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\noindent \textbf{Answers to Odd-Numbered Problems, 4th
Edition of
Games and
Information, Rasmusen }

\begin{center}
{\bf PROBLEMS FOR CHAPTER 13: Auctions}
\end{center}

\noindent      16 November 2006.  
Erasmuse@indiana.edu. Http://www.rasmusen.org.

\noindent
 This appendix contains answers to the odd-numbered problems in the
fourth edition of \textit{Games and Information} by Eric Rasmusen,
which came out in 2006. The answers
to the even-numbered problems are
available to instructors or self-studiers on request to me at
Erasmuse@indiana.edu.

 

\newpage


\begin{center}
{\bf PROBLEMS FOR CHAPTER 13: Auctions}
\end{center}



\noindent
 \textbf{13.1. Rent-Seeking} \\
 Two risk-neutral neighbors in 16th century England, Smith and Jones, have gone
to court and are considering bribing a judge. Each of them makes a gift, and the
one whose gift is the largest is awarded property worth \pounds 2,000. If both
bribe the same amount, the chances are 50 percent for each of them to win the
lawsuit. Gifts must be either \pounds 0, \pounds 900, or \pounds 2, 000.

\begin{enumerate}

 \item[(a)]
   What is the unique pure-strategy equilibrium for this game?

\noindent
\underline{\textit{Answer}.} \text{Each bids \pounds 900,} for expected profits
of 100 each (=-900 + 0.5(2000)).  This is an all-pay auction, but with restrictions on the bid amounts.   Table A13.1 shows the payoffs (but also
includes the payoffs for when the strategy of a bid of 1,500 is allowed). A
player who deviates to 0 has a payoff of 0; a player who deviates to 2,000 has a
payoff of 0. (0,0) is not an equilibrium, because the expected payoff is 1,000,
but a player who deviated to 900 would have a payoff of 1,100.

\textbf{Table A13.1:   Bribes I }

\begin{tabular}{lll|cccc}
&  &  & \multicolumn{4}{c}{\textbf{Jones}} \\
&  &  & \pounds 0 & \pounds 900 & \pounds 1500 & \pounds
2000 \\
&  &  & \multicolumn{4}{l}{} \\ \hline
&  & \pounds 0 & 1000,1000 & 0,1100 & 0,500 & 0,0 \\
& \textbf{Smith:} & \pounds 900 & 1100,0 & 100,100 & -900,
500 & -
900,0 \\
&  & \pounds 1500 & 500,0 & $500,-900$ & $-500,-500$ & $-
1500, 0$ \\
&  & \pounds 2000 & 0,0 & $0,-900$ & $0,-1500$ & $-1000, -
1000$ \\
\multicolumn{7}{l}{} \\
\multicolumn{7}{l}{\textit{Payoffs to: (Smith, Jones).}}
\end{tabular}

\item[(b)]
  Suppose that it is also possible to give a \pounds 1500 gift. Why does there
no longer exist a pure-strategy equilibrium?

\noindent
\underline{\textit{Answer}.}  
If  one player bids 0 or 900, the other would bid 1500, so we know 0 and 900 would not be used in equilibrium. 
If
both  player bid  1500,  payoffs would be negative (=  2000/2-1500 each), so one could deviate to 0 and increase his payoff.   If both bid 2000,   one can profit by
deviating to 0. If one player bids 1500 and the other bids 2000, the one with the bid of 1500 loses, for a payoff of -1500, and would be better off deviating to 0. This exhausts all the possibilities.

 

\item[(c)]
   What is the symmetric mixed-strategy equilibrium for the expanded game? What
is the judge's expected payoff?

\noindent
\underline{\textit{Answer}.} Let $(\theta_0$, $\theta_{900} $, $\theta_{1,500}$
,$\theta_{2,000})$ be the probabilities. It is pointless ever to bid 2,000,
because it can only yield zero or negative profits, so $\theta_{2,000}= 0$. In a
symmetric mixed-strategy equilibrium, the return to the pure strategies is equal
and the probabilities add up to one, so
\begin{equation}  \label{e127}
\begin{array}{l}
\pi_{Smith} (0) = \pi_{Smith} (900)= \pi_{Smith} (1500) \\
\\
0.5\theta_0(2000) = -900+ \theta_0(2000)+0.5 \theta_{900} (2000)\\
  = - 1500 + \theta_0(2000)+ \theta_{900} (2000) + 0.5\theta_{1500} (2000),\\
\end{array}
\end{equation}
and
\begin{equation}  \label{e128}
\theta_0 + \theta_{900} + \theta_{1500} = 1.
\end{equation}
Solving out these three equations for three unknowns, \textrm{the equilibrium
is} ${(0.4, 0.5, 0.1,0.0)}$.

\textrm{The judge's expected payoff is 1,200 } ($=2(0.5(900) + 0.1(1500))=
2[450+150]))$.

\noindent
\underline{Note:} The results are sensitive to the bids allowed. Can you
speculate as to what might happen if the strategy space were the whole continuum
from 0 to 2000?

\item[(d)]
  In the expanded game, if the losing litigant gets back his gift, what are the
two equilibria? Would the judge prefer this rule?

\noindent
\underline{\textit{Answer}.} Table A13.2 shows the new outcome matrix.
\textrm{There are three equilibria: ${\ x_1 = (900,900), x_2 = 1500, 1500)}$,
and } ${x_3 = (2000,2000)}$.

\textbf{Table A13.2:   Bribes II }

\begin{tabular}{lll|llll}
&  &  & \multicolumn{4}{c}{\textbf{Jones}} \\
&  &  & \pounds 0 & \pounds 900 & \pounds 1500 & \pounds
2000 \\
&  &  & \multicolumn{4}{l}{} \\ \hline
&  & \pounds 0 &   1000,1000  & 0,1100 & 0,500 &
0,0 \\
& \textbf{Smith:} & \pounds 900 & 1100,0 & {\bf 250,250}
& 0, 500 & 0,0 \\
&  & \pounds 1500 & 500,0 & $500,0$ & \textbf{250, 250}
& $0,
0$ \\
&  & \pounds 2000 & 0,0 & $0,0$ & $0,0$ & \textbf{  0,
0
} \\
\multicolumn{7}{l}{} \\
\multicolumn{7}{l}{\textit{Payoffs to: (Smith, Jones).}}
\end{tabular}

The judge's payoff was 1200 under the unique mixed-strategy equilibrium in the
original game. Now, his payoff is either 900, 1500, or 2000. Thus,
\textrm{whether he prefers the new rules depends on which equilibrium is played
out in it.}

\end{enumerate}

 

 



\bigskip

 
 \noindent
 {\bf 13.3. Government and Monopoly} (medium) \\
 Incumbent Apex and potential entrant Brydox are bidding for government favors
in the widget market.  Apex wants to defeat a bill that would require it to
share its widget patent rights with Brydox.  Brydox wants the bill to pass.
Whoever offers the chairman of the House Telecommunications Committee more
campaign contributions wins, and the loser pays nothing.  The market demand
curve for widgets is $P = 25-Q$, and marginal cost is constant at 1.


\begin{enumerate}

\item[(a)]
  Who will bid higher if duopolists follow Bertrand behavior? How much will the
winner bid?

\noindent
\underline{\textit{Answer}.} Apex bids higher, because it gets monopoly profits
from winning, and Bertrand profits equal zero. {\ \textrm{Apex can bid some
small ${\ \epsilon} $ and win. }}

\item[(b)]
  Who will bid higher if duopolists follow Cournot behavior? How much will the
winner bid?

\noindent
\underline{\textit{Answer}.} Monopoly profits are found from the problem
\begin{equation}  \label{e0c1}
   \stackrel{Maximize}{Q_a}  \;\;\;\;Q_a(25-Q_a - 1),
\end{equation}
which has the first order condition $25 - 2Q_a -1 = 0$, so
that $Q_a =
12$
and $\pi_a =144 \; (=12(25-12-1)) $.

Apex's Cournot duopoly profit is found by solving the
problem
\begin{equation}  \label{e0d1}
\stackrel{Maximize}{Q_a} \;\;\;\; Q_a(25-[Q_a + Q_b] - 1),
\end{equation}
which has the first order condition $25 - 2Q_a -Q_b-1 = 0$, so that if the
equilibrium is symmetric and $Q_b = Q_a$, then $Q_a = 8$ and $\pi_a = 64 \; (=
8(25- [8 + 8] -1) ) $.

Brydox will bid up to 64, since that is its gain from being a duopolist rather
than out of the industry altogether. Apex will bid up to $80 (=144-64) $, and so
\textrm{will win the auction at a price of 64. }

\item[(c)]
  What happens under Cournot behavior if Apex can commit to giving away its
patent freely to everyone in the world if the entry bill passes? How much will
Apex bid?

\noindent
\underline{\textit{Answer}.} \textrm{Apex will bid some small ${\ \epsilon} $
and win. } It will commit to giving away its patent if the bill succeeds, which
means that if the bill succeeds, the industry will have zero profits and Brydox
has no incentive to bid a positive amount to secure entry.

\end{enumerate}

% -----------------------------------------------


\bigskip
 \noindent
 {\bf  13.5.  A Teapot  Auction with Incomplete Information }\\
  Smith believes that Brown's value $v_b$ for a teapot  being sold at auction is
0 or 100 with
equal probability.  Smith's value of $v_s=400$ is known by both players.



 \begin{enumerate}

 \item[(a)]
     What are the players' equilibrium strategies in an open cry auction? You
may assume that in case of ties, Smith wins the auction.

\noindent
\underline{\it  Answer.} Brown bids up to his value of 0 or 100. Smith bids up
to his value of 400. Thus, Smith wins, at a price of 0 or of 100.


   \item[(b)]
     What are the players' equilibrium strategies in a first-price sealed-bid
auction? You may assume that in case of ties, Smith wins the auction.

\noindent
\underline{\it  Answer.} Brown bids either 0 or 100 in equilibrium.    Smith
always bids
$100$, because his value is so high that winning is more
important than paying a low price.




   \item[(c)]
 Now let $v_s =102$ instead of 400.  Will Smith use a pure strategy? Will Brown?
You need not find the exact strategies used.

\noindent
\underline{\it  Answer.}   Smith would use a mixed strategy, and while Brown
would still offer 0 if his value were 0, if his value were 100 he
would use a mixed strategy too.  No pure strategy can be part of a
Nash equilibrium, because if Smith always bid a value $x < 100$,
Brown would always bid $x+ \varepsilon$, in which case Smith would
deviate to $x +  \varepsilon$, and if Smith bid $x \geq 100$ he would
be paying 100 more than necessary half the time.




 \end{enumerate}










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