# Market Makers for Multi-Outcome Markets

Previous articles in this series have discussed market makers and how they differ from book order markets, how to improve Liquidity in multi-Outcome claims, and how to integrate a Market Maker into Book order systems. But none of those talked in any detail about how a multi-outcome market maker coordinates prices and probabilities. Those details turn out to be important for an upcoming article on Combinatorial Markets, so I&#8217-ll go through them carefully here.

Researchers use scoring rules as a laboratory tool to convince people to reveal their true expectations about some set of outcomes. Participants are asked to give estimates of the likelihood for a set of outcomes, their scores are some function of the value they gave for the actual outcome. Scoring Rules are called &#8220-Proper&#8221- if they are designed so the participant&#8217-s best strategy is to honestly reveal the probabilities that seem most likely. The Logarithmic Scoring Rule (one of the Proper rules) provides a reward that equals the logarithm of whichever estimate turns out to correspond to the actual value. Since the total of all the estimates must be 1, the participant can only increase some probabilities by decreasing others.

Robin Hanson described how an Automated Market Maker (AMM) that adjusts its prices based on a scoring rule can support unlimited liquidity in a prediction market. If each successive participant in the market pays the difference between the payoff for her probability estimate and that due to the previous participant, the AMM effectively only pays the final participant. If the AMM&#8217-s scoring rule is logarithmic, participants who only update some probabilities don&#8217-t effect the relative probabilities of others they haven&#8217-t modified. (This last effect is only valuable for Combinatorial Markets, which I&#8217-ll talk about in a later post.)

The change in the user&#8217-s payoff is `log(newP) - log(oldP)` (or equivalently `log(newP/oldP)`) for each state. For a binary question, the possible gain will be `log(newP/oldP)`, and the cost will be `log((1-oldP) / (1-newP))`. For the rest of this article, I&#8217-ll use gain and cost rather than the `log(...)` expressions, since there are only these two, and I&#8217-ll be using them a lot.In multi-outcome markets, the most common approach is to let the user specify a single outcome to be increased or decreased, and to adjust all the other outcomes equally, but this isn&#8217-t the only possibility. This design choice has the useful property that the probabilities of other outcomes will be unchanged relative to one another. Since the other outcomes are treated uniformly, they can be lumped together, which results in the same arithmetic as a binary market. Since those other cases sum to `1-P`, the price is cost. It is also reasonable to allow the user to specify either a complete set of probabilities, or particular cases to increase and decrease and how much to change them. Whatever the case, the LMSR adjusts the reward for each outcome to be `log(newPi/oldPi)`. I&#8217-ll describe more possibilities in this vein when I cover the Combinatorial Market.

I hope you found all this interesting in an intellectual sort of way, but you may have noticed that this description isn&#8217-t applicable to markets in which the traders hold cash and securities. The whole thing is couched in terms of participants who will receive a variable payoff, but they don&#8217-t pay for the assets, they merely rearrange their predictions in order to improve their reward.

In order to turn this into an AMM that accepts cash for conditional securities, we have to pay careful attention to the effects of the MSR on people&#8217-s wealth. The effects are easiest to describe in the binary case, and every other case is directly analogous, so I&#8217-ll start there. In a binary market, the participant raises one probability estimate (call it A) from oldP to newP and lowers the probability of the opposite outcome (not A) from `1-oldP` to `1-newP`. If the trader had no prior investment in this market, the reward will increase by gain.

In order to reproduce that effect in cash and securities, the AMM charges cost in exchange for gain + loss in conditional securities. Why does the trader get securities equal to the cost plus the potential gain? The effect of this is that if A occurs, the participant has paid cost, and received gain + cost, for a net increase of gain over the original position. If A is judged false, the participant has paid cost with no return, which is the effect we hoped to match.

When an AMM supports a multi-outcome market using the approach I described above, one outcome is singled out to increase (or decrease), while all other outcomes move a uniform distance in the opposite direction. If the single outcome is increasing, the exchange is trivial to describe: we charge the trader cost for gain + cost in securities. The effect looks just like the binary case. The user has spent some money and owns a security that will pay off in a situation the trader thought was more likely than its price indicated.

If the trader singles out one outcome to sell (and thus reduce its probability), the difference among the alternatives I described in the first article in this series on Basic Prediction Markets Formats becomes evident. The trader is betting against something, and the market can represent this using short selling, complementary assets, or baskets of goods. The market might allow short selling (like InTrade), a complementary asset (like NewsFutures and Foresight Exchange), or a basket of securities representing all the other outcomes (like IEM). Since there are distinctly different points of view on this question, different markets will make different choices.

In order to support the short sales model, the trader needs to receive the payment first along with a conditional liability. In our model, the trader would receive gain in cash immediately, and securities that required repayment of gain + cost if the outcome (which the trader bet against) occurs. The platform would presumably require the trader to hold reserves to ensure the repayment.

With baskets of goods, the trader would get the appropriate number of shares of each of the other outcomes. The charge would be cost, and that would purchase gain + cost of conditional assets in all other outcomes.

The complementary assets model would charge cost in currency, and provide gain + cost of an asset that paid off if the identified outcome didn&#8217-t occur. The complicated part of this representation is that traders can hold both positive and negative assets. In a 4 outcome market, a trader holding 3 units of A and 2 units of B who sold 4 units of C could be shown equivalent portfolios of either A: 3, B: 2, C: -4 or A: 7, B: 6, D: 4. I think either choice is defensible. The first resembles the transactions the user has made, and so is probably more recognizable- the second provides a more consistent view of possible outcomes. (And looks the same as baskets.) If both positive and negative numbers are shown, the trader has to realize that the negative holdings pay off in all other cases. On the other hand, displaying a portfolio in a 7-outcome market as A: 3, B: 3, C: 3, E: 5, F: 3, G: 3 doesn&#8217-t seem as clear as D: -3, E: 2.

I doubt this detail will be of much interest to most users of Prediction Markets. Luckily for them, the trade-off the logarithmic rule makes between cost and reward just happens to produce prices that match probabilities. But if you are implementing Hanson&#8217-s LMSR, you should understand the alternatives well enough to verify that your market maker correctly implements the design.
Zocalo Prediction Markets support binary and multi-outcome markets with a Market Maker based on the Logarithmic Market Scoring Rule. The design takes advantage of the parallels between the different markets by only implementing the logarithmic rule in one place.

### 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)
• Conditional and Combinatorial Betting (2007-03-06)

## 21 thoughts on “Market Makers for Multi-Outcome Markets”

1. Michael Giberson said:

Thanks for this explanation. I’ve been trying to work through some related ideas, and your post helps me understand a bit more. (And BTW, thanks for your other ‘explainers’, too, especially Liquidity in N-way Claims.)

Re: “the most common approach is to let the user specify a single outcome to be increased or decreased, and to adjust all the other outcomes equally, but this isn’t the only possibility.

What you describe as the common approach probably suits generic multi-outcome markets (i.e. which of four candidates will win an election), but in some cases there is an underlying structure to the event which may make some other adjustment to other outcomes more appropriate, or at least more elegant.

For example, predicting the winner of a single elimination tournament. Assume teams A and B play, then C and D play, then the winners in the first round play in the final. An increase in the likelihood of A may more appropriately fall heavier on B than on C and D. A market operator could presumably employ a bayes net, as Hanson suggests in Combinatorial Information Market Design, to specify relationships among outcomes for non-generic cases.

I think that, in principle, using a generic allocation when a non-generic allocation is called for means the AMM is mispricing the other outcomes. However, the mispricing will likely be small relative to the price change of the traded outcome. So long as a trader is not interrupted in his use of the market scoring rule, he can trade on the mispricings to attain the desired set of prices.*

In an actively traded non-generic market, one in which the trader is likely to be interrupted in his use of the MSR, this might be a problem demanding of a more elegant solution.

*Because a trader using a MSR in effect just “pays off the previous user” (to quote Hanson), when a trader repeatedly uses the MSR to adjust for the mispricings, all the “pay[ing] off the previous user” nets out except for the difference between the state of the MSR before the trader started and the state of the MSR when the trader ended. (If the trader is interrupted, then the interrupting trader can capture the small gains implied by the mispricing).

2. David Pennock said:

Thanks Chris!

To me, the most intuitive/natural interface for a multi-outcome market is:

Whenever you place any bet, you pay up front the maximum amount you could possibly lose.

Then, when the outcome is revealed, you may get none, some, all, or more than all of your money back.

This interface by definition does not allow short selling in the traditional (stock market) sense.

Every new bet is a buy order, and you can only sell things that you previously bought.

When you bet against something, you actually “buy the opposite” & still pay something up front (your max possible loss), in return for the possibility of getting it all back plus more. The mechanics of betting against something are exactly the same as betting for something. This makes sense, since the choice of what outcome is defined as “against” and what is defined as “for” is usually arbitrary anyway.

There doesn’t seem much point in allowing short selling *and* making sure the trader has enough money to cover her potential losses. Why not just make them put up the potential loss up front when the trade first happens?

The reason stock markets have to employ short selling is that the short seller’s potential loss is unbounded.

I believe that the average person finds short selling confusing. Since short selling is entirely unnecessary in an event market, why not get rid of it?

3. David Pennock said:

Michael, good point.

You are absolutely right if the outcomes of a two-level tournament are encoded in the straightforward way as “A wins”, “B wins”, “C wins”, and “D wins”.

In this case, if the current probabilities are 1/4, 1/4, 1/4, 1/4, and then tons of people start betting against B, the new probabilities would become 1/3, 0, 1/3, 1/3. What should happen is for the probabilities to go to something more like 1/2, 0, 1/4, 1/4.

However if the outcomes are defined better to reflect the tournament structure, we can get the “right” behavior, albeit making things more complicated.

Let’s define the outcomes as “A beats C in final”, “A beats D in final”, “B beats C in final”, etc., and assume the probabilities start off at 1/8 each.

Now if tons of people start betting against B to even make it to the final, then every outcome with B appearing in the final will go toward zero. Now the probabilities will adjust how we want. For example, the probability of A winning (prob of “A beats C” plus prob of “A beats D”) will be 1/2, probability of C winning will be 1/4, etc.

4. Michael Giberson said:

Nice approach for small tournaments. Defining the outcomes in that fashion allows the independence between related outcomes (like “A beats C in final” and “A beats D in final”) that allows expression of complex interactions. If the outcome is defined as “X wins final” then prior to the outcome of the C vs. D game, the “A wins final” contract would represent a weighted average of the probabilities that “A beats C” and “A beats D.” But maybe A is certain to beat C, and just as certain to lose to D — a price of 0.50 for “A wins final” hides underlying relevant information.

The only downside is that since you need to define all possible final game outcomes, the approach doesn’t scale well.

(Assume a tournament final pits the winner of the West bracket against the winner of the East bracket. If West has w teams and East has e teams, your platform will require 2*w*e outcomes. In the four team example only 8 outcomes are required, but for the 65 team NCAA men’s basketball tournament 2,112 outcomes are theoretically possible for the final game. )

5. Michael Giberson said:

Or more carefully state, if A is certain to beat C, and just as certain to lose to D, the price of “A wins final” becomes the same as the probability that C beats D.

A more elaborate approach, and I’m not suggesting that it is a good idea, would be to treat “A wins final” as a bundle of contracts (composed of “A beats C in final” and “A beats D in final”), and allow the trader to unbundle the package and resell the components. Such an approach allows traders the simplicity of just picking a winner, if that is what they want to do, but also allows more sophisticated trading.

6. Chris Hibbert said:

You east coasters have a real advantage in getting started with the comments early! Thanks for the comments.

to Mike Giberson on predicting the winner of a single elimination tournament:

That’s a good example; I hadn’t been thinking of that. I planned to cover the general case of allowing the user to specify how to adjust the other probabilities when I got to combo markets (which is mostly written. This explanation was actually pulled out of that one when it grew too big.) The more common case I was thinking of was when there is a “default” outcome, and it makes sense to for most bets to to be compared to that outcome rather than all others. (I don’t have a specific case in mind yet.)

To Dave Pennock on natural interfaces:

I agree that always buying is more natural. The LMSR naturally supports this, but not everyone seems to run it that way.

To Dave on “A beats C in final”.

That’s getting closer to the combo market. I think once you’re heading in that direction, it makes sense to go all the way. That write-up should be out soon. Unless the number of entrants is really small (your example with 4 teams in a tournament format turning into 8 questions is about the limit with a single multi-outcome market.) The number of questions is just as high with the combo-market, but the institution uses its liquidity better, and it preserves the conditional independence relations when betting on independent outcomes.

7. Chris. F. Masse said:

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8. Alex Forshaw said:

I didn’t read through the entire post (flame at your convenience), but automated market makers do not exist in real markets for a very good reason: liquidity is not free.

A lot of academic economics seems to take for granted perfect and free liquidity. It allows for algebraic pirouettes unintelligible to outside observers. It makes for very elegant solutions to very complex problems. But it all tends to break down when liquidity is very imperfect … and if liquidity is not generated by additional deposits, it must be generated by the exchange taking on liability via the AMM. It’s suicidal.

Just my cantankerous two cents.

9. Chris Hibbert said:

“I didn’t read through the entire post”

“automated market makers do not exist in real markets for a very good reason: liquidity is not free.”

Completely correct. Prediction Markets were invented to extract information from traders. Sometimes the trading is valuable enough to attract the traders on its own, sometimes not. If the information is valuable, someone may be willing to subsidize the market. We’ve seen with internal corporate markets that this is often the case. (Though there isn’t a long history or a huge number of examples yet.) With play money markets, subsidies are cheap and often useful.

So the value isn’t completely academic, but it may not apply to the markets you are interested in. Fair comment.

10. Chris. F. Masse said:

ALEX: “automated market makers do not exist in real markets”

Did HegeStreet use an AMM at the very beginning? I’m asking Chris Hibbert. I suspect “yes” is the asnwer.

11. Jason Ruspini said:

Where did phrases like “unlimited liquidity” become associated with specific market designs apart from the real/play money distinction.

12. Michael Giberson said:

“Liquidity” is a somewhat slippery term — in some cases it just refers to the ability to turn an asset into money, but in other cases there is an added implication of the ability of turning an asset into money without causing a significant adverse price movement.

The LMSR provides unlimited liquidity in the first sense, in that an AMM implementing the LMSR stands ready to convert assets to money (or money to assets) at any time.

The real money/play money distinction only affects whether the sponsor of the LMSR AMM provides the subsidy in real or play money. Nothing inherent in the LMSR depends upon the real money/play money distinction.

13. Michael Giberson said:

[Duplicated comment deleted. Sorry about that.]

14. Jason Ruspini said:

At any time?? Again, only if someone is providing the money. Whether the sponsor provides the subsidy in real or play money determines how unlimited the liquidity is, not the market design. We can also say that a CDA can support unlimited liquidity. (Expanding the set of tradeable propositions is another issue.)

15. Chris Hibbert said:

“Did HegeStreet use an AMM at the very beginning? I’m asking Chris Hibbert. I suspect “yes” is the asnwer.”

I have screen shots from November 2005 that show some markets that definitely didn’t have market makers and other markets that look like they probably did have an AMM. Some multi-outcome markets had commodities with no available offers; I can’t think of a reason to design an AMM that would do this. Other multi-outcome markets had a consistent bid-ask spread across all the offers. (.59 in one market, .99 in another) I don’t recognize the AMM algorithm, but this looks likely to have been automatic. I don’t remember trading to find out whether book orders were allowed in these markets, or whether it was a pure AMM.

16. Chris Hibbert said:

Jason, saying that a CDA “can support unlimited liquidity” is different from saying that the LMSR (with a subsidy, to be sure) “provides unlimited liquidity”. In the latter case, it’s part of the institution and omnipresent. In the former, all you are saying is that if the participants were interested, they could ensure that it was a thick market. Seems like an important distinction to me.

As to the distinction that Michael points out, I’d push it a little farther. In a thick market and normal circumstances, you can be reasonably sure that you’ll be able to buy without moving the price very far, but come exceptional times, and the liquidity may dry up. With the LMSR (and most other AMMs) the AMM doesn’t change it’s policy in volatile times. You can always buy without moving the price very much. In fact, the amount the price moves is constant over time. In volatile times and with a thin market, the price may move a long distance because the next available book order is not near by. No guarantees for the trader.

17. Jason Ruspini said:

I am contesting that your first point is an important distinction. Someone has to decide to risk money either way. A contract could obligate a CDA market-maker.

To the extent that traders can’t move the price very much, the information extraction may be compromised. There may be very good reasons why bids and offers evaporate at certain times in CDAs.. but I am not debating CDA vs LMSR. I was just observing that phrases like “unlimited liquidity” seem to be bandied about in unexamined ways in PM circles with surprising persistence.

18. Chris Hibbert said:

“Someone has to decide to risk money either way.”

Yes, but there’s a difference between deciding to endow a market maker (permanent decision) and deciding moment-by-moment whether to keep orders on the book.

“A contract could obligate a CDA market-maker.”

Ah. Yes. In real-money markets, there are often parties with obligations to make markets through thick and thin. I’m not sure what the details are when the market goes south, but I agree that in all normal times, there will be orders available. The price can move precipitously, which you appear to think is an advantage (I’m not calling it a disadvantage, I just don’t see requiring trading to move the price of an AMM a long way as a problem.) Of course, this isn’t a distinction between real- and play-money, just a facet of some markets that hasn’t been seen as important in others yet.

So the interesting thing about adding a market maker to a CDA in my mind is whether it’s built into the market or provided externally. The first effect is that in volatile times, you want the market maker to have priority access so the other traders don’t pre-empt him/her/it when times are toughest. And from the MM’s point of view, it may be crucial to have access to every trade in order to enable it to follow rules that constrain its losses. So I prefer a built-in MM. Zocalo supports (binary) markets with both a CDA and an AMM. There’s no reason other markets couldn’t do the same. It would be straightforward to implement other algorithms than the LMSR. But a manual MM wouldn’t be able to provide the same assurances to traders and wouldn’t be able to constrain its own expenses.

19. Chris Hibbert said:

“I was just observing that phrases like ‘unlimited liquidity’ seem to be bandied about in unexamined ways in PM circles with surprising persistence.”

I mean something pretty specific when I say it. What’s missing from my understanding or from the discussion? Just that it’s a disadvantage that the price doesn’t move quickly enough some of the time? I’ll grant that it doesn’t move quickly; I remain to be convinced that it’s a disadvantage. What else are we missing?

20. Jason Ruspini said:

Yes but the difference is more about the continuity of the liquidity than its depth and the total risk the sponsor takes on (which could also be bound with a binary CDA).

The problem isn’t what you’re missing Chris, it’s what readers might be missing when they see certain phrases — which by now have been clarified by these comments.

21. BetFair's brand-new matching-bet logic is endorsed by the Chairman of the Midas Oracle Advisory Board. | Midas Oracle .ORG said:

[…] in “Increasing Liquidity in Multi-Outcome Claims.” (For more, see Hibbert’s “Market Makers for Multi Outcome Markets” and Dave Pennock’s “Right Way to Implement a Multi-Outcome Market.”) If […]