4.2 Prediction markets4.2.1 IntroductionHere’s the Wikipedia introduction [49].2Prediction markets are speculative markets created for the purposeof making predictions. Assets are created whose final cash value istied to a particular event (e.g., will the next U.S. president be a Re-publican) or parameter (e.g., total sales next quarter). The currentmarket prices can then be interpreted as predictions of the probabil-ity of the event or the expected value of the parameter. Predictionmarkets are thus structured as betting exchanges, without any riskfor the bookmaker.Other names for prediction markets include predictive markets, in-formation markets, decision markets, idea futures, event derivativesand virtual markets.People who buy low and sell high are rewarded for improving themarket prediction, while those who buy high and sell low are pun-ished for degrading the market prediction. Evidence so far suggeststhat prediction markets are at least as accurate as other institutionspredicting the same events with a similar pool of participants.Many prediction markets are open to the public. Betfair is theworld’s biggest prediction exchange, with around $28 billion tradedin 2007. Intrade is a for-profit company with a large variety ofcontracts not including sports. The Iowa Electronic Markets is anacademic market examining elections where positions are limited to$500. TradeSports are prediction markets for sp orting events. ThesimExchange, 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 money. Bet2Give is a charity predictionmarket where real money is traded but ultimately all winnings aredonated to the charity of the winner’s choice.For concreteness we’ll consider Tradesports and consider baseball matches.The figure refers to a match between the New York Yankees and the Los AngelesAngels. There is a contract, which expires (at match end) at 100 if a specifiedteam (in this case, the Angels) win, and expires at 0 if this team loses. Theunits are arbitrary; the non-arbitrary aspect is that one contract expiring at 100is worth $10. So to buy one contract at the opening “offer” price of 57 wouldcost $5.70. You can bet on the other team by selling the contract. So you couldsell one contract at the opening “bid” price of 56.2The Wikipedia article isn’t great: rewriting it would be a goo d STAT 157 course project.32This setting differs from traditional gambling in that there isn’t a bookmaker;you are trading with other participants. What you see on the site3is bid andasked prices and quantities; you can either accept someone’s posted offer, orpost your own offer.xxx figure of bid/offer pricesTradesports makes its money from a 4% fee on net winnings. So in the twocases above (buy or sell one contract at opening bid/offer prices), when theAngels won the match, your profit or loss would have been[buy]: profit of (100 − 57)% × $10 × 96% = $4.13[sell] loss of (100 − 56)% × $10 = $4.40.Of course most trades involve a larger number of contracts, just as most stockmarket trades involve more than one share of the stock. At present (August2008), Tradesports emphasized three baseball games each day, and typicallyaround 4,000 contracts were traded on each game.Figure xxx. Times4and prices of trades on a particular baseball match.As an alternative to posting or accepting individual offers, a participant canset themself up as a “market maker” by posting bid and offer prices at the sametime.STAT 157 projects. I am collecting5data in the form of a Figure xxxgraph for about 100 baseball games; this, and other prediction market data youmight be able to find yourself, suggests a number of projects. In particular,does data correspond the predictions of theory in the next section?3What I describe is available freely without registration; obviously to actually trade needsregistration and depositing money.4Tradesports is based in Ireland and shows Irish time; the match time was really 7 - 10 pmin Los Angeles.5and would like some STAT 157 student to take over the task of collecting!334.2.2 Theory says: prediction market prices are martin-galesIt is natural to view prediction market prices as “consensus probabilities”; aprice of 63 on a contract for team A to win represents a probability 63% for Ato win; algebraicallyP (A wins) = x/100 where x is current price.Moreover it is natural to expect that prices fluctuate as a martingale.xxx tie up with discussion of martingales.In particular, the optional sampling theorem (xxx) says that the current pricex should be the expected value of the price at any future time; and this leadsto the theoretical prediction (4.1) below. Write a for some price less than thecurrent price x (maybe a = 0, maybe 0 < a < x) and write b for some price morethan the current price x (maybe b = 100, maybe x < b < 100). Write px(a, b)for the probability that the price reaches b before reaching a. Then theory sayspx(a, b)=x − ab − a. (4.1)(More precisely, this is exact assuming prices vary continuously; it’s only anapproximation when prices ca jump, but in the present context it’s a goodapproximation.)Formula (4.1) can be used to get more interesting formulas; we state themfirst and then outline the derivations of all these formulas.Maximum and minimum prices. For a contract starting at price x, either(i) team A wins, the contract expires at 100, and there is some overall minimumprice Lxsuch that Lx≤ x;(ii) or team A loses, the contract expires at 0, and there is some overall maximumprice Lxsuch that Lx≥ x.We can use formula (4.1) to get a formula for the distribution of Lx:P (Lx<a)=a(100 − x)100(100 − a), 0 < a < x (4.2)P (Lx>b)=x(100 − b)100b, x < b < 100. (4.3)Crossings of an interval. Fix a price interval, say [40, 60]. If the price isever in this interval, then there is some first time the price crosses 40 or 60 –suppose it crosses 40. Either it sometime later crosses 60, or it expires at 0without crossing 60, and from formula (4.1) the chance it reaches 60 equals 2/3.I it reaches 60, call this a first “crossing”. From 60, it may (with chance 2/3)cross 40 again (a second “crossing”) or it may expire at 100 without crossing 40.So there is some random number C ≥ 0 of crossings, and from the agrument34above this number has the (shifted) Geometric distribution (xxx tie up