# Prediction Markets, Fair Games and Martingales

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1 Chapter 3 Prediction Markets, Fair Games and Martingales Prediction markets are speculative markets created for the purpose of making predictions. The current market prices can then be interpreted as predictions of the probability of the event People who buy low and sell high are rewarded for improving the market prediction, while those who buy high and sell low are punished for degrading the market prediction. 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. Prediction market (W) 3.1 Some data from Tradesports Here I describe some data collected from Tradesports 1 in Figure 1 refers to a baseball match between the New York Yankees and the Los Angeles Angels. There is a contract, which expires (at match end) at 100 if a specified team (in this case, the Angels) win, and expires at 0 if this team loses. The units are arbitrary; the non-arbitrary 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 \$5.70. You can bet on the other team by selling the contract. So you could sell one contract at the opening bid price of 56. This setting differs from traditional gambling in that there isn t a bookmaker; you are trading with other participants. What you see on the web 1 which ceased business in October

3 3.2. FAIR GAMES Fair games Let me use the phrase known pure chance for a setting where probabilities and payoffs are explicitly known. Roulette is the iconic example, or blackjack using a specified strategy. Mathematical probability textbooks teach you how to calculate pretty much anything you want in such settings. As much as possible I avoid repeating such standard material in these lectures, whose focus is on understanding the real world outside the casino, where we generally do not have reliable prior knowledge of probabilities. In the known pure chance setting, a bet is called favorable to a player if that player s mean gain is positive, unfavorable if the mean gain is negative (i.e. a loss) and fair if the mean gain is zero. (Here mean is mathematical expectation). Note the word fair here has a different meaning from its everyday one. The rules of team sports are fair in the sense of being the same for both teams, so the more skillful team is more likely to win, so a bet at even odds is not a fair bet. The iconic example of a fair game is to bet on a sequence of coin tosses, winning or losing one dollar each time depending on whether you predict correctly. Of course this is too boring for anyone to actually do! In the corresponding mathematical model (often called simple symmetric random walk) write X n for your fortune (amount of money) after n tosses. 3.3 Martingales Before stating the definition, let me distinguish between two settings. The first is exemplied by the simple symmetric random walk model above, and the second by prices in a prediction market. Within a specified math model, a sequence (X n ) either is or isn t a martingale, depending on whether it satisfies the definition below. When X n denotes some real-world quantity at time n, we could model (X n ) as some unspecified martingale. In some contexts there are compelling reasons why this is reasonable; in some other contexts there are plausible but vague arguments why this is reasonable. In contexts where one has enough data, one can check empirically whether quantitative predictions from martingale theory are correct. Note this parallels closely the situation with stationary process in Lecture 2. There we gave plausible but vague arguments why it is reasonable to model English text as some (unspecified) stationary process.

4 28CHAPTER 3. PREDICTION MARKETS, FAIR GAMES AND MARTINGALES The formal definition of martingale is a process satisfying E(X n+1 X n = x n,x n 1 = x n 1,..., X 0 = x 0 )=x n, all x 0,x 1,..., x n. This (and the (W) page) are frankly incomprehensible if you haven t taken a course in mathematical probability, so let me try to explain in words, in the context of gambling. What makes a single bet fair is that you start with a known fortune x 0, you will get a random fortune X 1, and your gain X 1 x 0 has mean zero, that is EX 1 = x 0. The martingale property is saying that at each time the expectation of your next gain, conditional on what has happened so far, has mean zero. If the odds offered to you are fair, then however much you choose to bet, this will be true. For any underlying fair game, if you bet repeatedly, choosing how much to bet each time using some arbitrary strategy of your choice, which you may change and which may depend on what has happened so far, then the progress of your fortune X n is a martingale. What makes martingale a useful concept for doing calculations is a result known formally as the Optional stopping theorem (W) but which I will call the Fair Game Principle. Ignoring some technical restrictions, and still in the gambling context, Fair Game Principle. Whenever you choose to leave the game, your final fortune X T will be such that EX T equals your initial fortune, that is the same as some single fair bet. To illustrate why this is useful, here is Basic Formula for Fair Games. Start betting on a fair game with x 0 dollars and continue until your fortune reaches either 0 or a target value B > x 0, whichever comes first, and without making any bet whose success would cause you to overshoot B. Then the probability of reaching B is exactly x 0 /B. To see why, the mean value of your final fortune is pb, where p is the probability above, and by the fair game principle this must equal x Prediction markets and martingales. The axiomatic setup of mathematical probability assumes there are true probabilities for events, which depend on the information known. Within this setup one can consider, in a sports match setting, X n = probability team A wins, given the information available at time n

5 3.4. PREDICTION MARKETS AND MARTINGALES. 29 and this is a martingale, as a mathematical theorem. How does this apply to an actual baseball game? One view a common default view, I believe is that the set-up is conceptually correct, but that in complex settings like a baseball game there is just no automatic procedure to assign numerical values to these probabilities 3. Another view (to be developed further in Lecture yyy on Philosophy) concerns the formalization of information available at time n. Here information is intended in the everyday sense, not the letters comprising English text meaning in Lecture 2. In the axiomatic set-up, information is represented as follows. There was some previously specified collection of events whose outcomes (happen or not) would be known at time n, and now we do know these outcomes. But this is so far removed from the way we actually perceive and act upon information that (in this view) its conceptual utility is dubious. Setting aside such conceptual questions, 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 consensus opinion that there is a 63% for A to win. The point, of course, is that a person confidently assessing the chance as higher than 63% would buy the contract as a perceived favorable bet, pushing the price higher; another person cassessing the chance as lower than 63% would sell the contract as a perceived favorable bet, pushing the price lower; so the price we observe represents the balance of opinions. How such a process actually works in detail is hard to say without introducing an awful lot of supposes. But the approximate identifications prices consensus probabilities probabilities within axiomatic setup suggest as a plausible hypothesis that (H) prices in a prediction market should behave (that is, fluctuate in time) approximately as a martingale. Rather than examine further the assumptions or logic that went into stating this hypothesis, let us consider how to study it empirically. The mathematics of martingales, which we introduced in the previous section and will continue in the next section, provides a wide variety of theoretical predictions. Are they correct, in real prediction markets? This makes an interesting topic for course projects, to gather data and test theory for oneself, or to read academic papers which have done so. What is remarkable about this topic is that it is one of the very few cases, outside of games of chance based 3 This is not a frequentist vs. Bayesian issue both use the same mathematics.

6 30CHAPTER 3. PREDICTION MARKETS, FAIR GAMES AND MARTINGALES on artifacts with physical symmetry, where one can make quantitative predictions without needing much input data from the real world. In fact the theory uses only the current price. 3.5 Theoretical predictions for the behavior of prediction markets Given a historical database of prediction market price fluctuations until the contract expires, the most obvious prediction to test is that, out of all contracts starting at price around 40, about 40% do in fact end with the event happening. For this purpose, around 60 is the same as around 40 by considering the opposite event, and we can make more efficient use of data by considering each contract that ever crosses 40 or 60, taking the first such crossing as our initial time. Such data generally agrees with the theory, except in the case of very small initial prices (see section 3.7). A skeptic might argue maybe the real probabilities were sometimes 50% and sometimes 30% and these just averaged out to the predicted 40%. The slightly more elaborate calculations below give predictions which allow one to distinguish between a wide spread around 40% and a small spread around 40%, given enough data. Write a for some price less than the current price x (maybe a = 0, maybe 0 < a < x) and write b for some price more than the current price x (maybe b = 100, maybe x < b < 100). Write p x (a, b) for the probability that the price reaches b before reaching a. By the same argument as for the Basic Formula, p x (a, b) = x a b a. (3.1) (More precisely, this would be exact if prices varied continuously; it s only an approximation when prices can jump, but in the present context it s a good approximation.) Formula (3.1) can be used to get various interesting formulas, and we will give two of them. Crossings of an interval. Fix a price interval, say [40, 60]. If the price is ever 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 0 without crossing 60, and from formula (3.1) the chance it reaches 60 equals 2/3. If 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

7 3.5. THEORETICAL PREDICTIONS FOR THE BEHAVIOR OF PREDICTION MARKETS31 crossing 40. So there is some random number C 0 of crossings, and from the argument above this number has the (shifted Geometric) distribution P (C = i) = 1 3 ( 2 3 )i, i =0, 1, 2,.... (3.2) Remember this is all under the assumption that the price reaches 40 or 60 sometime. I have a collection of charts for 103 baseball matches like that in Figure 1 earlier. Of these, 89 reached 40 or 60, and here is the data for the observed number of crossings of the interval [40, 60] observed predicted The discrepancy for large values can perhaps be explained by lack of continuity (near the end of a close game a single hit may have a substantial effect, and there may be no trades between hits). 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 minimum price L x such that L x x; (ii) or team A loses, the contract expires at 0, and there is some overall maximum price L x such that L x x. The formula for the distribution of L x is P (L x <a) = a(100 x) 100(100 a), 0 < a < x (3.3) P (L x >b) = x(100 b), 100b x < b < 100. (3.4) Here s how to derive these formulas. With starting price x, consider the first time T that the price reaches a or b, whichever happens first. The chance b is reached first is p = p x (a, b), in which case the price at T is b, and with chance 1 p the price is a. So the optional sampling theorem says x = E(price at time t) =pb + (1 p)a and solving this equation for p gives formula (3.1). For 0 a x, P (L x < a, A wins) = P (price hits a before 100 initial price 100) P (A wins current price = a) = (1 p x (a, 100)) a 100 = 100 x 100 a a 100 which is formula (3.3); and formula (3.4) is derived similarly.

8 32CHAPTER 3. PREDICTION MARKETS, FAIR GAMES AND MARTINGALES 3.6 Other martingale calculations One can sometimes do mathematical calculations which are not directly about gambling by considering hypothetical bets. This is sufficiently striking that for the second example I will break my rule of not discussing unmotivated coin-tossing type problems. How many serious Presidential contenders? Consider the Intrade prediction market for the U.S Presidential election winner. This started in 2010 with Obama at around 50 and no other candidate above 10. At that time, what could one say about the number N of other candidates whose price would sometime reach at least 10? Theory says EN = 5. Imagine betting 50 on Obama and then 10 on each other candiate at the time their price reaches 10, if it ever does. Your total cost is N. Your final payoff is 100, because you have certainly bet on the ultimate winner. By the Fair Game Principle this is like a single fair bet, so E( N) = 100. Waiting times for patterns in coin-tossing. If you toss a fair coin repeatedly, how long before you see the pattern HTHT? It turns out the mean number of tosses equals 20. Here s the gambling argument we wish to spotlight (there are several other ways to solve this problem). Imagine a casino where you can bet at fair odds on the coin tosses. For each toss, say toss 7, imagine a gambler following the strategy bet 1 that toss 7 is H if win, bet all (now 2) that toss 8 is T if win, bet all (now 4) that toss 9 is H if win, bet all (now 8) that toss 10 is T If win, walk away with your 16. If lose sometime, walk away with 0. Recall that for each toss a different gambler starts this strategy. Look at what happens from the casino s viewpoint, up to and including the random toss (number S) on which the pattern HTHT is first completed. The casino has taken in 1 from each of S gamblers. It has paid out 16 to the gambler whose first bet was on toss S 3. It has paid out 4 to the gambler whose first bet was on toss S 1, but nothing to the other gamblers. So the casino s gain is S 20. But by the Fair Game Principle, the mean gain must be zero.

9 3.7. STOCK MARKETS AND PREDICTION MARKETS Stock markets and prediction markets Lecture 4 will discuss long term stock market investment, as the prime realworld instance of gambling on a favorable game. Let me compare and contrast the two. A prediction market is conceptually simpler than a stock market because the final value corresponds to a definite event. Saying a stock index ends the year at 1321 says nothing other than it ends at A prediction market is mathematically simpler because we need no empirical data to make the theoretical predictions; for the analogous predictions in a stock market one needs an estimate of variance rate. Compared to stock markets, prediction markets are very thinly traded, suggesting they will be less efficent and less martingale-like. In particular, for a year-duration contract with a small price (say 5) on Intrade, the true probability might be much smaller, because to make the sure thing profit of 5 by betting against the event requires you to cover the contract and tie up 95 capital for a year. Standard economic theory asserts that long-term gains in a stock market will exceed long-term rewards in a non-risky investment, because inverstors risk-taking must be rewarded. In this picture, a stock market is a positive sum game benefiting both investors and corporations seeking capital; financial intermediaries and speculators earn their share of the gain by providing liquidity and convenient diversification for investors. In contrast a prediction market is a zero-sum game, in fact a slightly negative-sum game because of transaction costs. The recent history of financial markets might provoke skepticism about standard economic theory in general, and in particular the efficient market hypothesis which reappears in Lecture 4. Prediction markets could in principle serve a non-gambling function via hedging real-world risks, but currently do not provide enough liquidity to do so. 3.8 Wrap-up The realization that (a) the general notion (as opposed to specific bets on specific games) of

10 34CHAPTER 3. PREDICTION MARKETS, FAIR GAMES AND MARTINGALES results of bets at fair odds has a precise formalization as martingale, (b) the way the current probability of a future event changes with time, as new information comes in, also behaves mathematically as a martingale was one of the triumphs of twentieth century mathematical probability. Prediction markets provide the most concrete real-world instance of observable quantities (prices, here) that can be directly interpreted as probabilities. The mathematical theory of martingales can be used to make testable predictions about the behavior of prices in prediction markets. On the other hand, the mathematical setup does not correspond well to our intuitive sense of how we actually make decisions under uncertainty. It is easy to make the assertion that the prediction market price obtained via different agents choices to act on their probability assessments gives a consensus probability that will behave as in the mathematical model of information. But hard to expand upon an argument for how this happens in detail. 3.9 Further reading Broader ideas underlying prediction markets are often discussed under the phrase The Wisdom of Crowds (W), from the title of a book by Surowiecki. Though a cute title for a book, it is another example of an il-chosen name for a concept. A prediction algorithm, whether based on software or wetware, does not constitute wisdom. As I have already mentioned several times, these lectures do not focus on games of pure chance, which are treated in many existing works. Two books I recommend are Haigh s Taking Chances at an elementary mathematical level, and Ethier s The Doctrine of Chances: Probabilistic Aspects of Gambling for material going beyond typical textbook material. A World of Chance: Betting on Religion, Games, Wall Street by Brenner, Brenner and Brown contains an interesting, mainly historical, account of the relation between the laws and culture surrounding gambling and the laws and culture surrounding business and market activity. Probability with Martingales by David Williams shows one can teach an advanced course on mathematical probability centered around martingales as a technical tool.

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