Crowdpark logoCrowdpark is an impressive, well-designed prediction market game that’s already attracted 500,000 monthly active users on Facebook, the 11th fastest growing Facebook app in April.

It’s a dynamic betting game with an automated market maker, not unlike Inkling Markets in functionality (or even Predictalot minus the combinatorial aspect). What stands out is the flashy UI, both literally and figuratively. The look is polished, slick, refreshing, and richly drawn. It’s also cutesy, animation-happy, and slow to load. Like I said, Flash-y in every way. The game is well integrated into Facebook and nicely incorporates trophies and other social rewards. Clearly a lot of thought and care went into the design: on balance I think it came out great.

Crowdpark is a German company with an office in San Francisco. In addition to their Facebook game, they have German and English web versions of their game, and white-label arrangements with gaming companies. They launched in English just last December.

Crowdpark’s stunning growth contrasts with decidedly more mixed results on this side of the Atlantic. I wonder how much of Crowdpark’s success can be attributed to their German roots, their product, their marketing, or other factors?

Crowdpark has an automated market maker they call “dynamic betting” that I can’t find any technical details about [1]. Here’s their well-produced video explanation:

They say it’s “patent pending”, though my colleague Mohammad Mahdian did some nice reverse engineering to show that, at least in their Facebook game, they’re almost certainly using good-old LMSR. Here is a graph of Crowdpark’s market maker price curve for a bet priced at 1%:

Crowdpark's automated market maker price curve

Here is the raw data and the fit to LMSR with b=20,000.

risk   to win (CP)   to win (LMSR)
1 91 91.079482
2 181 181.750593
5 451 451.350116
10 892 892.847929
20 1747 1747.952974
50 4115 4115.841760
100 7535 7535.378665
200 13019 13019.699483
500 23944 23944.330406
600 26594 26594.687310
700 28945 28945.633048
800 31059 31059.076097
900 32979 32979.512576
1000 34740 34740.000000


Still, there’s a quote buried in the video at 0:55 that caught my attention: “you’re current profit is determined by the fluctuation of the odds”.

There’s only one market maker that I know of where the profit fluctuates with the odds, and that’s my own dynamic parimutuel market, which by coincidence recently went from patent pending to inventor cube delivered. :-)

David Pennock's dynamic parimutuel market (DPM) patent cube - 4/2011

With every other market maker, indeed almost every prediction market, the profit is fixed at the time of the bet. Add to that the fact that Crowdpark bought a majority stake in Florida horse racing circuit Saratoga Racing Inc. and plans to operate all bets exclusively through their system, leads me to wonder if they may have some kind of parimutuel variant, the only style of betting that is legal in the US.

Of course, it may be that I simply misinterpreted the video.


[1] The technical exec at Crowdpark seems to be Aleksandar Ivanov. I found a trade press paper on (internal) prediction markets he wrote in 2009 for the Journal of Business Forecasting.

Cantor Gaming mobile device for in-running bettingLast January, a few friends and I visited the sportsbook at the M Casino in Las Vegas, one of several sportsbooks now run by Cantor Gaming, a division of Wall Street powerhouse Cantor Fitzgerald. Traditional sportsbooks stop taking bets when the sporting event in question begins. In contrast, Cantor allows “in-running betting”, a clunky phrase that means you can bet during the event: as touchdowns are scored, interceptions are made, home runs are stolen, or buzzers are beaten. Cantor went a step further and built a mobile device you can carry around with you anywhere in the casino to place your bets while watching games on TV, drink in hand. (Cantor also runs spread-betting operations in the UK and bought the venerable Hollywood Stock Exchange prediction market with the goal of turning it into a real financial exchange; they nearly succeeded, obtaining the green light from the CFTC before being shut down by lobbyists, er, Congress.)

Back to the device. It’s pretty awesome. It’s a Windows tablet computer with Cantor’s custom software — pretty well designed considering this is a financial firm. You can bet on the winner, against the spread, or on one-off propositions like whether the offensive team in an NFL game will get a first down, or whether the current drive will end with a punt, touchdown, field goal, or turnover. The interface is pretty nice. You select the type of bet you want, see the current odds, and choose how much you want to bet from a menu of common options: $5, $10, $50, etc. You can’t bet during certain moments in the game, like right before and during a play in football. When I was there only one game was available for in-running betting. Still, it’s instantly gratifying and — I hate to use this word — addictive. Once my friend saw the device in action, he instantly said “I’m getting one of those”.

When I first heard of Cantor’s foray into sports betting, I assumed they would build “betfair indoors”, meaning an exchange that simply matches bettors with each other and takes no risk of its own. I was wrong. Cantor’s mechanism is pretty clearly an intelligent automated market maker that mixes prior knowledge and market forces, much like my own beloved Predictalot minus the combinatorial aspect. Together with their claim to welcome sharps, employing a market maker means that Cantor is taking a serious risk that no one will outperform their prior “too much”, but the end result is a highly usable and impressively fun application. Kudos to Cantor.


P.S. Cantor affectionately dubbed their oracle-like algorithm for computing their prior as “Midas”, proving this guy has a knack for thingnaming.

In the Book of Odds, you can find everything from the odds an astronaut is divorced (1 in 15.54) to the odds of dying in a freak vending machine accident (1 in 112,000,000).

Book of Odds is, in their own words, “the missing dictionary, one filled not with words, but with numbers – the odds of everyday life.”

I use their words because, frankly I can’t say it better. The creators are serious wordsmiths. Their name itself is no exception. “Book of Odds” strikes the perfect chord: memorable and descriptive with a balance of authority and levity. On the site you can find plenty of amusing odds about sex, sports, and death, but also odds about health and life that make you think, as you compare the relative odds of various outcomes. Serious yet fun, in the grand tradition of the web.

I love their mission statement. They seek both to change the world — by establishing a reliable, trustworthy, and enduring new reference source — and to improve the world — by educating the public about probability, uncertainty, and decision making.

By “odds”, they do not mean predictions.

Book of Odds is not in the business of predicting the future. We are far too humble for that…

Odds Statements are based on recorded past occurrences among a large group of people. They do not pretend to describe the specific risk to a particular individual, and as such cannot be used to make personal predictions.

In other words, they report how often some property occurs among a group of people, for example the fraction all deaths caused by vending machines, not how likely you, or anyone in particular, are to die at the hands of a vending machine. Presumably if you don’t grow enraged at uncooperative vending machines or shake them wildly, you’re safer than the 1 in 112,000,000 stated odds. A less ambiguous (but clunky) name for the site would be “Book of Frequencies”.

Sometimes the site’s original articles are careful about this distinction between frequencies and predictions but other times less so. For example, this article says that your odds of becoming the next American Idol are 1 in 103,000. But of course the raw frequency (1/number-of-contestants) isn’t the right measure: your true odds depend on whether you can sing.

Their statement of What Book of Odds isn’t is refreshing:

Book of Odds is not a search-engine, decision-engine, knowledge-engine, or any other kind of engine…so please don’t compare us to Google™. We did consider the term “probability engine” for about 25 seconds, before coming to our senses…

Book of Odds is never finished. Every day new questions are asked that we cannot yet answer…

A major question is whether consumers want frequencies, or if they want predictions. If I had to guess, I’d (predictably) say predictions — witness Nate Silver and Paul the Octopus. (I’ve mused about using *.oddhead.com to aggregate predictions from around the web.)

The site seems in need of some SEO. The odds landing pages, like this one, don’t seem to be comprehensively indexed in Bing or Google. I believe this is because there is no natural way for users (and thus spiders) to browse (crawl) them. (Is this is a conscious choice to protect their data? I don’t think so: the landing pages have great SEO-friendly URLs and titles.) The problem is exacerbated because Book of Odds own custom search is respectable but, inevitably, weaker than what we’ve become accustomed to from the major search engines.

Book of Odds launched in 2009 with a group of talented and well pedigreed founders and a surprisingly large staff. They’ve made impressive strides since, adding polls, a Yahoo! Application, an iGoogle gadget, regular original content, and a cool visual browser that, like all visual browsers, is fun but not terribly useful. They’ve won a number of awards already, including “most likely company to be a household name in five years”. That’s a low-frequency event, though Book of Odds may beat the odds. Or have some serious fun trying.

What is a better investment objective?

  1. Grow as wealthy as possible as quickly as possible, or
  2. Maximize expected wealth for a given time period and level of risk

The question is at the heart of a fight between computer scientists and economists chronicled beautifully in the book Fortune’s Formula by Pulitzer Prize nominee William Poundstone. (See also David Pogue’s excellent review.*) From the book’s sprawling cast — Claude Shannon, Rudy Giuliani, Michael Milken, mobsters, and mob-backed companies (including what is now Time Warner!) — emerges an unlikely duel. Our hero, mathematician turned professional gambler and investor Edward Thorp, leads the computer scientists and information theorists preaching and, more importantly, practicing objective #1. Nobel laureate Paul Samuelson (who, sadly, recently passed away) serves as lead villain (and, to an extent, comic foil) among economists promoting objective #2 in often patronizing terms. The debate sank to surprisingly depths of immaturity, hitting bottom when Samuelson published an economist-peer-reviewed article written entirely in one-syllable words, presumably to ensure that his thrashing of objective #1 could be understood by even its nincompoop proponents.

Objective #1 — The Kelly criterion

Objective #1 is the have-your-cake-and-eat-it-too promise of the Kelly criterion, a money management formula first worked out by Bernoulli in 1738 and later rediscovered and improved by Bell Labs scientist John Kelly, proving a direct connection between Shannon-optimal communication and optimal gambling. Objective #1 matches common sense: who wouldn’t want to maximize growth of wealth? Thorp, college professor by day and insanely successful money manager by night, is almost certainly the greatest living example of the Kelly criterion at work. His track record is hard to refute.

If two twins with equal wealth invest long enough, the Kelly twin will finish richer with 100% certainty.

The Kelly criterion dictates exactly what fraction of wealth to wager on any available gamble. First consider a binary gamble that, if correct, pays $x for every $1 risked. You estimate that the probability of winning is p. As Poundstone states it, the Kelly rule says to invest a fraction of your wealth equal to edge/odds, where edge is the expected return per $1 and odds is the payoff per $1. Substituting, edge/odds = (x*p – 1*(1-p))/x. If the expected return is zero or negative, Kelly sensibly advises to stay away: don’t invest at all. If the expected return is positive, Kelly says to invest some fraction of your wealth proportional to how advantageous the bet is. To generalize beyond a single binary bet, we can use the fact that, as it happens, the Kelly criterion is entirely equivalent to (1) maximizing the logarithm of wealth, and (2) maximizing the geometric mean of gambles.

Investing according to the Kelly criterion achieves objective #1. The strategy provably maximizes the growth rate of wealth. Stated another way, it minimizes the time it takes to reach any given aspiration level, say $1 million, or your desired sized nest egg for retirement. If two twins with equal initial wealth were to invest long enough, one according to Kelly and the other not, the Kelly twin would finish richer with 100% certainty.

Objective #2

Objective #2 refers to standard economic dogma. Low-risk/high-return investments are always preferred to high-risk/low-return investments, but high-risk/high-return and low-risk/low-return are not comparable in general. Deciding between these is a personal choice, a function of the decision maker’s risk attitude. There is no optimal portfolio, only an efficient frontier of many Pareto optimal portfolios that trade off risk for return. The investor must first identify his utility function (how much he values a dollar at every level of wealth) in order to compute the best portfolio among the many valid choices. (In fact, objective #1 is a special case of #2 where utility for money is logarithmic. Deriving rather than choosing the best utility function is anathema to economists.)

Objective #2 is straightforward for making one choice for a fixed time horizon. Generalizing it to continuous investment over time requires intricate forecasting and optimization (which Samuelson published in his 1969 paper “Lifetime portfolio selection by dynamic stochastic programming”, claiming to finally put to rest the Kelly investing “fallacy” — p.210). The Kelly criterion is, astonishingly, a greedy (myopic) rule that at every moment only needs to worry about figuring the current optimal portfolio. It is already, by its definition, formulated for continuous investment over time.

Details and Caveats

There is a subtle and confusing aspect to objective #1 that took me some time and coaching from Sharad and Dan to wrap my head around. Even though Kelly investing maximizes long-term wealth with 100% certainty, it does not maximize expected wealth! The proof of objective #1 is a concentration bound that appeals to the law of large numbers. Wealth is, eventually, an essentially deterministic quantity. If a billion investors played non-Kelly strategies for long enough, then their average wealth might actually be higher than a Kelly investor’s wealth, but only a few individuals out of the billion would be ahead of Kelly. So, non-Kelly strategies can and will have higher expected wealth than Kelly, but with probability approaching zero. Note that, while Kelly does not maximize expected (average) wealth, it does maximize median wealth (p.216) and the mode of wealth. See Chapter 6 on “Gambling and Data Compression” (especially pages 159-162) in Thomas Cover’s book Elements of Information Theory for a good introduction and concise proof.

Objective #1 does have important caveats, leading to legitimate arguments against pure Kelly investing. First, it’s often too aggressive. Sure, Kelly guarantees you’ll come out ahead, but only if investing for “long enough”, a necessarily vague phrase that could mean, well, infinitely long. (In fact, a pure Kelly investor at any time has a 1 in n chance of losing all but 1/n of their wealth — p.229) The guarantee also only applies if your estimate of expected return per dollar is accurate, a dubious assumption. So, people often practice what is called fractional Kelly, or investing half or less of whatever the Kelly criterion says to invest. This admittedly starts down a slippery slope from objective #1 to objective #2, leaving the mathematical high ground of optimality to account for people’s distaste for risk. And, unlike objective #2, fractional Kelly does so in a non-principled way.

Even as Kelly investing is in some ways too aggressive, it is also too conservative, equating bankruptcy with death. A Kelly strategy will never risk even the most minuscule (measure zero) probability of losing all wealth. First, the very notion that each person’s wealth equals some precise number is inexact at best. People hold wealth in different forms and have access to credit of many types. Gamblers often apply Kelly to an arbitrary “casino budget” even though they’re an ATM machine away from replenishment. People can recover nicely from even multiple bankruptcies (see Donald Trump).

Some Conjectures

Objective #2 captures a fundamental trade off between expected return and variance of return. Objective #1 seems to capture a slightly different trade off, between expected return and probability of loss. Kelly investing walks the fine line between increasing expected return and reducing the long-run probability of falling below any threshold (say, below where you started). There are strategies with higher expected return but they end in ruin with 100% certainty. There are strategies with lower probability of loss but that grow wealth more slowly. In some sense, Kelly gets the highest expected return possible under the most minimal constraint: that the probability of catastrophic loss is not 100%. [Update 2010/09/09: The statements above are not correct, as pointed out to me by Lirong Xia. Some non-Kelly strategies can have higher expected return than Kelly and near-zero probability of ruin. But they will do worse than Kelly with probability approaching 1.]

It may be that the Kelly criterion can be couched in the language of computational complexity. Let Wt be your wealth at time t. Kelly investing grows expected wealth exponentially, something like E[Wt] = o(xt) for x>1. It simultaneously shrinks the probability of loss, something like Pr(Wt< T) = o(1/t). (Actually, I have no idea if the decay is linear: just a guess.) I suspect that relaxing the second condition would not lead to much higher expected growth, and perhaps that fractional Kelly offers additional safety without sacrificing too much growth. If formalized, this would be some sort of mixed Bayesian and worst-case argument. The first condition is a standard Bayesian one: maximize expected wealth. The second condition — ensuring that the probability of loss goes to zero — guarantees that even the worst case is not too bad.

Conclusions

Fortune’s Formula is vastly better researched than your typical popsci book: Poundstone extensively cites and quotes academic literature, going so far as to unearth insults and finger pointing buried in the footnotes of papers. Pounstone clearly understands the math and doesn’t shy away from it. Instead, he presents it in a detailed yet refreshingly accessible way, leveraging fantastic illustrations and analogies. For example, the figure and surrounding discussion on pages 197-201 paint an exceedingly clear picture of how objectives #1 and #2 compare and, moreover, how #1 “wins” in the end. There are other gems in the book, like

  • Kelly’s quote that “gambling and investing differ only by a minus sign” (p.75)
  • Louis Bachelier’s discovery of the efficient market hypothesis in 1900, a development that almost no one noticed until after his death (p.120)
  • Poundstone’s assertion that “economists do not generally pay much attention to non-economists” (p.211). The assertion rings true, though to be fair applies to most fields and I know many glaring exceptions.
  • The story of the 1998 collapse of Long-Term Capital Management and ensuing bailout is sadly amusing to read today (p.290). The factors are nearly identical to those leading to the econalypse of 2008: leverage + correlation + too big to fail. (Poundstone’s book was published in 2005.) Will we ever learn? (No.)

Fortune’s Formula is a fast, fun, fascinating, and instructive read. I highly recommend it.

__________

* See my bookmarks for other reviews of the book and some related research articles.

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