Seminar in Information Markets, TAU 1

Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation

[Robin Hanson, 2002]

Roi Meron

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Outline• Scoring Rules• Market Scoring Rules• Logarithmic Market Scoring Rule(LMSR)• Distribution == Cost function• Combinatorial Markets

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Event• finite set of outcomes (mutually exclusive and

exhaustive states of the world)– Example: all possible prime ministers in elections ’13

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Scoring Rule• - Agent’s belief about the probability that state

will occur.• ) is the payment made to agent who reports

distribution if outcome is .

• A proper scoring rule is when

– Strictly proper: when is unique

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Exercise• Is the following (binary)scoring rule proper?

where is the probability that will happen

(This is a variation of brier scoring rule)

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Logarithmic Scoring Rule

– to be big enough for agents to participate• The only strictly proper scoring rule in which the

score for outcome depends only on and not on the probabilities given to for

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Logarithmic Scoring Rule

– to be big enough for agents to participate• The only strictly proper scoring rule in which the

score for outcome depends only on and not on the probabilities given to for – Example: Quadratic Scoring Rule

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Properness - simple case proof• Assume we have 2 possible outcomes.

• Derivative w.r.t. gives:

• Second derivative is negative.• Logarithmic scoring rule is proper.

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Objective• We want multiple agents(traders) to share their

beliefs– Without paying each one– We want a single(unified) prediction.

• One option is a standard market(Double auction information markets)– What happens in thin markets?

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Market Scoring Rule

• Market maker starts with an initial distribution . At each step, an agent reports his distribution.

• The report should be honest if we use LMSR. Why?

• The agent pays to the previous agent according to the scoring rule

• The market maker finally pays

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Market Scoring Rule(2)

• Market maker subsidizes the reward for accurate predictions:– Gives incentive to participate and share your

knowledge– Increases liquidity– Easy to expand to multiple outcomes

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Market Scoring Rule(3)• If only one agent participates, it is equivalent to

simple scoring rule.• If many agents participate, it gives the same

effect of a standard information market, at the cost of the payment to the last agent.

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Logarithmic Market Scoring Rule (LMSR)

• Reminder: • is a parameter that controls:– liquidity– loss of the market maker– Adaptivity of the market maker.* Large values allow a trader to buy many shares at the current price without affecting the price drastically.

• Market maker’s worst case expected loss is the entropy of the initial distribution he gives,

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• Typically, initial distribution is uniform, i.e. where is the number of possible outcomes.– Market maker’s loss is bounded by .

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how do we implement a market?

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Distribution is equal to Buying and Selling shares

• We can think of the market maker scoring rule as an automated market maker that:– Holds a list of how many units of the form “pays 1$ if the state is ” were sold (outstanding shares).– Has an instantaneous price ( for any outcome).– Will accept any fair bet.

• Its main task is to extract information implicit in the trades others make with it, in order to infer a new rational price.– The rational: people buying suggest that the price is too low and selling suggest that the price is too high.

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LMSR price function• Let be the outcome which finally took place. – The market maker pays exactly . A Dollar to any share

holder.– On the other hand, it should be equal to the payment

using LSR.• We want a price function such that:

The price function is the current distribution “given” by the last agent.

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LMSR price function(2)• The price function is the inverse of the scoring

rule function

• A large trade can be described by a series of “tiny” trades between to . The price of this trade event is given by integrating over in the range of .

• Probabilities represent prices for (very) small trades.

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LMSR cost function

• For simplification, assume we have only 2 possible outcomes, then:

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“How much?”• Buying shares means Selling . • Say someone wants to buy 20 shares of outcome .

He pays:

• In general, if a trader changes the outstanding volume from to , the payment is:

• If , i.e. selling, then the cost is negative, as expected.• People might find this version of LMSR more natural.

Buying and Selling instead of probabilities estimation.

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Equivalence proofTrader’s profit if happens= a logarithmic scoring rule payment

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Example

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𝜋 ⅇ

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• Will Wile E. Coyote fall off a cliff next year?

– Uniform priors:

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Combinatorial Markets• Say we bet on the chances of rain of the following

week.– - will it rain on Sunday – - will it rain on Monday– - will it rain on Tuesday

• We can think of other events:– rain on Monday given it rains on Sunday…

• Ideally, trading on the probability of given should not result in a change in the probability of or a change in prob. “ given C”.

• But in fact…..07-Nov-12

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LMSR local inference rule• Logarithmic rule bets on “A given B” preserve

and, for any event preserve and .• The other direction also holds.• In other words: All MSR except LMSR might

change . LMSR preserve it and probabilities regarding any other event .

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Combinatorial Product Space• Given variables each with outcomes, a single

market scoring rule can make trades on any of the possible states, or any of the possible events.

• Creating a data structure to explicitly store the probability of every state is unfeasible for large values of .

• Computational complexity of updating prices and assets is NP-complete in worst-case.

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THANK YOU

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