After people have used Prediction Markets for a while and have gotten used to their ability to provide forecasts, they start thinking about different scenarios. Who would be the best Republican to face Clinton? How are the prospects for a market boom or crash effected by the winner of the election? How will poverty be affected by a proposed World Bank program? These kinds of questions can be posed in a number of ways using Prediction Markets. Markets can allow betting on conditional (if) or conjunctive (and) questions. Either one can be used to answer the what if questions, but they provide different choices to the bettors, and some make it easier for observers to decode the answers.

The easiest compound question to pose is a simple conjunction of two others. InTrade had separate markets in whether Bush would be reelected in 2004 (&#8221-BUSH&#8221-), and whether Osama bin Laden (&#8221-OSAMA&#8221-) would be captured before the election. Justin Wolfers and Eric Zitzewitz asked InTrade to add a single combined contract that would pay off if both came true. Their paper, Experimental Political Betting Markets and the 2004 Election shows how the prices on these three contracts can be combined to show how one event would be likely to effect the other.

InTrade created three separate claims to cover combinations of the two base questions. They were &#8220-Bush wins election&#8221- (BUSH), &#8220-Osama is captured before the election&#8221- (OSAMA), and the combination: BUSH&amp-OSAMA which would have paid out if both the others came true. Wolfers and Zitzewitz estimated the market&#8217-s conditional probability by comparing the price of OSAMA with the price of BUSH&amp-OSAMA. If the price levels were rational, the difference between the two prices had to equal the chance that Osama would be captured and Bush would not be reelected. Since the market price of BUSH&amp-OSAMA was 91% as high as the price of Osama, they concluded that that represented the conditional probability. A weakness of this conclusion is that while investors and arbitrageurs have an incentive to ensure that the price of BUSH is correct relative to ~BUSH, (and OSAMA with respect to ~OSAMA), there&#8217-s no bet that lets an arbitrageur exploit superior knowledge of the conditional probabilities.

Sometimes investors believe they know how one outcome will effect another, and want to bet directly on that linkage. If you were confident before the election that Osama&#8217-s capture would raise the probability of Bush&#8217-s reelection to 95% (above the level the the market prices implied), having the conjunctive bets didn&#8217-t provide a bet that would have looked beneficial to you. You might think you could buy Bush&amp-Osama (because you believe Bush&#8217-s chances are improved if Osama is captured) and sell ~Bush&amp-Osama (because this is the outcome your view says is least likely), but you&#8217-d lose both bets if Osama wasn&#8217-t captured (which is an outcome your prediction doesn&#8217-t specify.)

Conjunctive claims allow observers to deduce connections between claims, but since the investors aren&#8217-t directly rewarded based on the conditional probabilities, they have little incentive to ensure that the implicit conditional probabilities reflect their understanding of the connections between the outcomes. In order to evaluate different proposals we have to look at what investors would spend up-front, and then compare the possible outcomes and how the investor&#8217-s earnings change in each situation.

If Bush is a 60% favorite to be re-elected, and the market thinks there&#8217-s only a 10% chance Osama will be captured before the election, the odds on the conjunctions might be:

If you think Osama&#8217-s capture would improve Bush&#8217-s prospects to 95%, what should you buy or sell? Your prediction says that the ratio of Bush&amp-Osama to ~Bush&amp-Osama should be 19:1, but doesn&#8217-t have anything to say about Bush&amp-~Osama or ~Bush&amp-~Osama. If you buy Bush&amp-Osama and sell ~Bush&amp-Osama, you can make the prices match your beliefs better, but you&#8217-ll lose money if Osama isn&#8217-t captured. In order to support conditional bets directly, market operators have to find ways to allow traders to buy positions without exposing themselves to risks due to the independent cases.

A contract that acts like a conditional bet directly (written as BUSH|OSAMA, pronounced as &#8220-Bush given Osama&#8221- or &#8220-Bush conditional on Osama&#8221-) would pay off if Bush is elected, and return your investment if Osama bin Laden isn&#8217-t captured. That gives investors the right incentive.

In order to support betting on conditional probabilities, the bets have to be able to return the investors&#8217- money in particular cases. I know of three detailed proposals that have this property. They are: betting on arbitrary boolean expressions, representing the complete cross-product of possible outcomes (providing a complete set of Arrow-Debreu securities), and using the independent claim as currency for purchasing the dependent claim. There are two additional suggestions that might work, but haven&#8217-t been written down in sufficient detail to be sure.

Robin described and implemented Combinatorial Information Markets which represent probabilities and traders assets explicitly for all possible combinations of outcomes. Fortnow, Kilian, Pennock, and Wellman described how you might try to support bets on arbitrary boolean combinations of conditions. Their conclusion seemed to be that solving the general problem would be computationally infeasible. They didn&#8217-t describe how to address the problems they found, but I think it&#8217-s possible that a market that supported only binary combinations could be designed. And finally, Peter McCluskey built (and released as open source) USIFEX in 1999. It allows the user to use the coupons of the independent event as the currency. This combination allows traders to express conditionals directly. Unfortunately, that system didn&#8217-t attract a user base quickly enough, and Peter stopped development soon after the initial release.

For an article on Decision Markets written in 1999, Robin Hanson suggested creating markets using assets that pay off in &#8220-units of A if B passes&#8221- (and &#8220-&#8230- if B doesn&#8217-t pass.&#8221-), and allow traders to trade the assets for each other. The price of A|B in terms of B (which can be built from component assets) expresses the conditional bet. Robin didn&#8217-t explain how to set up a market in which people trade assets for assets and didn&#8217-t describe how to let the users see how various combination bets would express the conditional claims they might have been interested in. (This is the first of the two incomplete suggestions.)

Robin&#8217-s Combinatorial Information Market design uses a complex internal representation and can support arbitrary conditional bets. He built a prototype implementation that allows the user to explore these conditionals by choosing assumptions, and then adjusting probabilities in the resulting hypothetical situations. I wrote a prototype of my own in E. Neither prototype is more than a proof-of-concept that the institution works, and neither has been operated for any general market. The st
rength of this approach is that users can express conditional connections between arbitrary claims- this aspect has been shown to be effective in a laboratory experiment. Robin ran tests of this market after he proposed its use for PAM, and there were apparently no problems in running it with 6 traders estimating all outcome combinations for 8 events. The glaring weakness is that it doesn&#8217-t scale well. It&#8217-s not clear how to build a version that would work even with a market with dozens of questions and hundreds of users. I&#8217-ll describe this market in more detail in a future post in this series.

Peter McCluskey built USIFEX in 1999. It works quite differently and doesn&#8217-t seem to have the performance problems of the other proposals. The primary idea for supporting conditional trading is that you buy units of A|B using units of B as currency when betting on a conditional question. The effect is that when buying A|B, you end up with coupons of ~B as part of the purchase, and that&#8217-s what ensures you&#8217-ll be repaid if the independent event doesn&#8217-t occur. USIFEX is open source, but it hasn&#8217-t been maintained since it was released in 2000. The code was resurrected for use in the Swiss MarMix exchange, (AFAICT without making any use of the conditional betting features). The biggest weakness of Peter&#8217-s approach, as I recall, was that it would have taken a lot of users to ensure that the conditional markets weren&#8217-t extremely thin. A longer description of USIFEX is also in the works.

Todd Proebsting built an implementation of the Hanson design that works without conditionals. Dave Pennock wrote up a description of Todd&#8217-s approach, focused on the Market maker. I intend to describe the implications of Todd&#8217-s approach for betting on conditionals in a future post. (This is the second incomplete suggestion.) I think it might be straightforward to extend Todd&#8217-s approach to support conditional betting without running into the exponential growth of Robin&#8217-s solution. The drawback is that the market operator has to separately capitalize and enable every conditional question that you want the system to support, while Robin&#8217-s approach enables all of them by default. It&#8217-s also possible that Zocalo Open Source Prediction Market software would be compatible with this approach, where it&#8217-s clear that Zocalo would require substantial modification to support the Hanson proposal.

Other Articles in this series

• PM intro: basic formats (2005-12-30)
• PMs with Open-ended Prices (2006-01-05)
• Looking at Both Sides (2006-04-17)
• Book and Market Maker (2006-04-28)
• Liquidity in N-Way claims (2006-07-19)
• Continuous Outcomes using Bands and Ladders (2006-09-20)
• Integrating Book Orders and Market Makers (2007-01-10)

Cross-posted from pancrit.org.

As more prediction market enthusiasts in the U.S. reconcile themselves with eventual CFTC regulation, the Poker Players Alliance is making a bid to join the ranks of the privileged exemptions to UIGEA. Prediction markets have simply not been profitable enough to support a national lobby for a similar exemption. The CFTC route is cheaper and less controversial, though more restrictive.

Is there is room in the PPA&#8217-s tent for prediction market interests? Probably not, but it is worth looking into, especially since they are characterizing their request as a &#8220-skill game exemption&#8221-.

If not, while I don&#8217-t begrudge the PPA their relative progress, I would not be inclined to support their efforts to attain an exemption for the game. According to PPA President Michael Bocherek, &#8220-While we are working toward the short-term goal of a poker exemption, the PPA will also be laying the foundation for the eventual U.S. regulation of online poker. This is the only proven public policy for online gaming.&#8221-

Perhaps they ought to keep to their long-term goals, as in the short-term, poker is a game of chance.

You could say that these opinions are divisive, but I would counter that it&#8217-s up to the PPA to determine how specific their interests are. The more they act like a privilege-seeking special interest, the more my general libertarian support for all forms of legal gaming is trumped.

[Cross-posted from Risk Markets and Politics]

Previous blog posts by Jason Ruspini:

• The CFTC safe-harbor option for event markets
• CFTC regulation and election contracts
• Asymmetry in Obama nomination market
• Prediction Markets: Powerful enough to be dangerous?
• 2009 tax futures yielding 1.5%
• Intrade, with carry
• Talking tax futures on BNN, Canada’s business channel

[Cross-posted from Pancrit.org.]

Dave Pennock gave a gentle introduction to the Market Scoring Rule invented by Robin Hanson. In the comments, Sid asked for an explanation of how to integrate the MSR with an order book. Dave asked me privately if I&#8217-d be willing to tackle that, and this post is the result. Robin&#8217-s short note on integrating an order book and a market maker covers a lot of territory very quickly. In Robin&#8217-s defense, it was written to clarify some ideas in the midst of a conversation we were having at the time, and hasn&#8217-t been cleaned up for publication. I&#8217-ll expand on it here so it has a chance of making sense to others. The paper couches things in terms of the MSR, a particular AMM, but none of the implementation depends on which AMM is used.

There&#8217-s a working example of the integration we&#8217-re talking about in the code for Zocalo. The code that does this is currently in transition since I&#8217-m adding support for multi-outcome markets. For the moment, I recommend reading the code for version 375, since the current code is more complex and possibly incomplete. You can either download the complete source code for release 2006.5 of the Zocalo Prediction Market, or browse the code directly using the SVN interface.

The paper starts by giving a very compressed introduction to the idea of a prediction market and market maker (hereafter AMM for Automated Market Maker). Unless you&#8217-re very familiar with the details and the formalisms that Robin uses to describe them, you&#8217-d be better off reading the original papers (Logarithmic Market Scoring Rules, Combinatorial Information Market Design) than trying to pick anything up from the first four paragraphs of the note.

The fourth paragraph slips into the idea of integrating an order book with the AMM he&#8217-s talked about to that point. (&#8221-If instead [the AMM price resulting from buying the entire quantity is higher than the user’s max marginal price], a portion […] could be traded with the market maker, leaving a book order for the remaining quantity&#8221-). From that point, he talks about how to integrate the two markets.

If new orders get the advantage of any order price overlap

In book order systems, if orders arrive asynchronously, you will often see orders that &#8220-overlap&#8221-, i.e. orders to buy at a higher price than the best offer to sell, or orders to sell lower than the best offer to buy. The system has to have policy about what price to transact at in these cases. The system could tell each party that they got the price they requested, and pocket the difference- it could use the book order&#8217-s price or the new offer&#8217-s price- or it could split the difference in the interest of fairness. If any choice is made other than using the stated price of the order in the book, investors have an incentive to carefully submit bids a little at a time (aka &#8220-structure&#8221- their bids) so they won&#8217-t pay more than they have to if new orders should arrive. Robin argued elsewhere (I can&#8217-t find the reference at the moment) that you should just transact at the book order price so that people submitting market price orders don&#8217-t waste their resources and yours on this optimization.

That choice also simplifies the calculation for accepting new offers. As Robin says, &#8220-each book order […] imposes a constraint on the market maker price&#8221-. The AMM should fulfill orders up to that limit, then let trade continue with the book order. This requires a loop, in which you buy from the AMM until you reach the limit imposed by the best order(s), then trade up to the book order&#8217-s available quantity, then go back to the AMM until you reach the next book order. You can see the approach in Zocalo&#8217-s method Market.buyFromBothBookAndMaker(&#8230-). (The method starts at line 237.)

At every step,

• find the remaining quantity q of the new order
• find the price p available from the best existing order
• if the AMM&#8217-s price is no better than the book order, trade up to q with the book
• otherwise trade with the AMM to the lesser of p or q

The loop stops either when the new order is fulfilled or the price limit specified by the new order is reached.

That&#8217-s the simple version for a one-dimensional AMM. The multi-dimensional version arises if you implement the AMM as described in &#8220-Combinatorial Information Market Design&#8221-. There are two open source implementations of this approach available for reading by hard-core hackers. Robin built an implementation in Lisp, and I wrote a version in E. Neither is more than a demonstration of how the market engine works, since no serious user interface was written for either one.

Rather than attempt to explain how the approach translates to the multi-dimensional case now, I&#8217-d prefer to wait until after I write an explanation of the n-dimensional combination market, and that depends on a gentle introduction to conditional and combinatorial betting which I haven&#8217-t written yet. Having someone ask about Robin&#8217-s note raises my priority for writing these prerequisites.

Other Articles in this series

PM intro: basic formats (2005-12-30)
• PMs with Open-ended Prices (2006-01-05)
• Looking at Both Sides (2006-04-17)
• Book and Market Maker (2006-04-28)
• Liquidity in N-Way claims (2006-07-19)
• Continuous Outcomes using Bands and Ladders (2006-09-20)

We should try to be more careful about our distinctions and definitions. I&#8217-m less concerned about whether we use the phrase &#8220-prediction market,&#8221- &#8220-information market,&#8221- &#8220-decision market,&#8221- or whatever- I&#8217-m more concerned about what exactly these phrases are intended to mean. Here are some possible concepts to distinguish (I&#8217-ll avoid naming them).

1. Markets that make predictions – Pretty much every speculative market ever created does this.
2. Markets that make easily interpreted predictions – Most betting markets would be included here, while most financial markets would not. For financial markets one must understand and estimate risk premia, discount factors, and so on be able to say much what exactly the current price estimates.
3. Markets whose prices embody information – Also pretty much every speculative market ever made.
4. Markets whose price predictions are used as info by non-traders – Many financial markets meet this criteria.
5. Markets whose interpretable predictions are used as info by non-traders – Many betting markets meet this criteria.
6. Markets whose primary function is not hedging – A &#8220-primary&#8221- function would most explain the existence of the market and its volume of activity. Most financial markets might fit here, as most volume is from speculation.
7. Markets where the main trader motivation is not material gain – Most play money markets would fit this criteria.
8. Markets where the social value of allowing the market to exist outweighs the social cost – this would be the sort of market we want to legally allow to exist.
9. Markets where the social value of the info gained by non-traders, but not the social value of its use for hedging, outweighs any other social costs of allowing the market to exist – These markets could justify a legal regime empowered to allow markets to exist for info reasons, not just hedging reasons.
10. Markets whose primary function is to inform non-traders – Non-trader interest in the info would be the main explanation for the existence of the market and the volume of trade- such non-traders would somehow subsidize the existence of the market and trading activity in order to gain the info they desire.

These last two concepts are of the most interest to me – I would like to have names that clearly identify them and distinguish them from the other concepts.