Category Archives: prediction markets

Can prediction markets be right too often?

Some people are puzzling over the fact that TradeSports simultaneously called every individual state’s Senate election correctly, yet failed to call the Democratic takeover of the Senate. Lance explains why this is not a puzzle at all, and a commenter on Marginal Revolution provides a good analogy:

… the Indianapolis Colts are favored to win in every game remaining on their schedule, but they are not expected to finish the season undefeated.

Because the Democrats essentially had to go “undefeated” across three or four states each with tight races that were only slightly leaning Democratic, the individual and overall predictions were perfectly consistent.

But this begs another question: didn’t TradeSports call too many states correctly? If the markets were correctly calibrated, then among all of the Republican candidates that were given a 40% chance of winning, shouldn’t roughly 40% of them have actually won? The same “problem” occurred in the 2004 election when TradeSports correctly called all fifty states in the Bush-Kerry presidential election. Again, both the close calls and the blowouts turned out exactly the way each market was leaning even slightly. In other words, 0% of the candidates that were given a 40% chance of winning actually won, and 100% of candidates that were given a 60% chance of winning actually won. Does that mean that those marginal underdogs were way overpriced, and the marginal favorites were way underpriced?

Not necessarily. The reason is that state election outcomes are very far from statistically independent events. Because common factors (news, scandals, economic conditions, etc.) effect all states simultaneously, the elections are far more likely to break in unison than would be expected of the same number of completely independent events. In the Indianapolis Colts analogy, it’s fairly reasonable to assume that each game is roughly statistically independent, making sixteen wins without a loss incredibly unlikely even for the best team. But for individual states’ elections held on the same day, it’s much more likely for one party to string together an “undefeated” series of wins than if the elections were truly independent. So the fact that TradeSports correctly called 33 Senate elections and 50 electoral college outcomes should not be considered as 83 independent pieces of evidence about how well TradeSports is calibrated, but rather some much smaller amount of evidence of (mis)calibration. The bottom line is we need more data across many elections to truly test TradeSports’s accuracy and calibration.

If TradeSports had offered combinatorial markets, we could have explicitly seen the strong dependence between states’ election outcomes. In fact, in the 2004 election, TradeSports did list some combinatorial contracts, like “Ohio + Florida” which revealed very strong statistical dependence.

Although TradeSports’s individual state predictions and overall Senate prediction were entirely consistent, one might argue that traders underestimated the degree of dependence (correlation) among states’ elections. In fact, I made a few bucks selling the “GOP Senate control” contract on TradeSports using exactly that reasoning. The truth is, I probably just got lucky, and it’s nearly impossible to say whether TradeSports underestimated or overestimated much of anything based on a single election. Such is part of the difficulty of evaluating probabilistic forecasts.

Hurricane Futures Market + Election Day

I just discovered this real-money Hurricane Futures Market run by the University of Miami in conjunction with the University of Iowa Electronic Market. Looks very thinly traded so far but worth keeping an eye on.

Also, today is election day so “boffins” and a “staffters” like me are watching the prediction market maps that Lance created with great interest. It looks like Senate control is coming down to two close races: Missouri and Virginia, both leaning 60% toward Democrats as of this writing. If we assumed these outcomes were statistically independent (which they’re not), then that would give a 64% chance of Republicans maintaining control of the Senate. The TradeSports market for Republicans keeping Senate control stands at 70%.

Implementing Hanson's Market Maker

Robin Hanson invented a wonderful market maker well suited for use in prediction market applications with a long name: the logarithmic market scoring rule market maker, which I’ll abbreviate as LMSR. (In fact, Hanson invented an entire class of market scoring rule market makers, but the logarithmic variant seems the most useful.) Hanson’s two papers on the subject are excellent, but Hanson does not spend a lot of time explaining how LMSR functions as a market maker in the typical sense. Instead, Hanson mostly emphasizes a second, alternate way of thinking about his market maker, as a “sequential shared scoring rule”, which I will not try to explain here. Hanson prefers to describe trader behavior in terms of “changing the price” instead of “buying and selling shares”. In my opinion, most people who encounter LMSR for the first time don’t quite see how beautifully and naturally LSMR can be used as a market maker in a standard prediction market setting. In fact, I am embarrassed to admit that upon my own first reading of Hanson’s papers, I did not fully “get it”. It took my seeing LMSR implemented in practice, by Todd Proebsting at Microsoft Research for Microsoft’s internal prediction markets, to realize how elegantly LMSR can be used as a market maker in an otherwise typical prediction market. LMSR is now being used in several places, including an implementation at InklingMarkets with a wonderfully intuitive interface, the Washington Stock Exchange, BizPredict, and (reportedly) at YooNew. Net Exchange was one of the first to use LMSR, though they seem to favor Hanson’s “change the price” interface over the more widespread “buy and sell shares” interface. As Chris Masse is quick to point out, LMSR has achieved much more widespread use than my own competing invention, the dynamic parimutuel market maker, which so far is being used in only one place: our own Yahoo! Tech Buzz Game.

In this post I will try to explain how to implement LMSR in a way that I believe most people familiar with prediction markets will understand. This interpretation of LMSR is not new: it’s the way Proebsting thinks about LMSR and it’s implicit “between the lines” in Hanson’s papers. But I haven’t seen this interpretation of LMSR written up anywhere, so I’m hoping that others can benefit from this explanation. The following understanding of LMSR was developed over the past few months together with my colleague Yiling Chen.

Suppose there are two outcomes that traders can buy or sell shares of (bet on or against) such that one and only one of the two outcomes is guaranteed to eventually occur. For example, the two outcomes could be “a Democrat wins the 2008 US Presidential election” and “a Democrat does not win the 2008 US Presidential election”. Each share is worth exactly $1 if and only if the trader is correct. In other words, one share of “Democrat wins” pays $1 if, in 2008, a Democrat actually wins the election, and is worthless otherwise. The following description can be easily generalized to any number of (disjoint and exhaustive) outcomes, including the case of combinatorial markets, but for ease of exposition I’ll stick to the two-outcome case.

The market maker keeps track of how many shares have been purchased by traders in total so far for each outcome: that is, the number of shares outstanding for each outcome. Let q1 and q2 be the number (“quantity”) of shares outstanding for each of the two outcomes. The market maker also maintains a cost function C(q1,q2) which records how much money traders have collectively spent so far, and depends only on the number of shares outstanding, q1 and q2. For LMSR, the cost function is:

C = b * ln(eq1/b+eq2/b)

where “ln” is the natural logarithm function, “e” is the constant e=2.718…, and “b” is a parameter that the market maker must choose. The parameter “b” controls the maximum possible amount of money the market maker can lose (which happens to be b*ln2 in the two-outcome case). The larger “b” is, the more money the market maker can lose. But a larger “b” also means the market has more liquidity or depth, meaning that traders can buy more shares at or near the current price without causing massive price swings.

Traders arrive one at a time and tell the market maker how many shares they want to buy or sell of each outcome. Traders say, for example, “I want to buy 13 shares of outcome 1 — how much will that cost?”, or “I want to sell 250 shares of outcome 2 — how much will you pay me?”. The market maker uses the cost function to answer these questions. The cost to buy 13 shares of outcome 1 is simply C(q1+13,q2) – C(q1,q2). The “cost” to sell 250 shares of outcome 2 is C(q1,q2-250) – C(q1,q2), which will be a negative number (negative cost), meaning that the seller receives money in return for the shares. In general, if a trader wants to buy or sell shares of either or both outcomes so as to change the number of shares outstanding from (q1,q2) to (q1*,q2*), then he or she must pay C(q1*,q2*) – C(q1,q2) dollars. If this amount is negative it means the trader receives money instead of paying money.

Here’s a simple example. Suppose b=100 and no one has purchased any shares yet, so q1=q2=0. A trader arrives who wants to buy 10 shares of outcome 1. The trader must pay:

C(10,0)-C(0,0) = 100 * ln(e10/100+e0) – 100 * ln(e0+e0) = $5.12

Now suppose that at some time later, the number of shares outstanding for outcome 1 is q1=50 and the number of shares outstanding of outcome 2 is q2=10. Now the same trader above returns to the market and wants to sell her 10 shares. The trader’s “payment” is:

C(40,10)-C(50,10) = 100 * ln(e40/100+e10/100) – 100 * ln(e50/100+e10/100) = -$5.87

This is a negative number so it means the trader receives $5.87. So in the end the trader made a round-trip profit of $0.75.

That’s it! Well, almost. If the market maker wants to quote a “current price”, he can. The current price for outcome 1 is:

price1 = eq1/b/(eq1/b+eq2/b)

and similarly for price2. But note that the current price only applies for buying a miniscule (infinitesimal, in fact) number of shares. As soon as a trader starts buying, the price immediately starts going up. In order to figure out the total cost for buying some number of shares, we should use the cost function C, not the price function. (If you remember your calculus: The total cost for buying k of shares of outcome 1 is the integral of the price function from q1 to q1+k. The price function (“price1”) is the derivative of the cost function C with respect to q1, and the cost function is the integral of the price function.)

Finally, although I won’t go into the details here, one can generalize the above so that the market maker can handle limit orders, for example an order to “buy up to 100 shares of outcome 1, each at price less than or equal to $0.80”. But if unfilled limit orders like this are allowed to persist, the market maker logic can get a little complicated.

As I mentioned, Hanson actually invented an entire class of market makers: he shows how to turn any proper scoring rule into a market maker. Yiling Chen and I have derived the cost and price functions corresponding to the quadratic scoring rule. It turns out, however, that the quadratic scoring rule market maker is not very interesting or useful in practice. I’ll save the details for another day. We’re also working on additional classes of market makers that do seem useful, results we hope to report on soon [update: see our paper “A utility framework for bounded-loss market makers”].

Carving a legal niche for prediction markets in the US

In the wake of US authorities arresting two executives of prominent European online gambling companies, and the surprise passage of the Unlawful Internet Gambling Enforcement Act of 2006, the shares of publicly-traded online gambling firms with large US exposure are down 50% or more. Now these companies are selling off their US operations for as little as $1. And it’s not just offshore gambling execs and shareholders who are worried. Many people are lamenting the seemingly dulled prospects of operating real-money prediction markets in in the United States.

In the previous post, I discussed what is legal in the US and what is not. In this post, I’d like to explore the pros and cons of different strategies for carving out a legal niche for prediction markets.

My personal opinion, and likely the opinion of many readers, is that gambling should be legal in the US as a matter of personal freedom, and that the US should follow the lead of the UK in legalizing, regulating, and taxing online gambling. However, as a practical matter we cannot hope for anything close to blanket legalization anytime in the foreseeable future. Here are four less sweeping approaches to drawing the legal boundaries, some more realistic than others.

  1. Economic Purpose Vs. Entertainment
    Robert Hahn and Paul Tetlock have written an excellent op ed in the New York Times calling for special legal distinction for prediction markets apart from gambling laws. They propose an “economic purpose test”, which would legalize prediction markets that have some economic value: either value as an instrument for hedging risk, or “information” value as a predictor of outcomes of significant economic consequence. Hahn and Tetlock argue that presidential betting would pass their economic purpose test, and that sports betting would not pass their test. However, one can argue that sports teams, local sports bars, and even city governments could use sports betting markets to hedge risk. I believe that, as a practical matter, sports betting would simply have to be called out as an exception in any such test.
  2. Skill-Based Vs. Chance-Based
    One argument is to draw the legal lines to outlaw pure chance-based games with a proven mathematical house edge that cannot be overcome. Roulette, craps, lotteries, and other common casino games fall into this category. The flipside would be to argue that any game that might allow a mathematical edge to a player with superior information or superior strategy should be allowed. Sports betting, poker, and, of course, prediction markets fall into this category. There is some precedent for allowing skill-based “gambling” games in many US states, as discussed in the previous post.
  3. Exchanges Vs. Bookies
    Another argument is to distinguish the new betting exchanges from more traditional bookies. Betting exchanges, like BetFair and TradeSports, simply provide a central marketplace for people to trade bets with one another. They collect transaction fees, but their profit does not depend at all on which side of a bet wins or loses. In contrast, bookies can end up with imbalanced exposure and may stand to gain or lose depending on the outcome of the bet. Also, bookies effectively enforce an artificially large bid-ask spread (often operationalized as a “vig” or tax on winnings) to ensure their profitability, while exchanges do not. Executives at TradeSports argue that these distinctions put them in safer legal territory than more typical online gamling operations. I’m not sure that US prosecutors would agree. The argument can sound like Napster’s argument that they were not directly responsible for users of their service who were violating the law.
  4. Investment Caps or Investor Qualifications
    One might argue that by enforcing strict investment limits, say $500 per person, the risk to problem gamblers is sufficiently minimized. This is part of the “no action” agreement between the Commodity Futures Trading Commission (CFTC) and the Iowa Electronic Markets. An almost opposite approach, but with similar motivation, is to limit participation to individuals with a very large net worth (e.g., millions of dollars). This is the legal cover that many hedge funds use: the supposition is that these individuals “know what they’re doing”, understand the risks, and have enough money to survive the inevitable ups and downs. The CFTC weilds a lighter regulatory hand on exchanges that cater only to the super rich.

In my opinion, although all the above arguments make some sense, the only one with any chance of actually gaining ground in the current legal and political environment is the first one (the “economic purpose test”), perhaps with the additional cover of a low investment cap and special exceptions ruling out sports betting and other stigmatized topics. Many people in the US, including lawmakers, still harbor outdated notions that gambling is a religious sin or has the taint of organized crime. If prediction market advocates want to make progress toward legalization, I believe they will have to distance themselves from gambling and sports betting. Although there is no logical distinction between betting on sports and trading contingent contracts, there is a very real social, political, and legal distinction. Though it can seem unpalatable to support gray and illogical distinctions, the unfortunate reality is that gray and illogical distinctions are the only ones with any practical chance of becoming law.

US gambling laws: Bizarre, illogical, & hypocritical. So what's legal?

The gambling laws in the United States, as in many other countries, are a hodgepodge of inconsistent and hypocritical provisions. First, most of the laws are at the state level, not the federal level, so a true understanding requires familiarity with fifty different sets of rules. But even at the federal level, the picture is murky, convoluted, and full of seemingly nonsensical exceptions.

So what is currently legal?

Here are some forms of gambling that are legal in the US:

  • Betting on horse races
  • Betting on Jai Alai in Florida
  • Betting on other sports (e.g., professional football, college basketball, …) in Nevada
  • Betting on roulette, craps, slot machines, and other “unwinnable” casino games in Nevada, Atlantic City, riverboat casinos, Detroit, American Indian reservations, and other specially designated places
  • Betting in state-run lotteries (which, incidentally, usually have a much larger “house edge” than any casino game)
  • Betting in “skill-based money tournaments” in most states (see below)
  • Betting in financial markets: Betting on stocks, options, futures, derivatives, “hedgelets”, and other financial instruments officially sanctioned by either the Commodity Futures Trading Commission (CFTC) or the Securities and Exchange Commission (SEC)
  • Betting on presidential elections in the Iowa Electronic Markets
  • Betting for insurance purposes: Betting on death, injury, theft, etc.

Yet casino gambling and sports betting are illegal in most of the country, and gambling on the Internet is illegal — well, sort of. MSNBC has an “oldie but goodie” special report on online gambling, which includes an article about the murky legal waters of online gambling.

The central question — whether Internet gambling is legal, illegal or exists in a legal nether world where no rules apply — is as gray as lawyers can make it.

(Aside: The same special report also includes articles on the rise of betting exchanges, and opportunities for arbitrage.)

Examples: Companies surviving on the edge

Here are some examples of US-based companies that are successfully navigating the murky legal waters. Some of these companies’ operations seem to teeter on the edge of what the gambling laws say is legal and what is not.

The Ticket Reserve and YooNew both allow sports fans to purchase tickets to important playoff games (e.g., the US NFL Super Bowl) contingent on their favorite team making it to the game. So, for example, a “Steelers Super Bowl ticket forward” entitles its owner to a ticket to the Super Bowl if and only if the Steelers are one of the two final teams that play in the Super Bowl, but is worthless if the Steelers do not make it to the Super Bowl. This is a brilliant idea: most fans value attending a big playoff game much more highly if their favorite team is playing in the game. Once a “ticket forward” is purchased, it can be sold via an exchange back to another fan at a profit or loss. The price of a “Steelers Super Bowl ticket forward” of course is intimately related to the odds of the Steelers making it to the Super Bowl, so in theory a gambler can use the TicketReserve or YooNew as an alternate betting exchange, without any intention of keeping the ticket and going to the game. These two companies have very carefully and (so far) successfully distanced themselves from the stigma and legal headaches of US gambling laws. Still, they are sufficient close to “the edge” that I doubt many large companies would take such a risk; both The Ticket Reserve and YooNew are small startups. (Aside: rumor has it that YooNew uses Robin Hanson’s logarithmic market scoring rule market maker for pricing.)

WorldWinner is another company that has successfully navigated the legal boundaries. WorldWinner was originally a US-based company, now owned by the international company Fun Technologies, and at least at one point was featured on Yahoo! Games (Internet Archive of Y!Games circa 2004). WorldWinner collects entry fees from players and pays out money to winners of various tournament-style games, including some clearly designed to resemble common casino games, like “Catch-21” (resembles blackjack) and “Royal Flush” (resembles poker). So how do they operate legally in the US? They work hard to be able to claim that their games are games of skill, not games of chance or luck. Most US states allow these “skill-based money tournaments”, but not all. Another interesting special case are companies that insure challenge prizes like those often featured at halftime of major sporting events.

HedgeStreet is one of the few companies that has trudged down the lengthy, costly, and arduous path of obtaining official sanction from the US Commodity Futures Trading Commission (CFTC). This is the safest route for any new prediction market company, as a CFTC license immediately overrides the fifty states’ varied and convoluted gambling laws. But HedgeStreet often seems to emphasize speculation as opposed to hedging in their advertisements, information collateral, and choice of contracts (e.g., short-term instead of long-term housing prices). Reportedly InTrade (the non-sports arm of TradeSports) is attempting to go down the same path toward CFTC approval. I wonder if the new anti-gambling climate will adversely effect their chances of approval. The Iowa Electronic Markets have a “no action” letter from the CFTC allowing them to operate, but that is a special case unlikely to be repeated having to do with the university’s academic status and a $500 max investment limit. US-based Cantor Fitzgerald is the parent company of UK spread betting firm Cantor Index (which in turn owns the play-money Hollywood Stock Exchange). The main difference (legally speaking) between Cantor Index, BetFair, and TradeSports, is that while Cantor Index and BetFair appear to take pains to lock out US-based bettors, TradeSports seems to specifically target the US audience by listing US-centric contracts and actively seeking exposure in US press outlets. Certainly there are other companies I’ve missed that are interesting test cases for US gambling laws.

What’s the lesson? Although the waters are murky, there are some innovative companies that are so far successfully staying on the good side of the myriad prosecutors, regulators, and lawmakers putting up barriers. Let’s encourage and support these companies, and thank them for blazing trails for the rest of us. At the same time, let’s push for clearer, saner, and less restrictive laws. More on this last point in the next post.