Category Archives: gambling

Challenge: Derive the Kelly criteria for play money

The Kelly criteria is a money management strategy for gamblers and investors. The strategy says that, when faced with a positive-expectation bet, you should invest a fraction of your budget that is proportional to your expected profit. The more your expect to gain, the more you should risk, but you never risk your entire budget.

The Kelly strategy is optimal in several senses: (1) it minimizes your “doubling time”, or the time it takes to go from having X dollars to having 2X dollars; (2) it minimizes the time it takes to achieve any given level of wealth; (3) it maximizes your long-run wealth.

(It turns out that the Kelly strategy is equivalent to maximizing a logarithmic utility function.)

A key reason the Kelly strategy is optimal is that it is very careful to never take you completely bankrupt: you spend only a fraction of your money, always reserving a bit for tomorrow, however small. This is sound advice when dealing with real money. (Aside: this all assumes you have a strict budget cap, which is not entirely realistic: you can almost always borrow at least some amount, even in today’s economy.)

But what about maximizing your virtual “wealth” inside a play-money game like NewsFutures, InklingMarkets, HubDub, or MediaPredict? The problem is not quite the same, precisely because you cannot really go bankrupt. Almost every game offers an option to “recharge” your account if you go bust. Even if the option is not explicit, you can always just abandon your account and start a new one with a fresh initial bankroll they typically give to new players.

So what is the Kelly criteria for play money? What is the optimal strategy that minimizes your doubling time when you’re always allowed to recharge back to a fixed starting value any time you go bankrupt? The answer is not obvious to me, so I’m crowdsourcing the problem: can readers derive the right rule?

My only conjecture is that it might become optimal to go “all in” on every single bet. But I’m not sure. [Update: I’ve convinced myself this is not optimal. Imagine two sequential bets, the first with minuscule expected profit and the second with huge expected profit: surely you should not go “all in” on the first.]

Note that finding the optimal solution may not just help you win more bragging rights in online games. There is a fascinating sports betting site called CentSports that gives everyone ten real cents to start with. If you can turn that ten cents into twenty dollars, they’ll cut you a check. Moreover, if you ever go to zero, they’ll restore you right back to ten cents. In other words, the system works just like play-money games except the potential for profit is real. So another way to phrase the challenge question is: what strategy in CentSports minimizes the time it takes you to go from ten cents to twenty dollars?

Innovation (or lack thereof) in casino gambling

Casino floors from Macau to Mississippi look eerily similar. The slot machine seas. The table game islands. The high-limit oases. The restaurants, shows, buffets. The colorful currency. The slot machines. The excruciating check-in lines. Minimum bet forced scarcity. The bleeping beeping slot machines.

The games themselves are for the most part the same that people have played for centuries, with rare exceptions. People flock to the games they already know: blackjack, craps, baccarat. Is this a matter of making gamblers comfortable wherever they go, luring them into a wallet-emptying rhythm? Have casinos evolved to perfection, like sharks? It seems ironic that gamblers who clearly exhibit risky behavior only want to deal with games that are known and familiar. Is there room for innovation in casino gambling? Is this a fat satiated industry resting on its laurels ready for a spark of creativity to ignite a shakeup, or a smart, precisely tuned machine already operating at full throttle in optimized mode, thank you very much?

For example, innovation in slot machine design seems to involve replacing spinning wheels with LCD screens that display in gorgeous 3D detail… spinning wheels. The greatest advance in poker technology has been the hole-card camera, enabling more engaging television coverage.

Outside of the casino, companies like betfair and twinspires are shaking up their respective industries. Why do casinos seem to be standing still?

I’d love to see an experimental marketplace where people play and invent new gambling games, and where breakout winners move on to trials in the “big leagues”. Would it ever fly? Would gamblers bother to play, or are they by and large unimaginative creatures of habit?

P.S. Did I mention that the woblomo deadline is midnight Hawaii time?

time in Hawaii

What is (and what good is) a combinatorial prediction market?

What exactly is a combinatorial prediction market?

2010 Update: Several of us at Yahoo! Labs, along with academic researchers, have theorized and written about combinatorial prediction markets for several years, as you’ll see below. But now we’ve gone beyond talking about them and actually built one. So the best way to answer the question is to see the market we built and play with it. It’s called Predictalot. The first version was based on the NCAA Men’s College Basketball tournament known as March Madness.

Combinatorial Madness

March Madness is the anything-can-happen-and-often-does tournament among the top 64 NCAA Men’s College Basketball teams. The “madness” of the games is rivaled only by the madness of fans competing to pick the winners. In Las Vegas, you can bet on many things, from individual games to the overall champion to more exotic “propositions” like which conference of teams will do best. Still, each gambling venue defines in advance exactly what you are allowed to bet on, offering an explicit list of usually no more than a few thousand choices.

A combinatorial market maker fulfills an almost magical promise: propose any obscure proposition, click “accept”, and your bet is placed: no doubt and no waiting.

In contrast, a combinatorial market could allow you to make up nearly any proposition you want on the fly, for example, “Duke will advance further than UNC” or “At least one of the top four seeds will lose in the first round”, or “ACC conference teams will win every game they play against lower-seeded SEC conference teams”. How many such propositions are there? Let’s count. There are 63 games (ignore the new play-in game), each of which could go to either to the favorite or the underdog, so there are 263 or over 9,220,000,000,000,000,000 (9.22 quintillion) outcomes, or ways the tournament in its entirety could unfold. Propositions are collections or sets of outcomes: for example “Duke will advance further than UNC” is a statement that’s true in something less than half of the 9.2 quintillion outcomes. Technically, then, there are 2263 possible propositions, a number that dwarfs the number of atoms in the universe. Clearly we could never write down a list that long, even inside a computer. However that doesn’t necessarily mean we can’t operate such a market if we are a little clever about how we implement it, as we’ll see below.

So here is my informal definition: a combinatorial market is one where users can construct their own bets by mixing and matching options in myriad ways, sort of like ordering a Wendy’s hamburger. (Or highly customized insurance.)

The Details

Now I’ll try for a more precise definition.

Just to set the vocabulary straight, outcomes are all possible things that might happen: for example all five candidates in an election, all 30 teams in an NBA Championship market, all 3,628,800 (or 10!) finish orderings in a ten-horse race, or all 9.2 quintillion March Madness tournament results. Among the outcomes, in the end one and only one of them will actually occur; traders try to predict which.

Bids express what outcome(s) traders think will happen. Bids also contain the risk-reward ratio the trader is willing to accept: the amount she wins if correct and the amount she is willing to lose if incorrect.

There are two reasons why we might call a market “combinatorial”: either the bids are combinatorial or the outcomes are combinatorial. The latter poses a much harder computational problem. I’ll start with the former.

  1. Combinatorial bids. A combinatorial bid or bundle bid is a concise expression representing a collection or set of outcomes, for example “a Western Conference team will win the NBA Championship”, encompassing 15 possible outcomes, or “horse A will finish ahead of horse B” in a ten-horse race, encoding 1,814,400, or half, of the possible outcomes. Yoopick, our experimental sports prediction market on Facebook, features a type of combinatorial bidding called interval bidding. Traders select the range they think the final score difference will fall into, for example “Pittsburgh will win by between 2 and 11 points”. An interval bet is actually a collection of bets on every outcome between the left and right endpoints of the range.

    For comparison, a non-combinatorial bid is a bet on a single outcome, for example “candidate O will win the election”. The vast majority of fielded prediction markets handle only non-combinatorial bids.

    What are examples of combinatorial bids besides Yoopick? Abe Othman built an interval betting interface similar to Yoopick (he came up with it on his own, proving that great minds think alike) to predict when the new CMU computer science building will finish construction. Additional examples include Bossaerts et al.’s concept of combined value trading and the parimutuel call market mechanism [Baron & Lange, Lange & Economides, Peters et al.]. 2010 Update: Predictalot is our latest example of a market featuring both combinatorial bids and outcomes.

  2. Combinatorial outcomes. The March Madness scenario is an example of combinatorial outcomes. The number of outcomes (e.g., 9.2 quintillion) may be so huge that we could never hope to track every outcome explicitly inside a computer. Instead, outcomes themselves are defined implicitly according to some counting process that involves enumerating every possible combination of base objects. For example, the outcome space could be all n! possible finish orderings of an n-horse race. Or all 2n combinations of n binary events. In both cases, the number of outcomes grows exponentially in the number of base objects n, quickly becoming unimaginably large as n grows.

    A market with combinatorial outcomes is almost nonsensical without allowing combinatorial bids as well, since individual outcomes are like microbes on a needle on a cruise ship of hay in a universe-sized sea. No one wants to bet on these minuscule possibilities one at a time. Instead, traders bet on high-level properties of outcomes, like “Duke will advance further than UNC”, that encode sets of outcomes. Here are some example forms of combinatorics and corresponding bidding languages that seem natural:

    • Boolean betting. Outcomes are combinations of binary events. Bids are phrased in Boolean logic. So if base objects are “Democrat will win in Alabama”, “Democrat will win in Alaska”, etc. for all fifty US states, and outcomes are all 250 possible ways the election might swing across all 50 states, then bids may be of the form “Democrat will win in Ohio and Florida, but not Virginia”, or “Democrat will win Nevada if they win California”, etc. For further reading, see Hanson’s paper on combinatorial market makers and our papers on the computational complexity of Boolean betting auctioneers and market makers.
    • Tournament betting. This is the March Madness example and a special case of Boolean betting. See our paper on tournament betting market makers.
    • Permutation betting. Outcomes are possible finish orderings in a horse race. Bids are properties of orderings, for example “Horse B will finish ahead of horse D”, or “Horse B will finish between 3rd and 7th place”. See our papers on permutation betting auctioneers and market makers.
    • Taxonomy betting. Base objects are (discretized) numbers arranged in a taxonomy, for example web site page views organized by topic, subtopic, etc. Outcomes are all possible combinations of the numbers. Bets can be placed on the range of any number in the taxonomy, for example page views of a sports web site, page views of the NBA subsection of the web site, etc. Coming soon: a paper on taxonomy betting led by Mingyu Guo at Duke. [Update: here is the paper.]

    We summarize some of these in a short article on Combinatorial betting and a more detailed book chapter on Computational aspects of prediction markets.

    2009 Update: Gregory Goth writes an excellent and accessible summary in the March 2009 Communcations of the ACM, p.13.

Auctioneer versus market maker

So far, I’ve only talked about the form of bids from traders. Next I’ll discuss the actual mechanics of the marketplace, or how bids are processed. How does the market operator decide which bids to accept or reject? At what prices?

I’ll focus on two major possibilities: either the market operator acts as an auctioneer or he acts as an automated market maker.

An auctioneer only matches up willing traders with each other — the auctioneer never takes on any risk of his own. This is how most financial exchanges like the stock market operate, and how intrade and betfair operate. (A call market is a special case where the auctioneer collects many bids over a period of time, then processes them all together in a single batch.)

An automated market maker will quote a price for any bet whatsoever. Even lone traders can place their bet with the market maker as long as they accept the price, greatly enhancing liquidity. The liquidity comes at a cost though: an automated market maker can and often does lose money, though clever pricing algorithms can guarantee that losses won’t mount beyond a fixed amount set in advance. Hanson’s logarithmic market scoring rule market maker is far and away the most popular for prediction markets, and for good reason: it’s simple, has nice modularity properties, and behaves well in practice. We catalog a number of bounded-loss market makers in this paper. The dynamic parimutuel market used in the (now closed) Yahoo! Tech Buzz Game can be thought of as another type of automated market maker.

A market with combinatorial outcomes almost requires a market maker to function smoothly. When traders have such a mind-boggling array of choices, the chances that two or more of their bets will exactly counter each other seems remote. If trades are rarely filled, then traders won’t bother bidding at all, causing a no-chicken-no-egg spiral into failure.

One the other hand, a market maker allows anyone to get a price quote at any time on any bet, no matter how convoluted or specific, even if no other traders had thought about that particular possibility. Thus interacting with a combinatorial market maker can be highly satisfying: propose any obscure proposition, click “accept price”, and your bet is placed: no doubt and no waiting.

I’ll discuss one more technicality. An auctioneer must decide whether bids can be partially filled, giving traders both less risk and less reward than they requested, in the same ratio. This makes sense. If I’m willing to risk $100 to win $200, I’d almost surely risk $50 to win $100 instead. Allowing partial fills greatly simplifies life for the auctioneer too. If bids are divisible, or can be filled in part, the auctioneer can use efficient linear programming algorithms; if bids are indivisible, the auctioneer must use integer programming algorithms that may be intractable. For more on the divisible/indivisible distinction, see Bossaerts et al. and Fortnow et al. Allowing divisible bids seems the logical choice in most scenarios, since the market functions better and most traders won’t mind.

The benefits of combinatorial markets

Why do we need or want combinatorial markets? Simply put, they allow for the collection of more information, the life-blood of every prediction market. Combinatorial outcomes allow traders to assess the correlations among base objects, not just their independent likelihoods, for example the correlation between Democrats winning in Ohio and Pennsylvania. Understanding correlations is key in many applications, including risk assessment: one might argue that the recent financial meltdown is partly attributable to an underestimation of correlation among firms and securities and the chances of cascading failures.

Although financial and betting exchanges, bookmakers, and racetracks are modernizing, turning their operations over to computers and moving online, their core logic for processing bids hasn’t changed much since auctioneers were people. For simplicity, they treat all bets like apples and oranges, processing them independently, even when they are more like hamburgers and cheeseburgers. For example, bets on a horse “to win” and “to finish in the top two” are managed separately at the racetrack, as are options to buy a stock at “strike price 30” and “strike price 20” on the CBOE. In both cases it’s a logical truism that the first is worth less than the second, yet the market pleads ignorance, leaving it to traders to enforce consistent pricing.

In a combinatorial market, a bet on “Duke will win the tournament” automatically increases the odds on “Duke will win in the first round”, as it logically should. Mindless mechanical tasks like this are handled automatically, by algorithms that are far better at it anyway, freeing up traders for the primary task a prediction market asks them to do: provide information. Traders are free to express their information in whatever form they find most natural, and it all flows into the same pool of liquidity.

I discuss the benefits of combinatorial bids further in this post, including one benefit I don’t mention here: smarter accounting, or making sure no more is reserved from a trader’s balance than necessary to cover their worst-case loss.

The disadvantages of combinatorial markets

I would argue that there is virtually no disadvantage to allowing combinatorial bids. They are more flexible and natural for traders, and they eliminate redundancy and thus concentrate liquidity (again I refer the reader to this previous post). Allowing indivisible combinatorial bids can cause computational problems, but as I argue above, divisible bids make more sense anyway.

On the other hand, there can be disadvantages to markets with combinatorial outcomes. First, trader attention and liquidity may be severely fractured, since there are nearly limitless things to bet on.

Second, and perhaps more troublesome, running an auctioneer with combinatorial outcomes is computationally intractable (specifically, NP-hard, or as hard as solving SAT) and running a market maker is even harder (specifically, #P-hard, as hard as counting SAT), meaning that the amount of time needed to run is proportional to the number of outcomes, exponential in the number of objects.

It gets worse. Even if we place strict limits on what types of bets traders can make, the market may still be infeasible to run. For example, even if all bets are pairwise, like “Horse B will finish ahead of horse D”, the auctioneer and market maker problems for permutation betting remain NP-hard and #P-hard, respectively. Likewise, Boolean betting remains hard even if the most complicated bet allowed is joining two events, like “E will happen and F will not” [see Chen et al. and Fortnow et al.].

How to build one

Now for some good news: in some cases, fast algorithms are possible. If all bets are subset bets of the form “Horse A will finish in position 1,2, or 10” or “Horse B,C, or E will finish in position 3”, then permutation betting with an auctioneer is feasible (using a combination of linear programming and maximum matching), even though the corresponding market maker problem is #P-hard. If all bets are of the form “Team B will advance to round k”, tournament betting with a market maker is feasible (using Bayesian network inference). Taxonomy betting with a market maker is feasible (using dynamic programming).

Finally, even better news: fast market maker approximation algorithms are not only possible and practical, they work without limiting what people can bet on, fulfilling the almost magical promise I made at the outset of constructing any bet you can imagine on the fly. Approximation works because people like to bet on things that have a decent chance of happening, say between a 1% and 99% chance. Standard sampling algorithms, including importance sampling and MCMC, are good at approximating prices for such reasonable events. For the extreme (e.g., 1-in-a-billion) events, sampling may fail, so the market maker will have to round off in its own favor to be safe.

Wrapping up, in my mind, the best way to implement a combinatorial-outcome prediction market is as follows:

  • Use a market maker. Without one, traders are unlikely to find each other in the sea of choices. Specifically, use Hanson’s LMSR market maker.
  • Use an approximation algorithm for pricing. Importance sampling seems to work well. MCMC is another possibility. See Appendix A of this paper.
  • The interface is absolutely key, and the aspect I’m least qualified to opine on. I think Predictalot, WeatherBill, Yoopick, and WhenWillWeMove point in the right direction.

2010 Update: Predictalot is our first pass at carrying through on this vision of how to build a combinatorial prediction market. In building it, we learned a great deal already, for example that sampling is much much trickier than I had initially imagined, and that it’s easy to accidentally create arbitrage loopholes if you’re not extremely careful with the math.

I glossed over a number of details. For example, care must be taken for the market maker to always round approximations in its own favor to avoid opening itself up to arbitrage attacks. Another difficulty is how to implement smart accounting to allow traders maximum leverage when they place many interrelated bets. The assumption that traders could lose all their bets is far too conservative — they might have bets that provably cannot simultaneously lose — but may serve as a reasonable starting point in practice.

Predict Olympic medal counts on Yoopick

We just added a new feature to Yoopick designed especially for Frenchmen Chris and Emile and citizens of nineteen other countries to place their swagor* on how many Olympic medals they think their country will win.

We’ve argued that the Yoopick interface is useful for predicting almost any kind of number, and since medal count is indeed a number, we thought we’d give it a try.

Besides, Lance told us it would be a good idea.

Sign up, play, enjoy, and don’t forget to tell us what you think!

Sharad Goel
David Pennock
Dan Reeves

* Scientific wild-ass guess, on record

Yoopick: Olympic medal count: Select

Yoopick: Olympics medal count: France: Make pick

Gambling advertising legal silliness

Google AdSense ads on intrade.comThe absurdity of gambling laws in the US leads to such silliness as:

  • In 2007, Google, Microsoft, and Yahoo! paid millions in penalties for placing gambling ads, something they haven’t done since they were told to stop in 2004.
  • Yahoo! can quote prices from intrade, but can’t link to intrade.
  • Google can’t advertise for intrade/tradesports, but can place AdSense ads on and In other words, Google can’t sell eyeballs to gambling sites, but can sell eyeballs on gambling sites.

Checkers bot can't lose… Ever

Mathematicians, third graders, and talkative defense department computers alike all know that there is an infallible way to play tic tac toe. A competent player can always force at least a tie against even the most savvy opponent.

In the July issue of Science, artificial intelligence researchers from the University of Alberta announced they had cracked the venerable game of checkers in the same way, identifying an infallible strategy that cannot lose.1

It doesn’t matter if the strategy is unleashed against a bumbling novice or a flawless grandmaster, it can always eke out at least a tie if not a win. In other words, any player adopting the strategy (a computer, say) makes for the most flawlessy grandmasterest checkers player of all time, period.

The proof of correctness is a computational proof that took six years to complete and was twenty-seven years in the making.

Tic tac toe and checkers are examples of deterministic games that do not involve dice, cards, or any other randomizing element, and so “leave nothing to chance”. In principle, every deterministic game, including chess, has a best possible guaranteed outcome2 and a strategy that will unfailingly obtain it. For chess, even though we know that an optimal strategy exists, the game is simply too complex for any kind of proof — by person or machine — to unearth it as of yet.

The UofA team’s accomplishment is significant, marking a major milestone in artificial intelligence research. Checkers is probably the first serious, popular game with a centuries-long history of human play to be solved, and certainly the most complex game solved to date.

Next stop: Poker

Meanwhile, the UofA’s poker research group is building Poki, a computer player for Texas Hold’em poker. Because shuffling adds an element of chance, poker cannot be solved for an infallible strategy in the same way as chess or checkers, but it can in principal still be solved for an expected-best strategy. Although no one is anywhere near solving poker, Poki is probably the world’s best poker bot. (A CMU team is also making great strides.)

Poki’s legitimate commercial incarnation is Poker Academy, a software poker tutor. An unauthorized hack of Poker Academy [original site taken down; see 2006 copy] may live an underground life as a mechanical shark in online poker rooms. (Poki’s creators have pledged not to use their bot online unidentified.)

Poker web sites take great pains to weed out bots — or at least take great pains to appear to be weeding out bots. Then again, some bot runners take great pains to avoid detection. This is a battle the poker web sites cannot possibly win.

1Technically, tic tac toe is “strongly solved”, meaning that the best strategy is known starting from every game position, while the UofA team succeeded in “weakly solving” checkers, meaning that they found a best strategy starting from the initial game board configuration.
2The best possible guaranteed outcome is the best outcome that can always be assured, no matter how good the opponent.

Betcha loses a battle; Not the war?

That didn’t take long.

Betcha is (was) an honor-based peer-to-peer betting service based in Seattle. On July 9, the Washington State Gambling Commission swept into Betcha’s offices, Gestapo style, confiscating everything, right down to their Programming PHP manual. Founder Nick Jenkins is now staring straight in the face of our country’s unconscionable forfeiture laws: you know, the ones that give law enforcement the right to sell Nick’s stuff on eBay and keep the proceeds, without ever charging him with a crime.

The Seattle Post-Intelligencer reported on the raid. The vast majority of the commenters sided with Betcha, urging Washington State officials to find better uses for their time and tax money, lamenting Washington’s ever-growing “Nanny State” credentials, and decrying the seemingly corrupt and hypocritical gambling politics involved.

To Nick and other Betcha employees and investors: Thank you for taking this risk and putting your stake in the ground, even if the current outcome is not what you’d hoped for. I hope you have the wherewithal to see this through to your day in court so that, if nothing else, we can get some clarity in the law. Here’s to hoping you’ve simply lost a battle and not the war.

Readers: Go to Betcha’s site to sign up for email updates and find out how to help.

Betcha's gambit

Betcha is bold. To say the least. The founder Nick Jenkins is either crazy, brilliant, or, like many founders, both. Betcha is a platform for peer to peer betting not unlike gottabet, betfair, or intrade. Except for two (intimately related) details: (1) all debts are on the honor system, and (2) it’s based in Seattle, WA, UIGEA. Betcha makes no bones about it ( no “wink wink” here): they expect users to bet on anything and everything including sports. But because coughing up is not strictly enforced, the site evades the letter of the gambling laws. To engender trust, Betcha verifies its users’ credit cards and tracks their reputation scores, but in the end all payments are voluntary. The site earns money via listing fees.

I can’t help but admire Jenkins and Co., and I hope their gambit succeeds: my heart is with them even if my head is a step behind. (For more legal discussion see Tom Bell and The Boston Globe.)

And as much as I like the concept, I do have to ding Betcha for one of the most convoluted, head-scratching explainers I’ve heard in a long time:

“As an open, honor-based betting platform, Betcha is like an auction site, Las Vegas, a marketplace of ideas, and The Golden Rule — all rolled into one. [1]

[1] “The Golden Rule” refers to the idea that you should do unto others as you’d have them do unto you. It is the fundamental principle behind most of the world’s major religions. And while we aren’t here to push religion on anyone, doing well by others is a principle we’d like to see more of.

Whaa? Four (weak) analogies plus a long-winded footnote? C’mon, Betcha, please KISS.

Challenge: Low variance craps strategy

This is the first of a series of challenge posts. I’ll pose a problem in the hopes of convincing the wise Internauts to come forth with solutions. I intend the problems to be do-able rather than mind boggling: simply intriguing problems that I’d love to know the answer to but haven’t found the time yet to work through. Think of it as Web 2.0 enlightenment mixed with good old fashioned laziness. Or think of it as Yahoo! Answers, blog edition.

Don’t expect to go unrewarded for your efforts! I’ll pay ten yootles, plus an optional and unspecified tip, to the respondent with the best solution. What can you do with these yootles? Well, to make a long story short, you can spend them with me, people who trust me, people who trust people who trust me, etc. (In lieu of a formal microformat specification for yootles offers, for now I’ll simply use the keyword/tag “yootleoffer” to identify opportunities to earn yootles, in the spirit of “freedbacking”.)

dice So, on with the challenge! I just returned from a pit stop in Las Vegas, so this one is weighing on my mind. I’d like to see an analysis of strategies for playing craps that take into account the variance of the bettor’s wealth, not just the expectation.

Every idiot knows the best strategy to minimize the casino’s edge in craps: bet the pass line and load up on the maximum odds possible. The odds bet in craps is one of the only fair bets in the casino, so the more you load up on odds, the closer the casino’s edge is to zero. But despite the fact that craps is one of the fairest games on the casino floor, it’s also one of the highest variance games, meaning that your money can easily swing wildly up or down in a manner of minutes. So on a fixed budget, craps can be exceedingly dangerous. What I’m looking for is one or more strategies that have lower variance, and are thus less risky.

So that this challenge is not vague and open ended, let me boil this overall goal down into something fairly specific:

The Challenge: Suppose that I walk into a casino with $200. I arrive at a craps table that has a $5 minimum bet and allows 2X odds. I’m looking for a strategy that:

  1. Has at least some chance of making a profit (otherwise, why bother?), and
  2. Maximizes the expected amount of time (number of dice rolls) that my $200 will last.

I prefer if you ignore the center bets in your analysis. Bonus points if you examine what happens with different budgets, table limits, and/or allowed odds. Another way to motivate this is as follows: I have a small fixed budget but want to hang around a high-limit table for as long as possible, because I get a better atmosphere, more drinks, and a glimpse of life as a high roller.

As an example, here is a strategy that appears to have very low variance: On the come out roll, bet on both the pass line and the don’t pass line. If the shooter rolls 2, 3, 7, or 11 you break even. If the shooter rolls 4, 5, 6, 8, 9, or 10, you’re also guaranteed to eventually break even. The only time you lose money is when the shooter rolls a 12 on a come out roll, in which case you lose your pass line bet and keep your don’t pass bet (i.e., you lose half your total stake). There’s only one problem with this strategy: it’s moronic. You have absolutely no possibility of winning: you can only either break even or lose. One thing you might add to this strategy to satisfy condition (1) is to take or give odds whenever the shooter establishes a point. Will this strategy make my $200 last longer on average than playing the pass line only?

For bonus points, I’d love to see a graph plotting a number of different strategies along the efficient frontier, trading off casino edge and variance. Another bonus point question: In terms of variance, is it better to place a single pass line bet with large odds, or is it better to place a number of come bets all with smaller odds?

To submit your answer to this challenge, post a comment with a link to your solution. If you can dig up the answer somewhere on the web, more power to you. If you can prove something analytically, I bow to you. Otherwise, I expect this to require some simple Monte Carlo simulation. Followed of course by some Monte Carlo verification. 🙂 Have fun!

Addendum: The winner is … Fools Gold!

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.