1. Seminar in Information Markets, TAU
Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation [Robin Hanson, 2002]
Roi Meron
07-Nov-12
Seminar in Information Markets, TAU

2. Seminar in Information Markets, TAU
Outline
Scoring Rules
Market Scoring Rules
Logarithmic Market Scoring Rule(LMSR)
Distribution == Cost function
Combinatorial Markets
07-Nov-12
Seminar in Information Markets, TAU

3. Seminar in Information Markets, TAU
Event
Ω={ 𝑤 1 … 𝑤 𝑛 } finite set of 𝑛 outcomes (mutually exclusive and exhaustive states of the world)
Example: all possible prime ministers in elections ’13
07-Nov-12
Seminar in Information Markets, TAU

4. Seminar in Information Markets, TAU
Scoring Rule
𝑝 𝑖 - Agent’s belief about the probability that state 𝑤 𝑖 will occur.
𝑆 𝑖 ( 𝑟 ) is the payment made to agent who reports distribution 𝑟 if outcome is 𝑤 𝑖 .
𝐸 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 = 𝑖=1 𝑛 𝑝 𝑖 𝑆 𝑖 𝑟
A proper scoring rule is when
𝑝 = argmax 𝑟 𝑖=1 𝑛 𝑝 𝑖 𝑆 𝑖 𝑟
Strictly proper: when 𝑝 is unique
Forecaster’s Expected payment
07-Nov-12
Seminar in Information Markets, TAU

5. Seminar in Information Markets, TAU
Exercise
Is the following (binary)scoring rule proper?
1− 𝑥−𝑟 2
where 𝑟 is the probability that 𝑥 will happen
(This is a variation of brier scoring rule)
07-Nov-12
Seminar in Information Markets, TAU

6. Seminar in Information Markets, TAU
Exercise
Is the following (binary)scoring rule proper?
1− 𝑥−𝑟 2
where 𝑟 is the probability that 𝑥 will happen
1−𝑝 1−𝑟 2 − 1−𝑝 𝑟 2
Derivative w.r.t. 𝑟 and comparing to 0 gives:
−2𝑟𝑝+2𝑝−2𝑟 1−𝑝 =0 2𝑝=2𝑟 𝑝=𝑟
07-Nov-12
Seminar in Information Markets, TAU

7. Logarithmic Scoring Rule
𝑟 𝑖 ≤1
𝑎 𝑖 needs to be big enough for agents to participate
The only strictly proper scoring rule in which the score for outcome 𝑖 depends only on 𝑟 𝑖 and not on the probabilities given to 𝑟 𝑗 for 𝑗≠𝑖
An agent’s payoff depends only on the probability he assigned to the actual event and not the ones that didn’t happen
07-Nov-12
Seminar in Information Markets, TAU

8. Logarithmic Scoring Rule
𝑟 𝑖 ≤1
𝑎 𝑖 needs to be big enough for agents to participate
The only strictly proper scoring rule in which the score for outcome 𝑖 depends only on 𝑟 𝑖 and not on the probabilities given to 𝑟 𝑗 for 𝑗≠𝑖
Example: Quadratic Scoring Rule
𝑆 𝑖 𝑟 =2 𝑟 𝑖 − 𝑗=1 𝑛 𝑟 𝑗 2
07-Nov-12
Seminar in Information Markets, TAU

9. Seminar in Information Markets, TAU
LSR is local
Therefore, we can look at a 𝑛-valued event as 𝑛 binary events and score remains the same.
Instead of betting on the winner of elections 13’, we can bet whether candidate 𝑖 will win or not, for all candidates.
Complexity issue (open question)
07-Nov-12
Seminar in Information Markets, TAU

10. Properness - simple case proof
Assume we have 2 possible outcomes.
Agent’s expected payment:
𝑝 log 𝑟 + 1−𝑝 log (1−𝑟)
Derivative w.r.t. 𝑟 gives:
𝑝 𝑟 − 1−𝑝 1−𝑟 =0⇒𝑝−𝑟𝑝−𝑟+𝑟𝑝=0
𝑝=𝑟
Second derivative is negative.
Logarithmic scoring rule is proper.
07-Nov-12
Seminar in Information Markets, TAU

11. Seminar in Information Markets, TAU
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?
07-Nov-12
Seminar in Information Markets, TAU

12. Seminar in Information Markets, TAU
Market Scoring Rule
Market maker starts with an initial distribution 𝑟 0 . 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 𝑆
𝑥 𝑖 =Δ 𝑠 𝑖 𝑟 𝑡 , 𝑟 𝑡−1 = 𝑠 𝑖 𝑟 𝑡 − 𝑠 𝑖 ( 𝑟 𝑡−1 )
The market maker finally pays Δ 𝑠 𝑖 𝑟 𝑡 , 𝑟 0
LMSR is proper
Simple scoring rules do not give us common(joined) estimates.
Simple markets need people to bet on the same event and give them no incentive to do it.
MSR give incentive for experts to participate and do not require them to find a counterparty.
Assume that a single person knows something about an event and the rest know that they know nothing about the event, then, standard market cannot acquire the person’s information.
Irrational participation problem – once rational agents have hedged their risk, they should not want to trade each other even when they have substantial private information
07-Nov-12
Seminar in Information Markets, TAU

13. Seminar in Information Markets, TAU
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
07-Nov-12
Seminar in Information Markets, TAU

14. Seminar in Information Markets, TAU
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.
For any scoring rule, an agent should volentarily agree this kind of payment, because he can choose not to participate
simple information market: .001 estimates – thick market –gives good accuracy (with many traders focusing on a single event). When we have few traders, market becomes thin and traders do not participate and accuracy declines.
Simple Scoring Rule – Thick market = opinion pooling problem. Many estimates per trader = hard to guess what is correct
Market Scoring Rule – combine the 2. combining opinions and anyone can change opinion to maximize his profit – better prediction.
07-Nov-12
Seminar in Information Markets, TAU

15. Logarithmic Market Scoring Rule (LMSR)
Reminder: 𝑆 𝑖 𝑟 = 𝑎 𝑖 +𝑏 log ( 𝑟 𝑖 )
𝑏 is a parameter that controls:
liquidity
loss of the market maker
Adaptivity of the market maker.
* Large 𝑏 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, 𝜋
−𝑏 Σ 𝑖 𝜋 𝑖 log 𝜋 𝑖
Adaptivity – We will see that in terms of inventory, it reduces the cost of
Maximum expected payment in cases where the agent eventually becomes sure of the actual state i
07-Nov-12
Seminar in Information Markets, TAU

16. Seminar in Information Markets, TAU
Typically, initial distribution is uniform, i.e. 𝑝 𝑖 = 1 𝑛 where 𝑛 is the number of possible outcomes.
Market maker’s loss is bounded by blog 𝑛 .
07-Nov-12
Seminar in Information Markets, TAU

17. how do we implement a market?
07-Nov-12
Seminar in Information Markets, TAU

18. 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 𝑞={𝑞 𝑖 } of how many units of the form “pays 1$ if the state is 𝑖” were sold (outstanding shares).
Has an instantaneous price 𝑝 ( 𝑝 𝑖 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.
07-Nov-12
Seminar in Information Markets, TAU

19. Seminar in Information Markets, TAU
LMSR price function
Let 𝑖 be the outcome which finally took place.
The market maker pays exactly 𝑞 𝑖 $. 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:
𝑆 𝑖 𝑝 𝑖 = 𝑞 𝑖
The price function is the current distribution “given” by the last agent.
07-Nov-12
Seminar in Information Markets, TAU

20. Seminar in Information Markets, TAU
LMSR price function(2)
The price function is the inverse of the scoring rule function
𝑝 𝑖 𝑞 = 𝑒 𝑞 𝑖 − 𝑎 𝑖 /𝑏 𝑘 𝑒 𝑞 𝑘 − 𝑎 𝑘 /𝑏
A large trade can be described by a series of “tiny” trades between 𝑡 𝑠 to 𝑡 𝑒 . The price of this trade event is given by integrating over 𝑝 𝑖 in the range of [ 𝑡 𝑠 , 𝑡 𝑒 ].
Probabilities represent prices for (very) small trades.
07-Nov-12
Seminar in Information Markets, TAU

21. Seminar in Information Markets, TAU
LMSR cost function
𝐶 𝑞 =𝑏 log 𝑖=1 𝑛 𝑒 (𝑞 𝑖 /𝑏)
Let’s assume no short selling, so 𝑞 ∈ ℕ 0 𝑛
For simplification, assume we have only 2 possible outcomes, then:
𝐶 𝑞 0 , 𝑞 1 =𝑏 log ( 𝑒 (𝑞 0 /𝑏) + 𝑒 (𝑞 1 /𝑏) )
07-Nov-12
Seminar in Information Markets, TAU

22. Seminar in Information Markets, TAU
“How much?”
Buying 𝑞 𝑖 <0 shares means Selling 𝑞 𝑖 .
Say someone wants to buy 20 shares of outcome 0. He pays:
𝐶 𝑞 0 +20, 𝑞 1 −𝐶 𝑞 0 , 𝑞 1
In general, if a trader changes the outstanding volume from 𝑞 𝑖 to 𝑞 𝑖 ∗ , the payment is:
𝐶 𝑞 0 ∗ , 𝑞 1 ∗ −𝐶 𝑞 0 , 𝑞 1
If 𝑞 0 ∗ < 𝑞 0 , 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.
07-Nov-12
Seminar in Information Markets, TAU

23. Seminar in Information Markets, TAU
Equivalence proof
Trader’s profit if 𝑖 happens = 𝑞 𝑖 ′ − 𝑞 𝑖 − 𝐶 𝑞 ′ −𝐶 𝑞 = 𝑞 𝑖 ′ − 𝑞 𝑖 − log 𝑗 𝑒 𝑞 𝑗 ′ − log 𝑗 𝑒 𝑞 𝑗 = log 𝑒 𝑞 𝑖 ′ − log 𝑒 𝑞 𝑖 − log 𝑗 𝑒 𝑞 𝑗 ′ − log 𝑗 𝑒 𝑞 𝑗 = log 𝑒 𝑞 𝑖 ′ 𝑗 𝑒 𝑞 𝑗 ′ − log 𝑒 𝑞 𝑖 𝑗 𝑒 𝑞 𝑗 = log [𝑝 𝑖 𝑞 ′ ] − log [ 𝑝 𝑖 𝑞 ] = a logarithmic scoring rule payment
07-Nov-12
Seminar in Information Markets, TAU

24. Seminar in Information Markets, TAU
Example 𝜋 ⅇ
07-Nov-12
Seminar in Information Markets, TAU

25. Seminar in Information Markets, TAU
Will Wile E. Coyote fall off a cliff next year?
𝑏=100
𝑞 0 = 𝑞 1 =0
Uniform priors: 𝑝 0 = 𝑝 1 = 1 2
07-Nov-12
Seminar in Information Markets, TAU

26. Seminar in Information Markets, TAU
Will Wile E. Coyote fall off a cliff next year?
𝑏=100
𝑞 0 = 𝑞 1 =0
Uniform priors: 𝑝 0 = 𝑝 1 = 1 2
Bugs arrives and buys 10 shares of outcome 0
𝐶 10,0 −𝐶 0,0 =100⋅ ln 𝑒 𝑒 0 − 100⋅ ln 𝑒 0 + 𝑒 0 =74.44−69.31=5.13$
Daffy buys 10 shares of outcome 1
𝐶 10,10 −𝐶 10,0 =100⋅ ln 𝑒 𝑒 − 100⋅ ln 𝑒 𝑒 0 =79.31−74.44=4.87$
RoadRunner runs and buys 40 shares of outcome 0
𝐶 50,10 −𝐶 10,10 =100⋅ ln 𝑒 𝑒 − 100⋅ ln 𝑒 𝑒 =101.3−79.31=21.99$
Bugs returns and sell his 10 shares of outcome 0
𝐶 40,10 −𝐶 50,10 =100⋅ ln 𝑒 𝑒 − 100⋅ ln 𝑒 𝑒 =95.44−101.3=−𝟓.𝟖𝟔$
And the current quote of “0” is:
𝑝 0 40,10 = 𝑒 𝑒 𝑒 0.1 =0.57
07-Nov-12
Seminar in Information Markets, TAU

27. Combinatorial Markets
Say we bet on the chances of rain of the following week.
𝒜- will it rain on Sunday ∈{0,1}
ℬ- will it rain on Monday
𝒞- will it rain on Tuesday
We can think of other events:
B|𝐴 rain on Monday given it rains on Sunday…
Ideally, trading on the probability of 𝐵 given 𝐴 should not result in a change in the probability of 𝐴 or a change in prob. “𝐴 given C”.
But in fact…..
07-Nov-12
Seminar in Information Markets, TAU

28. LMSR local inference rule
Logarithmic rule bets on “A given B” preserve p(𝐵) and, for any event 𝐶 preserve p 𝐶 𝐴𝐵 𝑝(𝐶| 𝐴 𝐵) and 𝑝(𝐶| 𝐵 ).
The other direction also holds.
In other words: All MSR except LMSR might change 𝑝(𝐵). LMSR preserve it and probabilities regarding any other event 𝐶.
07-Nov-12
Seminar in Information Markets, TAU

29. Combinatorial Product Space
Given 𝑁 variables each with 𝑉 outcomes, a single market scoring rule can make trades on any of the 𝑉 𝑁 possible states, or any of the 2 𝑉 𝑁 possible events.
Creating a data structure to explicitly store the probability of every state is unfeasible for large values of 𝑁.
Computational complexity of updating prices and assets is NP-complete in worst-case.
07-Nov-12
Seminar in Information Markets, TAU

30. Seminar in Information Markets, TAU
Thank you
07-Nov-12
Seminar in Information Markets, TAU