v_i) = a_i + c_i v_i, \; i = s, b,
$$
with $a_i \geq 0$ and $c_i >0$.
\begin{enumerate}
\item Given the seller's linear strategy $p_s(v_s) = a_s + c_s v_s$, what is the buyer's expected payoff 
$$
E_{v_s} [\pi_b(p_s(v_s), p_b, v_b)]?
$$
\item To maximize $E_{v_s} \pi_b[p_b, p_s(v_s)| v_b]$, what is the buyer's best response?\footnote{You should be surprised and excited that the expression for buyer's best response is a linear function!}
\item Similarly, given the buyer's linear strategy $p_b(v_b) = a_b + c_b v_b$, what is the seller's expected payoff 
$$
E_{v_b} [\pi_s(p_s, p_b(v_b), v_s)]?
$$
\item To maximize $E_{v_b} [\pi_s(p_s, p_b(v_b), v_s)]$, what is the seller's best response?
\item Use all the previous part to find the linear Bayesian Nash equilibrium.
\item Compute the region where the trade occurs, and show that the linear Bayesian Nash equilibrium is not Pareto efficient.\footnote{Again, this is a special case of \href{https://en.wikipedia.org/wiki/Myerson?Satterthwaite_theorem}{Myerson-Satterthwaite theorem}}
\item Compute the expected total gains from trade for the linear Bayesian Nash equilibrium.
How does it compare to the one-price equilibria?\footnote{In fact, \href{https://doi.org/10.1016/0022-0531(83)90048-0}{Myerson and Satterthwaite (1983)} showed that, for the uniform valuation distributions, the linear equilibrium
yields higher expected total gains
than any other Bayesian Nash equilibria of the double
auction (including but far from limited to the
one-price equilibria).}
\end{enumerate}



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