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