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Roi Meron

Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation [Robin Hanson, 2002]. Roi Meron. Outline. Scoring Rules Market Scoring Rules Logarithmic Market Scoring Rule(LMSR) Distribution == Cost function Combinatorial Markets. Event.

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Roi Meron

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  1. Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation[Robin Hanson, 2002] RoiMeron Seminar in Information Markets, TAU

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

  3. Event • finite set of outcomes (mutually exclusive and exhaustive states of the world) • Example: all possible prime ministers in elections ’13 Seminar in Information Markets, TAU

  4. 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 Seminar in Information Markets, TAU

  5. Exercise • Is the following (binary)scoring rule proper? where is the probability that will happen (This is a variation of brier scoring rule) Seminar in Information Markets, TAU

  6. 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 Seminar in Information Markets, TAU

  7. 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 Seminar in Information Markets, TAU

  8. Properness - simple case proof • Assume we have 2 possible outcomes. • Derivative w.r.t. gives: • Second derivative is negative. • Logarithmic scoring rule is proper. Seminar in Information Markets, TAU

  9. 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? Seminar in Information Markets, TAU

  10. 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 Seminar in Information Markets, TAU

  11. 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 Seminar in Information Markets, TAU

  12. 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. Seminar in Information Markets, TAU

  13. 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, Seminar in Information Markets, TAU

  14. Typically, initial distribution is uniform, i.e. where is the number of possible outcomes. • Market maker’s loss is bounded by . Seminar in Information Markets, TAU

  15. how do we implement a market? Seminar in Information Markets, TAU

  16. 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. Seminar in Information Markets, TAU

  17. 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. Seminar in Information Markets, TAU

  18. 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. Seminar in Information Markets, TAU

  19. LMSR cost function • For simplification, assume we have only 2 possible outcomes, then: Seminar in Information Markets, TAU

  20. “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. Seminar in Information Markets, TAU

  21. Equivalence proof Trader’s profit if happens = a logarithmic scoring rule payment Seminar in Information Markets, TAU

  22. Example Seminar in Information Markets, TAU

  23. Will Wile E. Coyote fall off a cliff next year? • Uniform priors: Seminar in Information Markets, TAU

  24. 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….. Seminar in Information Markets, TAU

  25. 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 . Seminar in Information Markets, TAU

  26. 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. Seminar in Information Markets, TAU

  27. Thank you Seminar in Information Markets, TAU

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