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What is an Automated Market Maker?

By Dr. Abe Othman

One of the core tenets of Augur’s platform is that the crowd determines the correct probabilities of wagered events taking place. In traditional markets, price discovery happens through the interaction of buyers and sellers. In contrast, the Augur platform uses an automated market maker to interact with traders and adjust prices in response to their behavior. In this blog post, I’ll discuss the implications of Augur’s automated market maker microstructure.

Why automated?

Augur uses an automated market maker instead of a more traditional microstructure for several reasons.

  • Solves the buyer/seller matching problem. The automated market maker guarantees that the market is always available to trade, even at 2 in the morning in your local time zone, even at the market’s initiation, and even if no other traders are currently active. By giving traders confidence that they can obtain liquidity for their wagers whenever they choose, automated market makers create additional incentives to participate.
  • Enables large markets with many events. Consider a prediction market run over which US presidential candidate will win the election. Conceivably, there could be dozens of different actively traded candidates, bound by the constraint that their cumulative odds cannot sum to more than 100%. These large event spaces are very difficult for human participants to seed, requiring careful, accurate probability estimates for very small values. Consequently, existing human-mediated prediction markets with large numbers of outcomes tend to have very wide bid/ask spreads, particularly for more obscure candidates. Automated market makers mitigate this issue, having consistent behavior in markets with large or small numbers of outcomes.
  • Concisely expresses the market’s current state. One of the concerns in a distributed system like Augur is that each piece of shared state increases the chance of inconsistencies between the views of various participants and increases the amount of time for different views of the market to come into alignment. Automated market makers can be embedded and distributed, with a minimal amount of state. This can be contrasted to a traditional double auction where the full market state includes the entire order book and where operations can include participants withdrawing or modifying their existing orders.
  • Allows for precise control of worst-case loss by market sponsors. The initiator of a market takes on the risk of losing a set amount (which they can always increase) in return for a variable payout brought about from trading fees. As the amount risked by the sponsor increases, the market becomes deeper and individual actions cause less slippage on prices. One interesting result of this bound is that markets can enter states of unconditional profit for their sponsors, wherein an active market has collected enough in trading fees to make the market profitable for its sponsor regardless of future wagers and regardless of which outcome actually takes place.
  • Automated Market Makers prevent common trader errors and information redundancy. Research studies have shown that traders in the Iowa Electronic Markets often violate the “law of one price”. Consider wagering on which of two candidates. Logically, a bet against candidate A winning the election is equivalent to a bet for candidate B winning. However, in traditional double auctions the prices of these contracts can diverge, and moreover, the IEM administrators observed many traders taking positions in the irrational, worse contract. In contrast, automated market makers enforce the equality of logically equivalent wagers.

How does it work?

The most famous automated market maker is the Logarithmic Market Scoring Rule, or LMSR. The LMSR was designed by Robin Hanson from the notion of a “scoring rule”, which is a statistical tool for decision elicitation that rewards experts for revealing their true beliefs. (The Augur platform uses a similar market maker with a more complex form but better practical properties, the Liquidity-Sensitive Logarithmic Market Scoring Rule, or LS-LMSR. We will discuss the differences between the LMSR and the LS-LMSR in a future blog post.)

Consider wagering on an event where exactly one of outcome X or outcome Y must occur. The automated market maker functions by keeping track of the quantity they must pay out if either outcome occurs. For simplicity, we’ll call x the amount the market maker owes to traders if event X occurs, and y the amount the market maker owes to traders if event Y occurs. In the LMSR, the market’s state is encapsulated by an (x,y)-tuple, which is called a payout vector. The market maker’s behavior is defined by a cost function, C, which maps a payout vector into an amount of money the market maker must collect to accept that payout vector. The partial derivatives of the cost function---the marginal cost on x and y, respectively, are called marginal prices.

One of the nice things about the LMSR is that it has a concise, closed form expression that will enable us to easily work out some example trades. In the LMSR, the cost function is

C(x,y) = b*ln(e^(x/b) + e^(y/b))

and the marginal prices are given by:

px(x,y) = e^(x/b) / (e^(x/b) + e^(y/b))

py(x,y) = e^(y/b) / (e^(x/b) + e^(y/b))

In these equations, b is a number chosen by the market maker called the liquidity parameter. It controls how much marginal prices change in response to trades, where prices change quickly for small b and slowly for large b.

Observe how the marginal prices on the two events sum to 1, and are positive. This means that we can regard the marginal prices as state probabilities; furthermore, observe that myopic, risk-neutral traders are incentivized to trade with the market maker until the marginal prices match their beliefs.

To concretely instantiate the theory, consider an example trader where b=10, x=5, and y=3, and the trader wants a $1 payout if event X takes place. That is, the trader wants to change the automated market maker’s payout vector from (5,3) to (5+1,3) = (6,3).

The marginal price on X before the interaction is:

px(5,3) = e^0.5/(e^0.5 + e^0.3) = .550 (or 55 cents on the dollar, or event X has a 55% chance of occurring)

The cost of the transaction is the difference in the evaluation of the cost function from before and after the wager, which in this case is:

C(6,3)-C(5,3) = 10*ln(e^0.6+e^0.3) - 10*ln(e^0.5+e^0.3) = 56.2 cents for a dollar payout if X occurs.

The marginal price of X after the interaction is:

px(6,3) = e^0.6/(e^0.6 + e^0.3) = .574 (or 57.4 cents on the dollar, or event X has a 57.4% chance of occurring)

Observe how the market maker has adjusted the marginal prices in response to the trade: by increasing the probability that X occurs.

The LMSR has many interesting properties. One curious one is that the most money the market maker can lose, regardless of outcome or trading behavior, is b*log(N), where N is the size of the event space (the number of possible outcomes).

Finally, the LMSR is closely related to many phenomena from across applied mathematics, each of which provides a different light on its behavior:

  • Log-sum-exp from convex analysis. In this light, the LMSR is a “soft max” function around the market maker’s worst-case loss.
  • The canonical ensemble from statistical thermodynamics. In this light, the probability of each event is equal to the “energy” that traders inject into it.
  • The entropic risk measure from finance. The LMSR is equivalent to this risk measure, which is used to assess portfolios.
  • Exponential utility from economics. The LMSR market maker behaves like a myopic agent with exponential utility.
Do automated market makers encourage honesty?

Without considering market dynamics like signaling, information leakage, etc. myopic traders are incentivized to move the market to match their beliefs. This is because, as long as the market’s marginal prices do not align with their beliefs, myopic traders have a profitable trading opportunity by definition.

Of course, market dynamics can be incredibly sophisticated! Just having an automated market maker does not eliminate or exacerbate any of the incentives around trading psychology, anticipation, beauty contests, etc. Automated market makers function to provide liquidity, not to change the nature of the market itself.


Interested in finding out more?

Pennock blog post

http://blog.oddhead.com/2006/10/30/implementing-hansons-market-maker/

Pennock + Chen review article in AI Magazine:

http://dash.harvard.edu/handle/1/5027266

Pennock + Chen utility paper (technical):

http://arxiv.org/pdf/1206.5252.pdf

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