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Measuring IA (Info Aggregation)

Measuring IA (Info Aggregation). Markets produce price distribution p = { p i } i Start at a uniform distribution u , where u i = 1/ I We calculate a full IA distribution q Assume a Bayesian who uses everyone’s data IA is a percentage distance between p and q

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Measuring IA (Info Aggregation)

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  1. Measuring IA (Info Aggregation) • Markets produce price distribution p = {pi}i • Start at a uniform distribution u, where ui= 1/ I • We calculate a full IA distribution q • Assume a Bayesian who uses everyone’s data • IA is a percentage distance between p and q • Mainly use quadratic: D(p,q) = iqi (qi - pi)2 • Also Kulback-Leibler: D(p,q) = iqi log (qi / pi) • Percentage distance is: 1 - D(p,q)/D(u,q)

  2. Mechanism Performance Market Maker Combined Value Standard 1 0 -1 1 0 -1 3 vars IA 87%11% 69%38% 27%63% 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 0 -1 1 0 -1 8 vars 26%17% 22%23% 3%5% Period

  3. Performance – All Data Market Maker Combined Value Standard 1 0 -1 1 0 -1 3 vars IA 70%36% 68%36% 27%63% 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 0 -1 1 0 -1 8 vars 26%17% 18%20% 3%5% Period

  4. Performance – KL Measure Market Maker Combined Value Standard 1 0 -1 1 0 -1 3 vars IA 71%22% 29%105% 4%50% 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 0 -1 1 0 -1 8 vars 23%17% -15%60% 5%8% Period

  5. Performance – KL, All Data Market Maker Combined Value Standard 1 0 -1 1 0 -1 3 vars IA 56%40% 27%103% 4%50% 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 0 -1 1 0 -1 8 vars 23%17% -12%50% 5%8% Period

  6. Price Dynamics - KL Market Maker Combined Value 1 0 -1 1 0 -1 3 vars IA 0 5 10 15 1 2 3 4 1 0 -1 1 0 -1 8 vars Minutes Round

  7. Situations: Goals, Training (Actually: X Z Y ) • Want in Situation: • explainable, fast, neutral • many variables, few directly related • few people, each not see all data cases • compute rational share-info estimates • Training Situation: • 3 binary variables X,Y,Z, 23 = 8 combos • P(X=0) = .3, P(X=Y) = .2, P(Z=1)= .5 • 3 people, see 10 cases of: AB, BC, AC • random map XYZ to ABC Case A B C 1 1 - 1 2 1 - 0 3 1 - 0 4 1 - 0 5 1 - 0 6 1 - 1 7 1 - 1 8 1 - 0 9 1 - 0 10 0 - 0 Sum: 9 - 3 Same A B C A -- -- 4 B -- -- -- C -- -- --

  8. Test Situation (Really: W V X S U Z Y T ) • 8 binary var. STUVWXYZ • 28 = 256 combos • .2 = P(S=0) = P(S=T) = P(T=U) = P(U=V) = … = P(X=Y) = P(Y=Z) • 6 people, see 10 cases of: ABCD, EFGH, ABEF, CDGH, ACEG, BDFH • random map STUVWXYZ to ABCDEFGH Case A B C D E F G H 1 0 1 0 1 - - - - 2 1 0 0 1 - - - - 3 0 0 1 1 - - - - 4 1 0 1 1 - - - - 5 0 1 1 1 - - - - 6 1 0 0 1 - - - - 7 0 1 1 1 - - - - 8 1 0 0 1 - - - - 9 1 0 0 1 - - - - 10 1 0 0 1 - - - - Sum: 6 3 4 10 - - - - Same A B C D E F G H A -- 1 2 6 -- -- -- -- B -- -- 7 3 -- -- -- -- C -- -- -- 4 -- -- -- -- D -- -- -- -- -- -- -- -- …

  9. B A f1>1 f2<1 Prices + + q1 $1 if A&B - - q2 $1 if B User Assets A Simple Implementation States

  10. A&B A&B A Simple Implementation States Prices User Assets

  11. D A C G F B E H A Scaleable Implementation • Overlapping variable patches • A simple MSR for each patch • Arbitrage neighbor patches • Limits profits to users who find inconsistencies • Only allow trade if all vars in same patch? • User assets per patch, move via overlap • Regroup patches from request activity?

  12. A B C B C .065 1.000 B B A A .9 .734 .2 .1 B C .4 .6 Cash extracted Arbitraging Patches .02 .08 .3 .1 .2 .7 .3 .3

  13. A B C B A B B A B C C .214 .786 .214 .786 .786 Arbitraging Patches .043 .171 .214 .160 .053 .175 .611 .393 .393

  14. A B C C 1 0 0 A 2 B A C B B B Moving Assets Between Patches 1 0 2 1 3 2 0 4

  15. A B C C 0 0 2 1 A C B A B B 1 B Moving Assets Between Patches 2 1 1 0 3 2 0 4

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