The Top Ten Things That Math Probability Says About the Real World

Introduction The top ten list

The top ten things that math probability says about the real world

David Aldous

9 November 2009

David Aldous The top ten things . . . Games of chance Introduction Chance, in individual life The top ten list Chance: global risks Probability: academic disciples outside mathematics

This is part of an ongoing “probability and the real world” project. I teach a junior/senior course, and am accumulating disorganized material on my web site (Bing “David Aldous”). The whole program has much scope for undergraduate projects. There really will be a list of 10 (OK, actually 8, so the audience can suggest more!). But ﬁrst let me try to “frame” the whole topic of chance by describing 4 corners of the topic.

David Aldous The top ten things . . . 18 . IMS Bulletin Volume 38 . Issue 6

Other Meetings Around the World: Announcements and Calls for Papers

Financial Support for Participation in ICM2010 (August 19–27, 2010, Hyderabad, India) w www.icm.org.in !e International Mathematical Union (IMU) and the ICM Local Organizing Committee are currently making efforts to obtain financial support to enable as many mathematicians as possible from developing and economically disadvantaged countries to participate in ICM. !ere are three different support categories (travel, registration, living support can be applied for): ) Young mathematicians from developing and economically disadvantaged countries ) Senior mathematicians from developing and economically disadvantaged countries Games of chance Introduction ) Mathematicians from developing countries in Asia with emphasis on countries neighboringChance, India in individual life The top ten list All mathematicians who wish to apply for support are kindly asked to complete the correspondingChance: Application global risks Form at the ICM website (the same form is used for all three categories). Applications may be submitted fromProbability:  July academic –  January  disciples. Queries outside may mathematics be sent to the organization of the ICM at the address [email protected]

NEW

ICMICM SatelliteSatellite ConferenceConference onon Probability and Stochastic Processes Indian Statistical Institute, Bangalore August 13-17, 2010

Scientific Programme Committee David Aldous (Univ. of California, Berkeley) Vivek S. Borkar (TIFR, Mumbai) Mu Fa Chen (Beijing Normal Univ., Beijing) Alice Guionnet (ENS de Lyon, Lyon) Takashi Kumagai (Kyoto University, Kyoto) Edwin A. Perkins (Univ. of British Columbia, Vancouver) Rahul Roy (Indian Statistical Institute, Delhi) Marta Sanz-Sole (Univ. de Barcelona, Barcelona) Maria Eulalia Vares (CBPF, Rio de Janeiro) Ofer Zeitouni (Weizmann Inst. of Science, Rehovot)

web: http://www.isibang.ac.in/~statmath/icmprobsat e-mail: [email protected]

David Aldous The top ten things . . . Games of chance Introduction Chance, in individual life The top ten list Chance: global risks Probability: academic disciples outside mathematics Corner 1

Games of chance based on artifacts with physical symmetry – dice, roulette, lotteries, playing cards, tossing coins, etc Indeed the most common iconic visual image for randomness is a die (*). And the math of probability started, several centuries ago, in part with games of chance. BUT most of us spend little time on games of chance Digression; Is my claim (*) true? That’s a good future undergrad project! One resource is my list of, and brief reviews of, 70 non-technical books relating to Probability: just look at their cover graphics.

David Aldous The top ten things . . . Games of chance Introduction Chance, in individual life The top ten list Chance: global risks Probability: academic disciples outside mathematics

Dicing With Death. Chance, risk and health. Against the Gods: The Remarkable Story of Risk. Struck by Lightning: the curious world of probabilities. What are the Chances? Voodoo deaths, oﬃce gossip and other adventures in probability. Chances Are: Adventures in Probability. Chance Rules: An informal guide to probability, risk and statistics. The Jungles of Randomness. The Drunkard’s Walk: How Randomness Rules Our Lives. The Broken Dice, and other mathematical tales of chance. Randomness. These 10 books are in the “Popular Science” category; other categories such as “Sports and Gaming” and “Stock Market and Finance”.

David Aldous The top ten things . . . Games of chance Introduction Chance, in individual life The top ten list Chance: global risks Probability: academic disciples outside mathematics Corner 2

In what contexts does the concept of chance arise, in ordinary life as an individual? Easiest way to attempt an answer: search through blogs to examine casual usage of speciﬁc words or phrases, e.g. “one in a million chance”. Done as undergrad project. Sample results next slide.

Fun to compare with hypothetical examples invented by a philosopher (Nicholas Rescher) in a monograph Luck: The Brilliant Randomness Of Everyday Life.

Project: articulate diﬀerences [someone, please]?

David Aldous The top ten things . . . Games of chance Introduction Chance, in individual life The top ten list Chance: global risks Probability: academic disciples outside mathematics Chance – as seen in blogs

Seeing how its like a one in a million chance to ﬁnd that one person you connect with. I have .... syndrome. The fact that I ever became a mother was a “one in a million chance”. i’m waiting for the day they [upcoming movie/TV ﬁlming locations] say my city which is one in a million chance .... got this contest. It’s a one in a million chance to get some people .... to tell me what they think of my work. Of course if I don’t go [to the doctor about certain symptoms], there’s that one in a million chance that I’ll be sorry I didn’t. and they [adults] all start talking about how im too young to be going out by myself ..... But it’s not like im going to listen to them, what happened [witnessing a mall shooting] was a once in a million chance.

winning an heiress in competition with another suitor. contracting [a cold] on the evening of one’s opening night performance your secret benefactor’s sending you that big check ... being wounded by an assassin who mistakes one for someone else you were inadvertently delayed and just missed crossing on the Hindenberg burglar who breaks into a house just before its owner returns well-armed from a bear hunt coming down with a disease for which a cure has just been discovered author whose biography of a celebrity hits the bookshops just as its protagonist is enmeshed in a highly publicized scandal ...

Chance, in the human world, on a larger scale – Global Risks

Once a year you hear a news item about the World Economic Forum in Davos, Switzerland. If the attending World Leaders pay attention they will hear a discussion of upcoming Global Risks, prepared by OECD.

David Aldous The top ten things . . . Games of chance Introduction Chance, in individual life The top ten list Chance: global risks Probability: academic disciples outside mathematics

Figure 1: Global Risks Landscape 2009: Likelihood with Severity by Economic Loss n o i l l i r t

1 2 7 n 6 a h t e r o m n o i l l i 31 r t 1

- 29 5 n 1 o i l

l 19 i

b 4 0 5

2 34 ) $ S n 13 o U 3 i l

l 9 8

i 23 n i b ( 21 0 y 5 t

2 16 i 24

- 32 r 0 e 5 20 v

e 14 30 S 25 n 35 o i

l 18 l i

b 10 17 0 5

- 22 15 0 1 26 11 36

n 27 28 12 o i l l i b 0 1 -

2 33

below 1% 1-5% 5-10% 10-20% above 20%

Likelihood Based on an the assessment of risks over a 10 year time horizon by the Global Risk Network Key: Boxes indicate change since last year’s assessment Increased Decreased

Stable New risk for 2009 Likelihood Severity

Source: World Economic Forum 2009

ECONOMIC ENVIRONMENTAL 1 Food price volatility 20 Extreme climate change related weather 2 Oil and gas price spike 21 Droughts and desertification 3 Major fall in US$ 22 Loss of freshwater 4 Slowing Chinese economy (6%) 23 NatCat: Cyclone 5 Fiscal crises 24 NatCat: Earthquake 6 Asset price collapse 25 NatCat: Inland flooding 7 Retrenchment from globalization (developed) 26 NatCat: Coastal flooding 8 Retrenchment from globalization (emerging) 27 Air pollution 9 Regulation cost 28 Biodiversity loss 10 Underinvestment in infrastructure SOCIETAL GEOPOLITICAL 29 Pandemic 11 International terrorism 30 Infectious disease 12 Collapse of NPT 31 Chronic disease 13 US/Iran conflict 32 Liability regimes 14 US/DPRK conflict 33 Migration 15 Afghanistan instability 16 Transnational crime and corruption TECHNOLOGICAL 17 Israel-Palestine conflict 34 CII breakdown 18 Violence in Iraq 35 Emergence of nanotechnology risks 19 Global governance gaps 36 Data fraud/loss

Some academic topics in which the math of probability plays a role. Kinetic theory of gases (statistical physics). Random inheritance of genes by offspring from parents; random mutation; the bottom-level mechanism on which evolution by natural selection operates (population genetics). Statistical estimates of probabilities for an individual, based on data for that individual and population statistics concerning relationships between factors – e.g. credit scores; amazon.com suggestions for books you might like. (statistical learning theory). Short-term fluctuations of equity prices, exchange rates etc (stochastic finance). Systems governed by deterministic rules but so sensitive to initial conditions that long-run behavior is effectively random (chaos). Software using randomness (algorithms).

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8 The top 10 list

Start with a critical analysis of freshman statistics – what do textbooks say that’s non-obvious and actually true? 1. Opinion polls actually work rather well. Like airplanes, they only get into the news when things go wrong. I’m using opinion polls as a proxy for other textbook concepts like randomized controlled experiments – we really do know how to determine whether medical drugs work, but again the subject only gets into the news when things go wrong. Data. California Field Poll – elections for Senate, Governor, President – reverse order from 2008 (Obama - McCain) back to 1982 (Bradley - Deukmejian for Governor) – percent errors were

0101132312013313033102105

and the 5 was so unusual it acquired a name: “the Bradley eﬀect”.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8 2. Separating skill from luck

My favorite quotation relevant to probability is Chance favors the prepared mind (Louis Pasteur) which reminds us that luck and skill are not really separable. When we examine the speciﬁc history of “a success” (Bill Gates; military victories; Superbowl winners etc) we see some mixture of skill and luck. Is it possible to say (very approximately) how much was due to skill vs luck? Theory says: 2. You can measure the relative contributions of skill and luck in the aggregate but not at an individual level.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8

2009 baseball: the 3 teams with worst records were Washington (.364) Pittsburgh (.385) Baltimore (.395) Without knowing anything about baseball I’m willing to bet $100 that at least 2 of these three teams will have better records in 2010. This is a textbook instance of the regression eﬀect; above-average teams one year tend to do less well the next year. It is not primarily due to changes in a team’s ability from one year to the next. Sport Predictions for Proportion correct Hockey Top 3 72% Hockey Bottom 3 79% Football Top 3 83% Football Bottom 3 83% Basketball Top 3 77% Basketball Bottom 3 85% Baseball Top 3 66% Baseball Bottom 3 85%

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8 3. Coincidences

3. Coincidences are more likely than you think. The famous birthday paradox says: with 23 people in a room, there’s about a 50% chance that some two people have the same birthday. There’s nothing magic about 23. For any number of people there’s a formula for the chance of some shared birthday, it just turns out that to make this chance be 50% you need 23. This is a theoretical prediction; and you can check how well it works with e.g. Major League Baseball teams. And it works pretty well.

David Aldous The top ten things . . . Active Roster | phillies.com: Team http://philadelphia.phillies.mlb.com/team/roster_active.jsp?c_id=phi

MLB Sites

News Audio & Video Scoreboard Schedule Roster Stats Community Fans Citizens Bank Park Tickets Shop Auction Mobile Fantasy

1-3 Introduction 4-6 Active Roster | 40-Man Roster | Depth Chart Roster The top ten list 7-8 Active Roster Phillies Roster Active Roster Pitchers B/T Ht Wt DOB 40-Man Roster Depth Chart 58 Antonio Bastardo R/L 5-11 195 09/21/85 Search 56 Joe Blanton R/R 6-3 250 12/11/80 Coaches 37 Chad Durbin R/R 6-2 220 12/03/77 Transactions 47 Scott Eyre L/L 6-1 225 05/30/72 Injury Updates 35 Cole Hamels L/L 6-3 190 12/27/83 Draft Results 43 J.A. Happ L/L 6-6 200 10/19/82 Front Office 34 Cliff Lee L/L 6-3 190 08/30/78 Broadcasters 54 Brad Lidge R/R 6-5 215 12/23/76 Minor League Affiliates 46 Ryan Madson L/R 6-6 200 08/28/80 Phillies tickets through 45 Pedro Martinez R/R 5-11 195 10/25/71 39 Brett Myers R/R 6-4 240 08/17/80 61 Chan Ho Park R/R 6-2 210 06/30/73 Catchers B/T Ht Wt DOB 23 Paul Bako L/R 6-3 210 06/20/72 51 Carlos Ruiz R/R 5-10 205 01/22/79 Infielders B/T Ht Wt DOB 4 Eric Bruntlett R/R 6-0 200 03/29/78 19 Greg Dobbs L/R 6-1 210 07/02/78 7 Pedro Feliz R/R 6-1 210 04/27/75 6 Ryan Howard L/L 6-4 260 11/19/79 11 Jimmy Rollins S/R 5-8 170 11/27/78 26 Chase Utley L/R 6-1 190 12/17/78 Outfielders B/T Ht Wt DOB 10 Ben Francisco R/R 6-1 190 10/23/81 29 Raul Ibanez L/R 6-2 225 06/02/72 12 Matt Stairs L/R 5-9 215 02/27/68 8 Shane Victorino S/R 5-9 185 11/30/80 28 Jayson Werth R/R 6-5 215 05/20/79

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1 of 1 11/2/09 9:53 AM 1-3 Introduction 4-6 The top ten list 7-8

Mathematicians conclude “coincidences are more likely than you think” based on this type of “small universe” example, but it’s a long way away from the kind of real-life coincidences that people ﬁnd striking.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8

Some people assert: observed real-life coincidences are immensely too unlikely to be explicable as “just chance” – and so they assign spiritual signiﬁcance. Rationalists assert: oh no they’re not. Because there are gazillions of possible coincidence that don’t happen. I’m on the rationalist side, but must admit that (outside “small universe” models like the birthday paradox) there’s essentially no evidence (analogous to baseball teams evidence for the birthday paradox) that interesting real-life coincidences occur no more often than “pure chance” predicts.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8

Here are three essential features of real-life coincidences: coincidences are judged subjectively – different people will make different judgements; if there really are gazillions of possible coincidences, then we’re not going to be able to specify them all in advance – we just recognize them as they happen; what constitutes a coincidence between two events depends very much on the concrete nature of the events. Studying real everyday life seems too difficult. Is there any other context in which we can both get data with these features and compare to theory? Next slide shows my best effort.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8

article article specific coincidence chance ×10−8 1 Kannappa Vasishtha Hindu religious figures 12 56 2 Harrowby United F.C. Colney Heath F.C. Engl. am. Football Clubs 160 120 3 Delilah Paul of Tarsus Biblical figures 20 30 4 USS Bluegill (SS-242) SUBSAFE U.S. submarine topics 6 1 8 5 Kindersley-Lloydminster Cape Breton-Canso Canadian Fed. Elec. Dist. 110 23 6 Walter de Danyelston John de Stratford 14/15th C British bishops 1 81 7 Loppington Beckjay Shropshire villages 4 55 8 Delivery health Crystal, Nevada Prostitution 9 46 9 The Great Gildersleeve Radio Bergeijk Radio comedy programs 4 23 10 Al Del Greco Wayne Millner NFL players 3000 77 11 Tawero Point Tolaga Bay New Zealand coast 3 32 12 Evolutionary Linguistics Steven Pinker Cognitive science ??? 36 13 Brazilian battleship Sao Paulo Walter Spies Ironic ship sinkings < 1 28 14 Heap overflow Paretologic Computer security ??? 52 15 Werner Herzog Abe Osheroff Documentary filmmakers 1 92 16 Langtry, Texas Bertram, Texas Texas towns 180 53 17 Crotalus adamanteus Eryngium yuccifolium Rattlesnake/antidote < 1 80 18 French 61st Infantry Division Gebirgsjäger WW2 infantry 4 45 19 Mantrap Township, Minnesota Wykoff, Minnesota Minnesota town(ship)s 810 41 20 Lucius Marcius Philippus Marcus Junius Brutus Julius Caesar associate 4 91 21 Colin Hendry David Dunn Premier league players 150 62 22 Thomas Cronin Jehuda Reinharz U.S. College presidents 32 44 23 GöstaKnuttson Hugh Lofting Authors of children’s lit. 32 31 24 Sergei Nemchinov Steve Maltais NHL players 900 16 25 Cao Rui Hua Tuo Three Kingdoms people 37 18 26 Barcelona May Days Ion Mot¸a Spanish Civil War 5 116 27 GM 4L30-E transmission Transaxle Auto transmissions 3 37 28 Tex Ritter Reba McEntire Country music singers 8 24

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8 4. Long term investing

4. The Kelly criterion marks the borderline between aggressive and insane investing. Background: if you’re worth $500,000 then it’s irrational to be risk-averse for small amounts – should regard “gaining $250” and “losing $250” as equal-but-opposite. But it’s rational to be risk-averse for $250,000. In fact people are Predictably Irrational (title of recent Dan Ariely book, and item 7 on our list) in such matters, but ......

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8

I focus on long-term investment. Imagine you inherit a sum of money at age 25 and you resolve to invest it and not start spending it until age 65. We envisage the following setting. (i) You always have a choice between a safe investment (pays interest, no risk) and a variety of risky investments. You know the probabilities of the outcomes of these investments. [of course in reality you don’t know probabilities – unlike casino games – so have to use your best guess instead]. (ii) Fixed time period – imagine a year, could be month or a day – at end your take you gains/losses and start again with whatever you’ve got at that time (“rebalancing”). The Kelly criterion gives you an explicit rule for how to divide your investments to maximize long-term growth rate.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8

To illustrate, imagine day-trading scheme with stocks based on some statistical non-randomness; within one day 51% chance to double money; 49% chance to lose all money. Looks good – expected gain 2% per day – but don’t want to risk all your money on one day. Instead use strategy: bet ﬁxed proportion p of money each day. Theory says: long-term growth rate, depends on p, but in an unexpected way. growth rate

2 10,000

0.02p 0.04 Optimal strategy: bet p = 2% of your capital each day; this provides 2 growth rate 10,000 per day, which (250 trading days per year) becomes 5% per year.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8

The numbers above depended on hypothetical assumptions. But the conceptual point is completely general. We are not assuming you can predict the future, just that you can assess future probabilities correctly. Provided there is some risky investment whose expected payoff is greater than the risk-free payoff, the Kelly criterion is a formula that tells you how to divide your portfolio between different investments. There’s one remarkable consequence of using this strategy. To get the maximum possible long-term growth rate, using “100% Kelly strategy”, you must accept a short-term risk, of the specific form 50% chance that at some time your wealth will drop to only 50% of your initial wealth. And 10% − 10% too! Of course, if not comfortable with this level of risk, you can use “partial Kelly strategy” combining with risk-free assets.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8

This story is told in the popular book Fortune’s Formula by William Poundstone. Maybe nothing in this story seems intellectually remarkable, but in fact something is. Consider an analogy: the light speed barrier. [Common sense says objects can be stationary or move slowly or move fast or move very fast, and that there should be no theoretical limit to speed – but physics says in fact you can’t go faster than the speed of light. And that’s a very non-obvious fact. ] Similarly, we know there are risk-free investments with low return; by taking a little risk (risk here equals short-term ﬂuctuations) we can get higher low-term reward. Common sense says this risk-reward trade-oﬀ spectrum continues forever. But in fact it doesn’t. As a math fact, you can’t get a higher long-term growth rate than you get from the “100% Kelly strategy”.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8 5. Coding

5. Coding for secrecy is essentially the same as coding for efficient communication or storage. The fact that most letter strings JQTOMXDW KKYSC have no meaning is what makes most simple letter substitution ciphers easy to break. In a hypothetical language in which every “monkeys on typewriters” string had a meaning, a letter substitution cipher would be impossible to break, because each of the 26 x 25 x 24 x .... x 2 possible decodings might be the true message. Now if you want to transmit or store information efficiently, you want every string to be possible as a coded string of some message (otherwise you’re wasting an opportunity) and indeed you ideally want every string to be equally likely as the coded string of some message. This is “coding for efficiency”, but with such an ideal public code one could just apply a private letter substitution cipher and get an unbreakable “code for secrecy”.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8 6. Genetics

Genetics is one of the classic applications of probability but rather than attempting to say anything portentious, let me just ask 6. Are you related to your 10th generation ancestors? Genetically, that is. You have 2 parents, 4 grandparents, 8 great-grandparents, and back 10 generations you have somewhat less than 1,024 ancestors (likely not all diﬀerent). How many have you actually inherited DNA from? You have 46 chromosomes, so if you inherited whole chromosomes you could only be related to 46 of the ancestors. In fact there is “crossover”, and under a simpliﬁed model, one can calculate that you inherited DNA from about 370 of your 10’th generation ancestors. Less than half of them. So even if you can prove that one of your ancestors was a relative of King George III, this doesn’t mean you have royal blood.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8 7. Predictably irrational

7. Predictable irrationality, in decisions under uncertainty. Much psychology research since 1980 (Amos Tversky et al) involves experiments on “decisions under uncertainty”. Here’s a famous example: decisions can be strongly aﬀected by how information is presented. Imagine a rare disease is breaking out in some community. if nothing is done, 600 people will die. There are two possible programs. To some subjects you describe the alternatives as (A) will save 200 people (B) will save everyone with chance 1/3 and save no-one with chance 2/3 to others you describe the alternatives as (C) 400 people will die (D) no-one will die with chance 1/3; 600 people will die with chance 2/3. Given “A or B” choice, most people choose A. Given “C or D” choice, most people choose D.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8

In my undergraduate course, students do course projects, and one option is to repeat some classic experiment. Here’s a fun example. Subjects: college educated, non-quantitative majors. Equipment: bingo balls (1 – 75) and 10 Monopoly $500 bills. Draw balls one at a time; subject has to bet $500 on whether next ball will be higher or lower than last ball; prompt subject to talk (recorded) about thought process. Repeat for 5 bets. Say: we’re doing this one last time; this time you have option to bet all your money. Prompt talk.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8

What is the point of this experiment? In ﬁrst part, everyone “plays the odds” – behaves and explains rationally: if this ball is 43 then more likely that next ball is less than 43, so bet that way. Point: what explanations do people give for their choice (last stage) of whether or not to bet all their money. In our experiments, about 50-50 split between risk-aversion; good or poor chances to win feeling (or have been) lucky or unlucky. Conclusion: even when “primed” to think rationally, people have innate tendency to revert to “luck” explanations.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8 8. Eﬃcient (?) markets

The efficient market hypothesis says (roughly) “current price reflects rational assessement of future value based on known information” – so price fluctuations reflect new information. Events of the last 15 months have shaken this academic orthodoxy. I don’t want to enter the debate over financial markets, which have many complicating features, so will consider something simpler in a moment. One point to remember is that time scale matters. Our previous Kelly criterion discussion dealt with the long term, say > 10 years, where compounding matters. Let’s illustrate widely varying time scales.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8

High frequency traders use computers that execute trades within milliseconds, or “with extremely low latency” in the jargon of the trade. In the U.S., high-frequency trading firms represent 2.0% of the approximately 20,000 firms operating today, but account for 73.0% of all equity trading volume. As of the first quarter in 2009, total assets under management for hedge funds with high frequency trading strategies were $141 billion, down about 21% from their high. The high frequency strategy was first made successful by Renaissance Technologies. (Wikipedia)

Contrast with what’s perhaps the world’s largest explicit bet. Warren Buﬀett versus Protege Partners, LLC Stakes $1,000,000 [to charity] Over a ten-year period commencing on January 1, 2008, and ending on December 31, 2017, the S&P500 will outperform a portfolio of funds of hedge funds, when performance is measured on a basis net of fees, costs and expenses.

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8

Instead of regular ﬁnancial markets – what do you learn by being told Microsoft stock is at 27? – it’s conceptually simpler to consider prediction markets in which you can bet (via trading contracts) on future events happening or not by a speciﬁed deadline. For instance

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8

David Aldous The top ten things . . . 1-3 Introduction 4-6 The top ten list 7-8 2012.REP.NOM.PALIN 16:47 https://www.intrade.com/jsp/intrade/common/c_cd.jsp?conDetailID=...

2012 Republican Presidential Nominee (others upon request) Sarah Palin to be Republican Presidential Nominee in 2012 New: Request A Quote

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1 of 2 11/4/09 8:48 AM 1-3 Introduction 4-6 The top ten list 7-8

In a prediction market, a price of 25 means there’s a consensus estimate that the chance of the event is 25%. One can’t say whether this is the “true” chance in individual cases, but theory says if you look at many different contracts with today’s price around 25, then for about 25% of them the event will in fact happen. What’s interesting is there are also less obvious math calculations. If today’s price is 25, math says the chance that the Palin contract price will sometime go above 75 but then she is not selected as candidate equals 1/12. Again one can test such predictions over many contracts. 8. The actual behavior of “closed-end” speculative markets is generally consistent with the predictions of an “efficient market hypothesis”, that prices reflect true probabilities.

David Aldous The top ten things . . .