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\begin{center}
{\bf    Errata for Eric Rasmusen's Games and Information,
Third
Edition, arranged by page number. Updated  July 18, 2003.
}
 \end{center}

 I apologize for the   errors in this book, but I have tried to keep
this errata current as partial compensation. I  extend my gratitude to
those readers who have pointed errors out to me. Specifically, I thank
Kyung Baik, David Collie (Cardiff Business School), Ralf Elsas (Goethe
U.),   Diego Garcia (Dartmouth),
Bettina Kromen (U. Koeln), Eva Labro (London School of Economics),
Frank P. Maier-Rigaud,  Ron Mallon (U. of Utah),   Alexandra Minicozzi
(U. Texas), Luis Pacheco (U. Portucalense),     Pedro Sousa (U.
Portucalense),  and Charles Tharp.   Martin Caley  of the Isle of Man
Treasury was particularly helpful, and I thank him for his very
careful reading.

 




If you find any  new errors, please let me know, so future readers can
be warned. Do not be shy-- if you think it might be an error, do not
feel you have to check it out thoroughly before letting me know. It's
my duty to make sure and to be clear, not yours.

 I can be reached at    Eric Rasmusen, 	  Indiana University, Kelley
School of Business, Rm. 456, 1309 E 10th Street, Bloomington, Indiana,
47405-1701. Office: (812) 855-9219.  Fax: 812-855-3354.
Erasmuse@Indiana.edu. Php.indiana.edu/$\sim$erasmuse. The webpage for
{\it Games and Information} is at\\ http:
//Php.indiana.edu/$\sim$erasmuse/GI/index.html

 To view Acrobat (.pdf) files, you will need to download  the Adobe
Acrobat Reader from http://www.adobe.com/acrobat/readstep.html.
 If  you don't use the web, just let me know and I'll send you
hardcopy.


 There was a reprint in October 2001, so you  may find some of these
errors corrected in your printing of
the book.

\newpage

\bigskip
\noindent
{\bf Chapter 1 The Rules of the Game   }

p. 16. second paragraph, line 3  (fixed in
October 2001 reprint). OldCleaner's expected profit is 32, not 38,
with a Low price. This does not affect the suboptimality of that
strategy, though.

p. 24. Near Table 1.4. Drop
Table 1.4 and the discussion of the Swiss Cheese Game,since this does
not fit the definition of ``weakly dominated'' that I  defined
earlier. So drop:


``The easiest example is table 1.4's  Swiss Cheese Game.
Every strategy is weakly dominated for every player.
Thus, one iterated dominance equilibrium is {\it (Up,
Left)}, found by first eliminating Smith's {\it Down}
and then Jones's {\it Right}, but {\it (Down, Right)} is
also an iterated dominance equilibrium. And, in fact,
every strategy combination is also a weak dominant
strategy equilibrium as well as an iterated dominance
equilibrium.

 \begin{center}
{\bf Table 1.4     The  Swiss Cheese Game   }


 \begin{tabular}{lllccc}
  &       &             &\multicolumn{3}{c}{\bf Jones}\\
  &       &             &  {\it  Left}    &   &  $ Right$
\\
  &   &  $ Up$      &    {\bf  0,0} & $\leftrightarrow$  &
{\bf 0,0} \\
 & {\bf Smith:} &&$\updownarrow$& & $\updownarrow$ \\
&  &    {\it Down }      &      {\bf  0,0}  &
$\leftrightarrow$  & {\bf
0,0} \\
\multicolumn{6}{l}{\it Payoffs to: (Smith, Jones) }
\end{tabular}
\end{center}

  The Swiss Cheese Game is pathological,  but it is not hard to come
up with less obvious examples, such as...''

\bigskip

Then start the next paragraph:

``Consider   the    Iteration Path Game...''


 p. 32. In the book: 

 Suppose the pareto-superior equilibrium ({\it Small},
{\it Small}) were chosen as a focal point in { Ranked Coordination},
but
the game was repeated over a long interval of time. The numbers in
the payoff matrix might slowly change until ({\it Small, Small}) and
({\it Large, Large}) both had payoffs of 1.6, and ({\it Large, Large})
 started to dominate.


Should be: 

Suppose the pareto-superior equilibrium ({\it Large},
{\it Large}) were chosen as a focal point in { Ranked Coordination},
but
the game was repeated over a long interval of time. The numbers in
the payoff matrix might slowly change until ({\it Small, Small}) and
({\it Large, Large}) both had payoffs of 1.6, and   shortly thereafter
({\it Small, Small})
might start to dominate.



p. 52, Figure 2.6 (fixed in October 2001
reprint)   Should be payoffs to ``(Smith, Jones)'', not  ``(Smith,
Brown)'' .

p. 52, Figure 2.6 (fixed
in October 2001  reprint) Jones's nodes should be labelled $J_1$ and
$J_2$, not $B_1$ and $B_2$.

p. 55.   (fixed in October 2001
reprint). ``Jones   uses the   likelihood and his priors.'' INSTEAD OF
``Jones   uses the the   likelihood and his priors.''


p. 57, Figure 2.8. Line (4)
is missing its heading, which should be $(B)|Large$.

p. 76. Paragraph starting ``First, it is
possible...''    Replace $Up$ with $Down$ and (3.14) with
(3.12). Also change ``$\theta^*>1$''  to ``$\theta^*>1$, or $\theta^*
\leq 0$''.

p. 77. (fixed in
October 2001  reprint)  The second
paragraph, third line, should have $z<y$, not $z>y$, to fit the
picture in Table 3.5.


  \bigskip
\noindent
{\bf Chapter 4: Dynamic Games with Symmetric Information   }


p. 93, Figure 4.2.
 $Exit$ should read $Out$ and $Remain$ should read $In$. Also
'Payoffs to (Smith,Jones)' needs to be added.

p. 107. Problem 4.2d, first line. Should be  ``Union after Lenin
died'' (missing word).

  \bigskip
\noindent
{\bf Chapter 5: Reputation and Repeated Games }



 p. 109 (fixed in October 2001  reprint). Line
12: ``known'', not ``knowns''.

  p. 113  (fixed in October 2001  reprint).
Line 2: ``always '', not ``alwyas ''.


p. 116, paragraph starting ``It is important to remember that...''
(Nov.2001). Replace ``Fuderberg'' with
``Fudenberg''.

 p. 118, Table 5.2b. The arrow on
the line {\it High Quality} should be pointed left towards 5,5 and not
to the right.




\bigskip
\noindent
{\bf Chapter 6: Dynamic Games with Incomplete Information}

p. 148, Figure 6.3. ``Strong'' entrant, not ``Stong'', fixed in
October 2001  reprint.

p. 149, line 4.  Delete ``$+0.05[40]$''.

 My explanation here is bad, so I will elaborate on it.   Figure
6.3 is abbreviated, and contains within it the game in Figure 6.1. The
(-10,300) and (- 10, 0) payoffs  indicate what happens if  the
incumbent chooses FIGHT depending on whether the  entrant is weak
(300) or strong (0). In either case, the entrant gets -10 when the
incumbent chooses FIGHT.

 If, however, the incumbent chooses COLLUDE, then the entrant gets a
payoff of 40, from Figure 6.1.

Suppose the entrant is strong and Nature told the incumbent  that. But
suppose the entrant   does not know whether Nature told the incumbent.
Nature did tell the incumbent with probability 0.1,  and if the
entrant then  enters, the incumbent will collude and the entrant's
payoff will be 40. Nature was silent with probability 0.9, and if the
entrant then enters, the incumbent will fight and the entrant's payoff
will be -10. The expected payoff is thus $-5 (= 0.1[40] + 0.9 [-10])$.



p. 157. 
The error is that in the form it takes in the book, there is no
equilibrium with limit pricing, and the answer I had posted on the web
made a mistake in its Low-price pooling equilibrium. For the question
to make sense, the low-cost incumbent (C=20) must suffer some loss if
entry occurs. In the original question, the low-cost incumbent is
indifferent about whether entry occurs, so a High price is a weakly
dominant strategy for it. To fix the problem, I have below added a
competition cost of 50. I've also clarified the wording a bit.

\bigskip
 \noindent \textbf{Problem 6.2: Limit Pricing.}\footnote{ See Milgrom
and Roberts (1982a).} An incumbent firm operates in the local computer
market, which is a natural monopoly in which only one firm can
survive. In the first period, the incumbent can price $Low$, losing 40
in profits, or $High$, losing nothing. It knows its own operating cost
$C$, which is 20 with probability 0.75 and 30 with probability 0.25. A
potential entrant knows those probabilities, but not the incumbent's
exact cost. In the second period, the entrant can enter at a cost of
100, and its operating cost of 25 is common knowledge. If there are
two firms in the market, each incurs a loss of 50, but one then drops
out and the survivor earns the monopoly revenue of 200. There is no
discounting; $r=0$.




\bigskip
\noindent
{\bf Chapter 7: Moral Hazard: Hidden Actions }

p. 163. Screening should be Figure 7.1e, not 7.1d.

 p. 167.  In the paragraph
starting ``Let's now fit out...'' replace ``then'' with ``let'' in two
places. This is more a clarification than a correction; this paragraph
is laying out specific functional forms for the problem, which
includes a payoff linear in output and wages for the principal.


 p. 168.    I made some mistake
in Mathematica, and the computation of $e^*$ is  slightly off (and so
all the resulting numbers are off too). Thus, replace the paragraph,

 ``but this cannot  be solved analytically.   Using the computer
program  Mathematica, I found that    $e^* \approx 0.84934$, from
which, using the formulas above, we get  $q^*  \approx
100*log(1+.84934) \approx 61.48$ and $w^* \approx  41.32$.''

with

 ``but this cannot  be solved analytically.   Using the computer
program  Mathematica, I found that    $e^* \approx 0.77$, from which,
using the formulas above, we get  $q^*  \approx 100*log(1+ 0.77)
\approx 57.26$ and $w^* \approx  36.50$.''

 The $e^*$ miscalculation continues in the discussion of the three
contracts at the bottom of the page. Change that section to read:


1 The  {\bf forcing contract}  sets $w(e^*) = w^*$ and $w(e \neq 0.77)
=0$.  Here,  $w(0.77) = 37$ (rounding up) and $w(e \neq e^*) =0$.

2  The  {\bf threshold contract}  sets $w(e \geq e^*) = w^*$ and $w(e
<e^*) =0$.  Here,  $w(e \geq 0.77) = 37$ and $w(e <0.77) =0$.

3 The  {\bf  linear contract}     sets $w(e) = \alpha + \beta e$,
where  $\alpha$ and $\beta$ are chosen so that $ w^* = \alpha + \beta
e^*$ and   the contract line is tangent to the indifference curve $U=
\bar{U}$ at $e^*$.      The slope of that indifference curve is the
derivative of the  $\tilde{w}(e)$ function, which is
$$
  \frac{\partial  \tilde{w}(e)}{\partial e}  =    2e* Exp(3+e^2). \;\;
\;\;\;\;\;(7.14)
$$
  At   $e^*=0.77$, this takes the value 56.   That is the $\beta$ for
the linear contract. The $\alpha$ must solve $w(e^*) = 37 = \alpha  +
56  (.77)$, so $\alpha \approx -7$.


 p. 169. In the first paragraph
replace $e=0.84$ with $e=0.77$.


p. 173. Just after equation (7.19), the cite should be to equations
(7.18a) and (7.18b) on the opposite page, not to (7.11a) and (7.11b),
which don't exist.


\bigskip
\noindent
{\bf Chapter 8: Further Topics in Moral Hazard    }

p. 200. In Figure 8.2, the contract $C_3$ is point (6, 4), but it
looks more like (6,3.1) the way the figure is drawn. For clarity, in
the next edition I  should mark 4 on the vertical axis and not 2, and
note  that $C_1 = (6,6), C_2=(3,3), C_3= (6,4)$. I have a better drawn
figure up on the Web at \\
http://pacioli.bus.indiana.edu/erasmuse/GI/pages/fig08.2.jpg.


\bigskip
\noindent
{\bf Chapter 9: Adverse Selection }

p. 223, fourth line from the bottom. Replace ``steeper'' with ``less
steep''.

p. 234. N9.2, second point  (extra explanation).  The service stream
seems to  depreciate by 33 percent, even though the car's price falls
by 66 percent, because one of the two years of the car's life is over.
Thus the service stream's apparent value only falls from \$3,000 to
\$2,000.

p. 238.   (fixed in October 2001 reprint) Although there is
no actual error in this problem, I have clarified the wording in
problem 9.3b and added a new part to the question.


\begin{itemize}
 \item[(9.3b)]   In equilibrium, the employer tests workers with
probability $\gamma$ and pays those who pass the test $w$, the
talented workers all present themselves for testing, and the
untalented workers present themselves with probability $\alpha$, where
possibly $\gamma=1$ or $\alpha=1$.  Find an expression for the
equilibrium value of $\alpha$ in terms of $w$. Explain why $\alpha$ is
not directly  a function of $x$  in this expression, even though the
employer's main concern is that some workers have a  productivity
advantage of $x$.

 \item[(9.3c)]  If $x=9$, what are the equilibrium values of $\alpha$,
$\gamma$, and
$w$?

\item[(9.3d)]  If $x=8$, what are the equilibrium values of $\alpha$,
$\gamma$, and
$w$?
\end{itemize}


\bigskip
\noindent
{\bf Chapter 10:  Mechanism Design}

p. 251, Figure 10.5. The shaded area should be  $r^*(x_u) - cx^*_u$,
not $r^*(x_u)$.

p. 254, paragraph starting ``Under perfect price discrimination...''.
Line 2 should be Figure 10.6a, not 10.4a. Line 3 should have $r(x_p) =
A+B+J+K+L$, not $r(x_p) = J+K+L$.


p. 255, top of page. The dash between $v(x_u)$ and $r(x_u)$ looks like
a minus sign. It would be better replaced by ``, because''.

The next line should start $v(x_p) - r(x_v)$, not  $v(x_p) - r(x_v u)
$.

p. 261, line 5. Should be ``commit'', not ``commits''.


\bigskip
\noindent
{\bf Chapter 11: Signalling }

p. 267. Line 4 should have Figures 7.1d and 7.1e, not 7.1e and 7.1f.




p. 282. (fixed in
October 2001  reprint) In the first paragraph of Section 11.5, delete
``with models of warranty issue by Matthews \& Moore (1987) and of''
and replace with ``for example, the multiple signal model used to
explain''. (Matthews and Moore (1987) is about product warranties, not
financial warrants.)

p. 297. In Figure 12.1, the captions are reversed. 12.1(a) is the Nash
Bargaining Game, and 12.1(b) is Splitting a Pie. Also, in Figure
12.1(a), the horizontal axis should be $U_{Smith}$, not
$\theta_{Smith}$.

\bigskip
\noindent
{\bf Chapter 12: Bargaining }

p. 309. Step (3) in the shaded
box and (3') below should say ``and to the seller otherwise'' rather
than ``and to the buyer otherwise''.

p. 310. The inequality in the equation in the
paragraph starting ``This is not efficient...''  should be reversed,
so it looks like $v_b < \frac{1+v_s}{2}$.

p. 312.  The minus sign at the start of  the payoff of
the seller, (12.19), should be deleted.

p. 322. (fixed in October 2001  reprint) In the last
line of
question 12.7, replace
``$p_s \geq p_b$'' with
 ``$p_s \leq p_b$''.

\bigskip
\noindent
{\bf Chapter 14:  Pricing}

p. 350. The shaded box for the Hotelling pricing game   has the
payoffs for Apex   if he captures the entire market as $ p_a(1)=1$.
It should be  $ p_a(1)=p_a$.

 p. 350 (clarification) When I say, ``Price aside, Apex is
most attractive for the customer at $x=0$...'' I mean that Apex is the
most attractive of the two sellers   for the customer at $x=0$. The
customer who most prefers Apex, another way to read that sentence,  is
the one at $x=x_a$.

p. 353. Figure 14.5: In example 2, both firms should be  located at
$x=0.9$, not
$x=0.7$.

p. 357. Line 2 should say ``asymmetric'', not ``asymemtric''.

p. 359.  A good addition to  the second paragraph  is  a sentence to
make it read:

``If $c_n$ increases enough, then the nature of the
equilibrium changes drastically, because firm $n$ goes out of
business. Even if $c_n$ increases a finite amount, the implicit function
theorem is not applicable, because then the change in $p_n$ will cause
changes in the prices of other firms, which will in turn change $p_n$
again.''

p. 360. A5' should say ``Increasing parameter $\tau$'', not
``Increasing parameter $c$''.


p. 365.   The caption of Figure 14.8 should  have
``Player'' capitalized.  The last sentence before equation (14.66) has
a reference
to equation (14.66) which should be (14.63) instead. The first
sentence
after equation (14.69) has a reference
to equation (14.63) which should be (14.66) instead.


 p. 369. the word ``be'' should be added to the second note on 14.4,
so it reads,
``...because his product can be recycled...''
 



\bigskip
\noindent
{\bf Chapter 15: Entry   }

p. 391. In question 15.1, Replace ``preying on
Brydox''  with ``preying on Brydox (and he does not learn from
experience) ''

 Equation (15.29) should have $-p_a +m < 2d$, not $-p_a +m \leq 2d$.

 Problem 15.1b should say, ``the equilibrium can be pooling'', not
``the equilibrium will be pooling''.

 Equation (15.30)  should be
 $$
 \theta  \geq \frac{d_b}{p_b+d_b}.
$$



p. 392.
 In question 15.1c: replace ``after observing Apex chose Prey in the
first period.'' with ``after observing that Apex    chose $Prey$ in
the first period. Show that the equilibrium values of $\alpha$ and
$\beta$ are:''

 In question 15.1c, equation (15.31) should be:
$$
\alpha = \frac{ \theta p_b   }{(1-\theta)d_b}
$$

Question  15.1d should say ``following phenomenon?'', replacing the
period with a question mark.

 In problem 15.3, clarify by replacing

``Set $f(x) =  log(x)$, $g(y) = .5(1+ y/(1+y)$ if $ y \geq
0$, $g(y) =
.5(1+ y/(1-y)$ if $ y \leq 0$,  $y=2$, and  $z = 1$.''

 with

``Set $f(x) =  log(x)$, and for $w = f(x_i) -f(x_e)$,
$g(w) = .5[1+
w/(1+w)]$ if $ w \geq 0$, $g(w) = .5[1+ w/(1-w)]$ if $ w
\leq 0$,  $y=
2$, and  $z = 1$.''


\bigskip
\noindent
{\bf  Mathematical Appendix  }

p. 394. In the definition of Maximum, it should say $Maximum (x-x^2 =
1/4$, not $Minimum (x-x^2 = 1/4$.

p. 396. In Figure A.1, draw in a dotted line to illustrate the
convexity definition similar to the one for the concavity
definition.


p. 397. In the definition of Metric, the second line should say
``$d(w,z)=0$ if and only if $w=z$'', not ``$d(w,z)=0$ if  $w=z$''.

The definition of Metric Space should begin ``Set X is a metric space
if ...'' rather than ``A non-empty set X is a metric space...''

   The definition of ``one-to-one'' is wrong. It should be:

``The mapping $f: X \rightarrow Y$ is one-to-one if every point in set
$X$ maps to a different point in  $Y$, so   $x_1 \neq x_2$  implies
$f(x_1) \neq f(x_2)$.''

Note also that a one-to-one {\it correspondence} is conventionally
defined as a correspondence which is a mapping that is  both   one-to-
one and onto.

\end{document}







