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# Seminar on Topics in Probability and Statistics STAT 157

GPA 3.64

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This 4 page Class Notes was uploaded by Floy Kub on Thursday October 22, 2015. The Class Notes belongs to STAT 157 at University of California - Berkeley taught by D. Aldous in Fall. Since its upload, it has received 26 views. For similar materials see /class/226727/stat-157-university-of-california-berkeley in Statistics at University of California - Berkeley.

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Date Created: 10/22/15

42 Prediction markets 421 Introduction Here7s the Wikipedia introduction 492 Prediction markets are speculative markets created for the purpose of making predictions Assets are created whose nal cash value is tied to a particular event eg will the next UiSi president be a Re publican or parameter eigi total sales next quarter The current market prices can then be interpreted as predictions of the probabil ity of the event or the expected value of the parameter Prediction markets are thus structured as betting exchanges without any risk for the bookmaker Other names for prediction markets include predictive markets in formation markets decision markets idea futures event derivatives and virtual marketsi People who buy low and sell high are rewarded for improving the market prediction while those who buy high and sell low are pun ished for degrading the market predictioni Evidence so far suggests that prediction markets are at least as accurate as other institutions predicting the same events with a similar pool of participants Many prediction markets are open to the public Betfair is the worlds biggest prediction exchange with around 28 billion traded in 20 7 ntrade is a forpro t company with a large variety of contracts not including sports The Iowa Electronic Markets is an academic market examining elections where positions are limited to 500 TradeSports are prediction markets for sporting events The simExchange Hollywood Stock Exchange NewsFutures the Pop ular Science Predictions Exchange Hubdub and the Foresight Ex change Prediction Market are virtual prediction markets where pur chases are made with virtual moneyi BetZGive is a charity prediction market where real money is traded but ultimately all winnings are donated to the charity of the winners choice For concreteness welll consider Tradesports and consider baseball matches The gure refers to a match between the New York Yankees and the Los Angeles Angels There is a contract which expires at match end at 100 if a speci ed team in this case the Angels win and expires at 0 if this team losesi The units are arbitrary the nonarbitrary aspect is that one contract expiring at 100 is worth 10 So to buy one contract at the opening offer price of 57 would cost 570 You can bet on the other team by selling the contract So you could sell one contract at the opening bid price of 56 2The Wikipedia article isn7t great rewriting it would be a good STAT 157 course project 1hssettlng dl exsfmm tnadatnanal gambhngmthat thene 181W ahmhn aken you are tnadang wath uthen Panxmpams What you se en the sltea ls hld and asked hues and quantities yDu can elthen amegt menu s Pmed D e est yum awn D ex b lees untsmalaesltsmuney hum a A fee unnet wannlngs sh mthe twu coy have hw a sell une mntnaet at ubemng hldD ex buses when the Angels wan the match yaur bna t m s wuuld have been buy bna t Df100797 x llo x 96 7 lm sell Ins loo e awn x llo o amuse mm tnades mvulve a langen numben 3139 contacts lust as must stuuk manlaet tn s mvulve male than m2 share 3139 the stock At present August 2003 Thdespuns embhasazed the baseball games each day and tybleally arulmd loco euntnaets wene tnaded um each game MLBNWLAALAA Aug 9 zone 3 us AM lsr e 5 55 AM lsr e Pvlce pnee an new MAM MSAM toow 522m SSSAM Aug a zone Sam2 www1rdesnnrlscnm Flgme me 39hmes and 1211625 bl tnades an a Panmular baseball match As an altennatwe to 12ng m ameptmg mdmdual 95215 a bantmlbamt can set themsellub as a market maker by pmmg hld andu ez Pnaes at the same 5 T 157 laments lam cullaetmgS data m the run bl a ngue xgtolt mph a about we baseball games thlss anduthen Predmunn market data you mlght be able to nd yoursell wang a numben of bnuleets ln bantmulan dues data mnespund the bnedmtms of theory m the next when 422 Theory says prediction market prices are martin gales It is natural to View prediction market prices as consensus probabilities a price of 63 on a contract for team A to Win represents a probability 63 for A to Win algebraically PA Wins 1100 Where I is current price Moreover it is natural to expect that prices fluctuate as a martingale xxx tie up With discussion of martingalesl In particular the optional sampling theorem xxx says that the current price I should be the expected value of the price at any future time and this leads to the theoretical prediction 4 1 belowl Write a for some price less than the current price 1 maybe a 0 maybe 0 lt a lt z and Write In for some price more than the current price 1 maybe I 100 maybe I lt b lt 100 Write p75 a b for the probability that the price reaches 12 before reaching a Then theory says I a bia Ma b 41 More precisely this is exact assuming prices vary continuously its only an approximation When prices ca jump but in the present context it s a good approximation Formula 41 can be used to get more interesting formulas we state them rst and then outline the derivations of all these formulasl Maximum and minimum prices For a contract starting at price I either i team A Wins the contract expires at 100 and there is some overall minimum price Lac such that Lac S 1 ii or team A loses the contract expires at 0 and there is some overall maximum price Lac such that Lac 2 I We can use formula 41 to get a formula for the distribution of Lac a100 7 z PL lta m 0ltaltz 4 2 100 7 b PL gt b z lt b lt100 4 3 Crossings of an interval Fix a price interval say 4060 If the price is ever in this interval then there is some rst time the price crosses 40 or 60 7 suppose it crosses 40 Either it sometime later crosses 60 or it expires at 0 Without crossing 60 and from formula 41 the chance it reaches 60 equals 23 I it reaches 60 call this a rst crossing From 60 it may With chance 23 cross 40 again a second crossing or it may expire at 100 Without crossing 40 So there is some random number C 2 0 of crossings and from the agrument above this number has the shifted Geometric distribution XXX tie up with discussion of Geometric 1 2 i PCz777 z0l2uu 4 4 3 3 Remember this is all under the assumption that the price reaches 40 or 60 sometimei lf instead of the interval 4060 we used some interval a 100 7 a we get a similar formula with a nonsymmetric interval a z we get a slightly more complicated formula Derivation of formulas With starting price I consider the rst time T that the price reaches a or 127 whichever happens rst The chance I is reached rst is p p75 a b7 in which case the price at T is 127 and with chance 1 7 p the price is a So the optional sampling theorem says I Eprice at time t pb l 7 pa and solving this equation for p gives formula 41 For 0 S a S I PL lt a A wins Pprice hits a before lOOlinitial price 100gtltPA winslcurrent price a a 7 10071 X a 100 7 1007 a 100 which is formula 42 and formula 43 is derived similarlyi 17 p a7100 gtlt xxx discussion simpler than stock market because de nite endi

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