2.3 Sequential Market Making
2.3.2 Market Scoring Rules (MSR)
The sequential market mechanism called Market Scoring Rules (MSR) introduced by Hanson [2003] is based on an automated market maker who uses scoring rules and solves the above problems inherent in sequential markets, thus encourages trading. Here I give a brief overview of MSR markets and the use of scoring rules to elicit trader beliefs.
Scoring rules are a measure of the quality of probabilistic predictions. Suppose a forecaster assigns probabilities to multiple outcomes N and gives a report vector
r ∈ RN. A scoring rule defines the payouts
n(r), if the nth outcome occurred. Use
of a proper scoring rule (which maximizes the expected score for well calibrated probability assessments) as the payoff function will incentivise the forecaster to re- port his true private beliefs (p). For example if the scoring rule sn(r) determines
the cash amount to be paid for report r for outcome n ∈ [N], the expected payoff will be ∑nN=1pn.sn(r). If the scoring rule is strictly proper, the agent can maximise
§2.3 Sequential Market Making 24
the expected payoff only by reporting his true belief (p) in the report (assuming risk neutral traders i.e., traders with linear utility). Some examples of proper scoring rules for reported vectorr ∈ RN and outcome probability vector p ∈ RN are given
in Table 2.1. For unnormalizedr(i.e. ∑nN=1rn 6=1), the scoring rules can be extended
for a normalized r by using ri/∑nN=1rn for ri [Hanson, 2007]. See Jose et al. [2008]
and Chen and Pennock [2007] for more examples including weighted power and pseudo-spherical scores (also given in Appendix A ).
Table 2.1: Proper Scoring Rule Examples [Hanson, 2007]
Scoring Rule Payoff for outcomen∈[N] Notes
Logarithmic sn(r) =an+b. log(rn) Special case of power law rule
when α= 1 and only depen-
dent on the probability report for outcome n
Quadratic sn(r) =an+2brn−b.hr,ri Special case of power law rule
whenα=2
Brier sn(r) =1−(ρn−rn)2 The original Brier score
[Brier, 1950] which can be derived from the quadratic scoring rule
Power Law sn(r) =an+b.α.R0rn pnα−2dρn−b∑nN=1rαn Proper whenα≥1
Spherical sn(r) =an+b.rn/(hr,ri)1/2
Scoring rules do not suffer from the irrational participation and thin market prob- lems, but they have the problem of being unable to produce a single consensus when different people give differing estimates, known as thethick market problem. [Hanson, 2003]
Market Scoring Rules (MSR) are in essence sequentially shared scoring rules that ad- dress the thick market problem. It is used by an automated market maker where a trader can change the current published report (prices) of the market, and be paid according to the new report, as long as he agrees to pay the last trader ac- cording to his report. For any proper scoring rule sn(r), if the trader changed the
published distribution from p to r, he should accept a (net) payment of the form
∆sn(r,p) = sn(r)−sn(p) for outcome n. When there is only one trader, the market
scoring rule reduces to a simple proper scoring rule which will elicit a true probabil- ity estimate from the trader [Hanson, 2003].
§2.3 Sequential Market Making 25
The most popular example of MSR markets is the LMSR (Logarithmic Market Scor- ing Rule) which uses the Logarithmic scoring rule (sn(r) = an+b. log(rn)). LMSR is
used by a number of companies including Inkling Markets, Consensus Point, Yahoo! and Microsoft.
Thus the market scoring rules provide incentives for truthful revelation and the se- quential nature of the use of the scoring rule also solves the irrational participation problem seen in prediction markets because traders can always adjust the published report and be paid according to a proper scoring rule. The introduction of an au- tomated maker also solves the thin market problem, because traders can trade with the market maker at any time, so the need to match trades is eliminated. This over- comes the problems of prediction markets and allows to incorporate the information of even a solitary trader [Chen and Pennock, 2007]. It also solves the thick market (information aggregation) problem of scoring rules since the traders share the scor- ing rule. Thus it has the combined advantages of scoring rules and the information aggregation characteristic of prediction markets [Hanson, 2003].
In traditional markets, market makers are human decision makers seeking to earn a profit (while also providing more liquidity), but in a prediction market the role of the market maker is to elicit information from the traders (by providing incentives), pro- vide liquidity (encourage and facilitate trading) and price discovery (aggregation), so he is expected to lose some money in the process [Chen and Pennock, 2007]. If the market maker published his initial beliefs r0, and there were T subsequent updates before the outcome was revealed to ben, the total loss for the market maker would be∑Tt=1(sn(rt)−sn(rt−1) =sn(rT)−sn(r0). The market maker incurs the max-
imum loss when the final probability estimate (rT) assigns probability 1 to the true outcome, thus the market maker has bounded worst case loss. So in the case of a Logarithmic Market Scoring Rule (LMSR) using sn(r) = an+b. log(rn), the maxi-
mum expected loss for the market maker would be the entropy, −b∑Nn=1r0nlog(r0n)
of the initial distributionr0, withbrepresenting the liquidity of the market [Hanson, 2007]. If the market maker started by publishing a uniform distribution (asr0), then his worst case loss is bounded by b. log(N) in the case of a logarithmic scoring rule and by (N−N1)b for a quadratic scoring rule (for N discrete outcomes) [Chen and Pen- nock, 2007].
Even though the MSR assumes that traders update the market prices to reflect their own beliefs, Hanson [2003] has also outlined how this setup can be implemented in
§2.3 Sequential Market Making 26
a prediction market framework with an automated market maker who determines fair prices for the contracts for an infinitesimal amount of trade and who will accept any finite fair bet that is an integral of such infinitesimal trades. Here the traders are asked to give their opinion (in the form of a report r) and the change of contracts required to change the current report is decided accordingly.