How A Log Market Scoring Rule Works thread
i was gonna do this at some point, now seems like a good time
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i was gonna do this at some point, now seems like a good time
https://twitter.com/prerationalist/status/1421581918710247427
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if you're not familiar with prediction markets in general, see this quick reply thread i just did with an overview
a log market scoring rule implements the market part of a prediction market. it's great because it's automated & always liquid
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a log market scoring rule implements the market part of a prediction market. it's great because it's automated & always liquid
https://twitter.com/prerationalist/status/1476611456120598531
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to quickly recap, the first piece of a prediction market is a "conditional ATM"
you put in regular money, and it prints out "conditional money"
e.g. for a single yes/no event, you can put in $1 of regular money and get out both Y$1 (worth $1 if "yes") and N$1 (worth $1 if "no")
you put in regular money, and it prints out "conditional money"
e.g. for a single yes/no event, you can put in $1 of regular money and get out both Y$1 (worth $1 if "yes") and N$1 (worth $1 if "no")
to create an LMSR machine, you do need to front some money. u can think of it like a subsidy for the information you want to get out of the prediction market.
you take your initial subsidy $, feed it through the conditional-ATM, and put all the conditional$ into the LSMR
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you take your initial subsidy $, feed it through the conditional-ATM, and put all the conditional$ into the LSMR
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the state of the LMSR machine is determined by the ratio between its current contents compared to its initial contents
these ratios can be converted into probabilities by raising 1/2 to the power of the ratio
(concrete example in the next tweet, if the above is confusing)
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these ratios can be converted into probabilities by raising 1/2 to the power of the ratio
(concrete example in the next tweet, if the above is confusing)
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for example, say you initialize an LMSR with $10
then it will contain Y$10 and N$10 initially
at the start, its current state will equal its initial state, so Y and N will both have probability (1/2)^(10/10)=50%
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then it will contain Y$10 and N$10 initially
at the start, its current state will equal its initial state, so Y and N will both have probability (1/2)^(10/10)=50%
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the core rule of the LMSR is simply:
the probabilities must always sum to 100%
if u want to take some Y$ out of the LMSR, u have to put in enough N$ so that the probabilities still sum to 100%
this completely specifies the behavior.
(don't worry; examples still coming!)
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the probabilities must always sum to 100%
if u want to take some Y$ out of the LMSR, u have to put in enough N$ so that the probabilities still sum to 100%
this completely specifies the behavior.
(don't worry; examples still coming!)
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say you want to take out Y$5 from the LMSR.
if you didn't put in any N$ to balance it out, the probabilities would then exceed 100% - no good!
in order to balance out the machine having only Y$5, it would need a total of N$17.7
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if you didn't put in any N$ to balance it out, the probabilities would then exceed 100% - no good!
in order to balance out the machine having only Y$5, it would need a total of N$17.7
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so for the example above, you need to put in N$7.7 in order to get out Y$5
the LMSR doesn't create or destroy any Y$ or N$; it just trades them in such a way that it always keeps its own stock of Y$ and N$ representing probabilities that sum to 100%
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the LMSR doesn't create or destroy any Y$ or N$; it just trades them in such a way that it always keeps its own stock of Y$ and N$ representing probabilities that sum to 100%
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you can also see that you will never be able to get *all* of the Y$ out of the LMSR, because that would set its probability to 100%. then you would need to put in infinite N$ to get its probability to 0%, and you don't have infinite money.
this means the LMSR never run out!
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this means the LMSR never run out!
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typically, you probably just want to buy all Y$ or all N$. you can accomplish this by using both the conditional-ATM and the LMSR machine together.
say you want all N$. first, put some regular money into the conditional-ATM. then trade your extra Y$ for N$ at the LMSR.
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say you want all N$. first, put some regular money into the conditional-ATM. then trade your extra Y$ for N$ at the LMSR.
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if the amount you are buying is small compared to the original subsidy, then the price to buy Y$1 using the above method will be ~equal to the current *probability* corresponding to Y$1
that is, if the LMSR reports 30% probability for "yes", then you can get Y$1 for ~30¢
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that is, if the LMSR reports 30% probability for "yes", then you can get Y$1 for ~30¢
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the internal contents of the LMSR also have a neat theoretical interpretation—the *suprisal*, or self-information
the number of Y$ is the suprisal of seeing "yes"; the number of N$ is the suprisal of seeing "no"
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the number of Y$ is the suprisal of seeing "yes"; the number of N$ is the suprisal of seeing "no"
https://twitter.com/johncarlosbaez/status/1424408476764106756
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this setup also generalizes really simply to N cases instead of just yes/no
say you want to see which of 5 candidates will win an election
then the conditional ATM prints 5 different bills (instead of 2) & the LMSR base for calculating probabilities is 1/5 instead of 1/2.
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say you want to see which of 5 candidates will win an election
then the conditional ATM prints 5 different bills (instead of 2) & the LMSR base for calculating probabilities is 1/5 instead of 1/2.
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if you want more details, the best source imo is still this paper by @robinhanson where he introduces the concept: mason.gmu.edu/~rhanson/mktsc…
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as usual, i did all the drawings and animations for this thread using blender grease pencil. go #b3d!
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