\documentstyle[12pt,epsf] {article} \parskip 10pt \textheight 8in \begin{document} \parindent 24pt \begin{center} {\bf August 21, 1999 errata for Eric Rasmusen's Games and Information, Second Edition, arranged by page number. Updated December 8, 2000. } \end{center} This list of errors is arranged by page number. A separate file arranges them by date of discovery. I have tried to star the mistakes most likely to cause trouble, since most of the mistakes listed here are fairly obvious typos. 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. Email: Erasmuse@Indiana.edu. Web: Php.indiana.edu/$\sim$erasmuse. Materials for for the second edition of {\it Games and Information} can be found at: http://pacioli.bus.indiana.edu/erasmuse/GI/edition2.htm To view Acrobat (.pdf) files, you will need to download Adobe Acrobat Reader If you don't use the web, just let me know and I'll send you hardcopy. Many of the game descriptions lack the gray boxes that they are supposed to have. This includes the games described on pages 260, 276, 280, and 349. p. 23. Teaching note: Boxed Pigs'' illustrates Nash equilibrium, but it is also possible to find the equilibrium of this game by iterated deletion of dominated strategies. There is no error here, but students may get confused. p. 33. Question 1.2c. should be changed to:\\ \hspace*{16pt} (1.2c) Is every iterated dominance equilibrium made up of weakly dominant strategies? \\ p. 43. (found by Kyung-Hwan Baik, Appalachian State/Sung Kyun Kwan U., March 29, 1996). In Table 2.3, $J$ should be $J_4$. *p. 51. In Figure 2.7, the labels on the two moves proceeding from node $S_1$ should be switched: Small to Large, and Large to Small. * p. 53: (Feb 1997, Chad Zutter) the very first word on this page should be Jones'', not Smith''. p. 61. (found by Kyung-Hwan Baik, Appalachian State/Sung Kyun Kwan U., March 29, 1996). The two equilibria that p. 59 says are boldfaced in Table 2.7 do not have the boldfacing. Those two equilibria are $\{Sue, Settle, Try), (Offer, Offer)\}$ and $\{Sue, Refuse, Try), (Resist, Resist)\}$. p. 61, the payoff under {(Sue;Settle,Try),(Offer,Offer)} should be 0.15 (-0.15,-0.15), not 0.15 (0.15,-0.15) p. 90 : Curiatius'', not Curatius''. p105. (David Rosenbaum) In paragraph beginning "In determining the settlement...", third line should have defendent, not plaintiff. p. 111 (Kyung Hwan Baik, May 20, 1996). On page 111, 13th line from the top, "In section 3.4,..." should be "In section 3.5,..." *p. 113. Second full paragraph, should be $V C$ (thanks to Michael Mesterton-Gibbons) *p. 118. Patrick Chen (Feb 1997) The probability next to $Wait$ for Smith should be $1-\theta$, not $\theta$. p. 119. Question 4.4d should be added to the book:\\ \hspace*{16pt}(4.4d) Which three games that have appeared so far in the book resemble Grab the Dollar''? p. 130. Four lines of text from the bottom: change Table 5.2'' to Table 5.3'' . p. 141, problem 5.2. Should be where $x \in (0,c]$ and the seller becomes liable for $x$ at the time of sale'', not where $x \in (0,c]$''. (clarifying addition) Page 141. (David Rosenbaum) Question 5.2 refers to the quality game in section 5.8. The quality game is in section 5.4. p. 151: (Feb 1997, Chad Zutter) Under Passive Conjectures, a parenthesis is missing. It should be $Prob (Hater|Apply) =0.9$. *p. 152. (January 8, 1996. Found by Hal Wasserman of Berkeley.) The sentence, The following is the unique perfect Bayesian equilibrium.'' should become The following is the unique perfect Bayesian equilibrium in pure strategies.\footnote{There exists a plausible mixed- strategy equilibrium too: $(Entrant: Enter if Strong, Enter with probability m=.2 if weak; Incumbent: Collude with probability n=.2)$. The payoff from this is only 150, so if the equilibrium were the one in mixed strategies, ignorance would $not$ help.}'' %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ January 8, 1996. Found by Hal Wasserman of Berkeley. *p. 154. Even if the entrant is weak and Nature tells this to the incumbent, the entrant would choose {\it Stay Out}, because he does not know that the incumbent knows, and his expected payoff from {\it Enter} would be $-7.5$ ($= [0.9 + 0.05][-10] + 0.05$).'' should become Even if the entrant is strong and Nature tells this to the incumbent, the entrant would choose {\it Stay Out}, because he does not know that the incumbent knows, and his expected payoff from {\it Enter} would be $-5$ ($= [0.9 ][-10] + 0.1$).'' %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ p. 156, first paragraph. Should be payoff is just 0.1'' not payoff is just 1'' p. 183 (April 1999, Axel Adam-Mueller and Stamen Gortchev). Figure 7.4. The line labelled $\pi_i = 6$ should be labelled $\pi_i = 3$. p. 185, line 1. It should not read $\pi_i = 6$, but rather $\pi_i = 3$. p. 186, Figure 7.5. $C_3$ should be immediately above the 6 on the horizontal axis. p. 196, Order of play for PG VI, move (1). Should be the agent a wage'', not the worker a wage''. p. 208 (November 1998, Kyung Hwan Baik). Equation (9) has (1-0) in the last exponent, but it should have $(1-\theta)$. p. 216. Change: Each student $i$ reports a number evaluating other students in the class.'' to Each student $i$ reports his numerical evaluation of the other students in the class.'' p. 219. First line of Problem 8.1. Change $U=\sqrt { w-\alpha e}$'' to $U=\sqrt { w}-\alpha e$'' p. 225, Order of Play. Should be The seller accepts or rejects'' not The buyer accepts or rejects.'' p. 227 (August 1998, Wendy Liu). Figure 9.2 should have $\overline{theta}$ on the horizontal axis, not $\theta$. p. 233. (November 1998, Kyung Hwan Baik). In some printings, Figure 9.5 might have the lavels switched on the dotted and solid indifference curves for Uunsafe and Usafe. To check whether it is correctly labelled, see if C2 is outside (away from the axis) of the Usafe curve, as it should be. p. 233 (November 1998, Kyung Hwan Baik). Line 8 says, "The insurance company is risk neutral, so its indifference curve is the straight line $\omega F$ if Smith is a customer regardless of his type.'' This is correct, but ambiguous. A better phrasing would be:\\ "The insurance company is risk neutral, so its indifference curve is a straight line. If Smith will be a customer regardless of his type, the company's indifference curve based on its expected profits is $\omega F$ (although if the company knew that Smith was Safe, the indifference curve would be steeper, and if it knew he was Unsafe, the curve would be steeper).'' p. 259. (November 1998, Kyung Hwan Baik). The screening game order of play is different here than in the general screening extensive form on page 167. I should redo the form on p. 167. It does not affect the analysis at all, but I should be consistent. p. 265. The Matthews and Moore article was published in 1987, not 1981. p. 268. Substitute I will use specific numbers for concreteness. The entrepreneur could signal that the stock has the high mean value, $\mu=120$, in two ways: (a) retaining a high percentage, $\alpha=0.4$, and making the initial offering at a high price of $P_0=90$, or (b) retaining a low percentage, $\alpha=0.3$, and making the initial offering at a low price, $P_0=80$. Figure 10.4 shows the different combinations of initial price and fraction retained that might be used. If the stock has a high variance, he will want to choose behavior (b), which reduces his risk. Investors deduce that the stock of anyone who retains a low percentage and offers a low price actually has $\mu=120$ and a high variance, so stock offered at the price of 80 rises in price. If, on the other hand, the entrepreneur retained $\alpha=.3$ and offered the high price $P_0=90$, investors would conclude that $\mu$ was lower than 120, but the variance was low also, so the stock would not rise in price. The low price conveys the information that this stock has a high mean and high variance rather than a low mean and low variance.'' for: Using particular numbers ... smaller discount and be willing to hold a larger fraction. Figure 10.4 shows the different combinations of initial price and fraction retained that might be used.'' (Peter Gordon, University of the West Indies) p. 314. Top line of the equations should have $p_a < p_b$'', not $p_i < p_b$''. p. 315. Equation (18a) should have $p_a < p_b$'', not $p_a< p_a$''. p. 315 (clarification, August 1998) Add this paragraph after equation (18). Here is why equations (18c) and (18d) look the way they do. If Brydox has the lower price, all consumers will want to buy from Brydox if they buy at all, but only 70 will be able to. If Brydox's price is more than 50, then less than 70 will want to buy at all, and so 0 customers will be left for Apex-- which is equation (18c). If Brydox's price is less than 50, then Brydox will sell 70 units, and the residual demand curve facing Apex is as in equation (17), yielding equation (18d). p. 326: The first line of equation (52) should have a negative sign in front of it (the second and third lines are correct). p. 328. What is now  From equation (13.\ref{fox}), $\frac{\partial \pi_n} {\partial p_n}$ is increasing in $\kappa$, so $\pi_n(p_n, p_{-n}, \kappa ) - \pi_n( p_n', p_{-n}, \kappa$'' should be   From equation (13.\ref{fox}), $\frac{\partial \pi_n}{\partial p_n}$ is increasing in $\kappa$, so $\pi_n(p_n, p_{-n}, \kappa ) - \pi_n( p_n', p_{-n}, \kappa )$'' p. 334. Add to the Appendix B glossary: {\bf maximand} A maximand is what is being maximized. In the problem Maximize $f(x, \theta)$ by choice of $x$'', the maximand is $f$. p. 335. Should be: but that the lower durability makes it credible to high-valuation buyers that the seller expects their business in the future and will not lower his price.'' not but that the lower durability makes it so credible to high-valuation buyers that the seller expects their business in the future and will not lower his price.'' (drop the so'') p. 336. The Kreps and Scheinkman article was published in 1983, not 1985. p. 338. Should be: In the durable monopoly model this would happen if the high-valuation buyers bought in the first period and thus were absent from consideration by the second period. In the bargaining model this would happen if the buyer rejected the first-period offer and the seller could conclude that he must have a low valuation and act accordingly in the second period.'' not In the durable monopoly model this would happen if the high-valuation sellers bought in the first period and thus were absent from consideration by the second period. In the bargaining model this would happen if the seller rejected the first-period offer and could conclude that he must have a low- valuation and act accordingly in the second period.'' p. 339. (found by Kyung Hwan Baik) Underneath equation (81) it should read where $\alpha > \beta$'' rather than where $\alpha - \beta$''. p. 346. Should be:  Because equation (6) uses the difference between the two firm's values of $f$, it is relative effort which determines the winner.'' not Using the difference between the $f$ functions for each firm makes it relative effort which matters.'' p. 348, 1st paragraph of 14.2. Should be predatory pricing is charging a low price'', not predatory pricing is charging a high price''. * p. 351 (March 1997, Peter-John Gordon): In the middle of the page it should be: Equating these two payoffs and solving for $\beta$ yields $\beta = \frac{MR_1 -R_2}{(M-1) R_2}$, which is...'' p. 351 (March 1997, clarification): Just above inequality (21), say: ...either of two conditions, both of which are found by substituting the equilibrium value of $\beta$ into expression (20). The first is if $R_0$ is small enough, a sufficient condition for which is'' p. 370. Formula (2) should be $s^*= \frac{2 \delta \theta}{n + \theta}$. %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ p. 382. ( Erik Johannessen ) The problem answer on p. 382 is wrong. The arrows and explanation for Scarface I are wrong. * {\bf 1.3: Timmy and Scarface.} Players Timmy and Scarface are caught in a game like the Prisoner's Dilemma'' of Table 1.1, except that Scarface already has a criminal record, so he will always get a prison term at least 5 years greater than Timmy, regardless of who confesses and who denies. Construct an outcome matrix (with Scarface as Row) and find the Nash equilibrium for this game. (Note: There is more than one game that reasonably fits this story.)\\ \hspace*{72pt} \underline{ {\it Answer.}} The story is too vague to tell us exactly how the payoffs change from Table 1.1, so I will give two possibilities. Table A.2 is constructed by just subtracting 5 from each of Scarface's payoffs in the original Prisoner's Dilemma'' in Table 1.1, except for subtracting 15 from his payoff for {\it (Confess, Deny)}. In equilibrium, Scarface denies and Timmy confesses. \begin{center} {\bf Table A.2 Scarface I'' } \begin{tabular}{lllccc} & & &\multicolumn{3}{c}{\bf Timmy}\\ & & & {\it Deny} & & $Confess$ \\ & & $Deny$ & $-6,-1$ & $\rightarrow$ & {\bf -15,0 } \\ & {\bf Scarface:} &&$\uparrow$& & $\downarrow$ \\ & & {\it Confess } & $-15,-10$ & $\rightarrow$ & $-13,-8$ \\ \multicolumn{6}{l}{\it Payoffs to: (Scarface, Timmy).} \end{tabular} \end{center} Table A.2 is a little far-fetched, because it implies that when Scarface confesses, Timmy's denial increases {\it Scarface}'s punishment, as well as Timmy's. This is possible. Maybe the judge wants to punish Timmy more (for denying), but must always punish Scarface more than Timmy. Table A.3 shows another game to fit the story, one which preserves the Prisoner's Dilemma'' property that a prisoner is treated more leniently for providing useful evidence. \begin{center} {\bf Table A.3 Scarface II'' } \begin{tabular}{lllccc} & & &\multicolumn{3}{c}{\bf Timmy}\\ & & & {\it Deny} & & $Confess$ \\ & & $Deny$ & $-6,-1$ & $\rightarrow$ & $-30,0$ \\ & {\bf Scarface:} &&$\uparrow$& & $\downarrow$ \\ & & {\it Confess } & $-13,-8$ & $\rightarrow$ & {\bf -20,-5}\\ \multicolumn{6}{l}{\it Payoffs to: (Scarface, Timmy).} \end{tabular} \end{center} In both new games, $(Confess, Confess)$ is the Nash equilibrium, even though $Confess$ is not a dominant strategy for Scarface (he would $Deny$ if he thought Timmy would go along with him). %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ p. 395. (Clarification) Problem 5.3e. Add: if payoffs are received at the beginning of each period''. It would have been better to specify payoffs at the end of each period. p. 395. The answer to question 5.5a should have $r = R/(T-R)$ rather than $r= (T-R)/R$. p. 400. In 6.1f, the figure 26 2/3 should be replaced by 23 1/3. p. 404. (Clarification) The problem answer currently says: \hspace*{16pt} (7.5a) Will the worker be paid anything if he makes a mistake?\\ \hspace*{72pt} \underline{\it Answer}. Yes. He is risk averse, unlike the principal, so his wage should be even across states.'' It would be clearer as: \hspace*{16pt} (7.5a) Will the worker be paid anything if he makes a mistake?\\ \hspace*{72pt} \underline{\it Answer}. Yes. He is risk averse, unlike the principal, so his wage would be even across states ideally; that is, whether he made a mistake or not. Paying him zero when a mistake occurs would stop him from taking the job, since the $-10/w$ term in his utility function means he would have a positive probability of a utility of negative infinity.'' p. 408. In problem 9.1d: compared to 1.57 (=$0.5 log(10.99) + 0.5 log(2.01))$'' should become compared to 1.50 (=$0.5 log(9.99) + 0.5 log(2.01))$'' . (found by Richard Tucker, Indiana U. poli sci) p. 410 (April 1999), Francisco Galera) In 9.3c, the answer should be $\alpha^* = 1$, not 0.5. Also, I have rewritten this problem and its answer generally, and can send the new, enlarged version to anyone who desires it. p. 418. Should be \underline{ {\it Answer}.} The utility point at which Jones has all the molasses and Smith has the muffins is now (1000, 350),'' not \underline{ {\it Answer}.} The utility point at which Jones has all the molasses and Smith has the molasses is now (1000, 350),'' p. 419. Clarification: Add a sentence at the end , changing, \\ {\bf 11.5: A Fixed cost of Bargaining and Incomplete Information. } Smith and Jones are trying to split 100 dollars. In bargaining round 1, Smith makes an offer at cost $c$, proposing to keep $S_1$ for himself.''\\ to {\bf 11.5: A Fixed cost of Bargaining and Incomplete Information. } Smith and Jones are trying to split 100 dollars. In bargaining round 1, Smith makes an offer at cost $c$, proposing to keep $S_1$ for himself. Smith does not have the option to refrain from making this first offer. ''\\ p. 433. This is OK as it stands. (The $R$s refers to two 1-dimensional real lines) p. 449. Chuan-Yang, not Chuan-Yank. p. 440. Theorem B.2, first paragraph, should have: $y \geq z$ (not $y \leq z$), so $y$ is the big'' equilibrium. (found by Mathias Erlei, U. Muenster, March 29, 1996) p. 457. McMillan, John (1992) {\it Games, Strategies, and Managers: How Managers can use Game Theory to Make Better Business Decisions}. Oxford, Oxford University Press, 1992. not McMillan, John , {\it Games, Strategies, and Managers: How Managers can use Game Theory to Make Better Business Decisions}. Oxford, Oxford University Press, 1992. p. 463. Scarf, Herbert.... The `63n'' should be deleted. Not yet paginated: If I talk about Rohm and Haas and arsenic in denture plastic, I am probably relaying a mistaken story. See McAfee and Deneckere, p. 158-9, JEMS, 1996, for a better story. \end{document}