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4 Barry Nalebuff 181 The question for tournament designers is: which system provides the better player with the biggest chance of winning? Outside the squash courts, economists can think more broadly about designing contest incentive schemes where effort and skill count more than luck. The perception from the New Scientist article (quoted in the Summer 1989 column) was that English rules were more favorable to the better player. The reason is that the receiver has to win two rallies in a row to score a point and this is much harder for the weaker player. Is that correct? For simplicity, you may assume that American squash played to fifteen points takes the same time to complete a game as scoring with the British rules to nine. Puzzle 4: Running Around in Squares This problem calls for some cleverness. But before I describe the puzzle at hand, it will help put you in the right frame of mind if I first relate a well-known problem concerning two trains and a fly. The two trains are initially 100 miles apart and are approaching each other at 50 miles per hour. The fly flies back and forth at 100 mph between two oncoming trains. The question is how far does the fly fly before it is squashed by the two trains? There are two ways to solve this problem. One is to sum up an infinite series. On the first leg, the fly starts at one train and flies 66 2/3 miles before running into the other train and turning around. But now the trains are only 33 1/3 miles apart so the fly's next leg is only 1/3 the previous distance. Thus the total distance travelled is simply the sum of the infinite series 66.66[1 + 1/3 + 1/9 + 1/ ] = 66.66(3/2) = 100. The cleverer way to solve the problem to see that it takes the two trains exactly one hour to meet and a fly travelling at 100 mph covers 100 miles during that time. The fame of this problem among economists comes from a story about John von Neumann. He was given this problem at a cocktail party and instantaneously responded with the answer: 100 miles. The person posing the question complimented von Neumann on recognizing the clever solution, for his experience showed that most people labor to solve the infinite series. Professor von Neumann responded: But that's how I did it! After being told this story, Richard Palmer (a polymath at Duke) offered a physicist's version. There are four individuals, located at the four corners of a 100 by 100 square. Each person begins walking towards the person to their right. How far do they walk before meeting in the center? The hard way to solve the puzzle is to integrate the distance along the spiral path they follow. Of course, there is an easier way. The only thing more impressive than figuring out the easy solution instantaneously is to figure it out the hard way just as quickly. Avinash Dixit (Princeton) showed off his Cambridge Math Tripos training and solved it right away. Even better, he offered a variation of the puzzle which makes it

7 184 Journal of Economic Perspectives more slot machine play? Would this be worse for the gamblers or for the casinos?) In Atlantic City, the slots near the doors and other prominent places have the highest payback ratios. Perhaps they also are tuned differently so as to pay back more frequently (and more noisily) but with smaller prizes. Answer to Puzzle 2 The answer.57 is awfully tempting but completely wrong. The reason is that it ignores the fact that this patient, unlike most other patients, died. This information should be used to update our prior belief about the likelihood she took the prescribed pain-killer. On the other hand, to apply the standard Bayesian rule, we need to know the unconditional probability that the patient died and that number doesn't seem immediately available. Here is where some creativity is needed. Consider a group of 1000 patients. Of the 600 who take the pain medication, say that X people die. How many will die from the group of 400 who didn't take the pain medication? For the first group, we are told that 0.95X died from the drug and 0.05X died from other causes. The people who died from other causes should be just as likely to die if they didn't take the drug as if they did! For example, the chance of a fatal bee sting should not depend on whether or not the pain medication was used. (Actually, the pain medication could have some synergistic effects that either increase or decrease the other competing risks, but it seems reasonable to ignore this possibility.) Thus, the best guess for the number of deaths in the group of 400 who didn't take the pain medication is 400(0.05X/600). Now all the necessary information is on the table. The number of deaths due to zomepirac compared to all deaths is 0.95X/[ X + (2/3)0.05X] = 95/103.3, which is approximately 92 percent. The odds are very strong that she took the drug and that the drug killed her. For those of you who are wondering whether this seems like a pretty dangerous medication, take comfort in the fact zomepirac has been taken off the U.S. market. Answer to Puzzle 3 Actually, given that the two scoring systems lead to games that involve roughly the same number of rallies, there should be no difference between the two in favoring the better player. The reason is that all the unscored points in English squash cancel out. The easiest way to see this is to pretend that every rally in English squash was scored. When the receiver wins a rally, that's one point she wouldn't otherwise get. But then when she loses a rally when serving, it's one point she wouldn't otherwise lose. Every time a player loses a serve and gets it back is simply one point added to both scores. When determining who wins, this effectively cancels out. There may be some slight correction for the fact that there may be a temporary imbalance giving the initial server a one-point edge. But the important point is that whoever is ahead at the end is the one who wins and since the missing points essentially cancel each other out, the scoring rules don't have any significant influence on who is leading. Thus the problem reduces to which game is longer. A longer game gives the better player a greater chance of being ahead. Here is where the simplifying assumption that games played under the different scoring rules take the same time

8 Barry Nalebuff 185 comes into play. More generally, one could adjust the winning levels to make this true or for the current levels simply find out which one takes longer to play and recognize that that's the one favoring the better player. If instead it were always necessary to win two rallies in a row in order to score a point, then the unscored points would no longer cancel out. The better player would have a great advantage. When the better player's chance of winning a rally is p > 1/2, the odds of winning two rallies in a row (and thus a point) are p2: (1 - p)2. For example, a player with a 20 percent edge (p = 0.60) finds the advantage roughly doubled-the player's chance of scoring a point grows to 69 percent and the chance of winning the game grows even more. The intuition that a rule which requires winning two rallies in a row before scoring helps the better player is correct provided that is true for all rallies, and not just the transition from server to receiver. Answer to Puzzle 4 The trick is to look at the problem from the frame of reference of one of the individuals, say person A. From A's perspective, the person to her right, B, is always walking at a right angle to the line connecting A and B. Thus, to the first order, B 's movement has no effect on the distance between A and B. Since A is continuously adjusting her direction towards B, it is always as if A is walking directly towards B. A B I Initially A and B are a distance of 100 apart. Thus A walks 100 before meeting B at the center. More generally, the picture below shows that the distance travelled when there are n participants beginning at the vertices of a symmetric n-gon equals the side length over [1 + sin(90-0)], where 6 = (n - 2)180/n, the measure of an interior angle for a symmetric n-gon. As A moves an infinitesimal amount A towards her right-hand neighbor B, B's A movement towards her right-hand neighbor C contributes an amount A sin(90-6) towards the distance between A and B. Thus the

9 186 Journal of Economic Perspectives distance A must travel is scaled by the factor 1/[1 + sin(90-0)]. Note that the scaling factor is smallest for a triangle (2/3), is unity for a square, and rises to infinity as the number of sides, n, approaches infinity. The reason is straightforward. In the limiting case, it is as if everyone is positioned on an infinitely large circle. It appears that you and your neighbor are walking in exactly the same direction and hence you never get any closer: an infinitely large circle has the same property as an infinite line! A AS sin(90 -e ) Answer to Puzzle 5 I don't know. Reader Mail and Comments on Previous Puzzles In the giving credit where credit's due department, the solution I described to Rafael Hassin's waiting line problem is due to John Geanakoplos (Yale). Hassin showed by construction that a LIFO allocation rule resulted in efficient line lengths. As he presented this result at an IMSSS workshop, Geanakoplos offered an intuitive argument generalizing the result: any rule where the most recent arrival is placed at any position other than at the end of the line leads to efficient line lengths since the last person in line (and marginal waiter) imposes no externality on anyone who comes later. Geanakoplos's argument is the one presented in the Summer 1989 column. In the better-late-than-never department, Dr. Aris Ikkos (Alexander, Young & Co. in Greece) provided a correct solution to the "Keeping One Step Ahead" puzzle published in Winter John Baffes (Univ. of Maryland, College Park) offered one more perspective on Joe Farrell's puzzle about competitive equilibrium as a welfare minimum. His point was that if the government subsidizes some goods and taxes others, the welfare optimization problem looks convex-unless appropriate constraints

10 Puzzles 187 are added. In a closed economy, supply must equal demand as otherwise there will be rationing. In an open economy, supply need not equal demand, but there must be a balanced budget. Adding this constraint restores the concavity of the maximization problem and thus the optimality of the competitive equilibrium. * Once again my thanks go to Carl Shapiro and Tim Taylor for keeping me honest; they test and improveach puzzle before you see it. References Kramer, Michael, "A Bayesian Approach to Assessment of Adverse Drug Reactions: Evaluation of a Case of Fatal Anaphylaxis,"Drug Informatzon Journal, 1986, 20,

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