Scalable Bayesian Updating of Combinatorial Information Markets

1. Scalable Bayesian Updating of Combinatorial Information Markets Robin Hanson, Kathryn Laskey, David Porter, Vernon Smith

2. Who We Are • Vernon Smith - 50yrs exper. econ. • David Porter - 20yrs market design • Kathryn Laskey - 16yrs Bayesian nets • Robin Hanson - 14yrs info markets

3. Markets Can Advise Decisions $1 if War & Move Troops P(M) $1 if Move Troops P(W | M) $1 Compare! P(W | not M) $1 if Not Move Troops $1 if War & Not Move Troops P(not M)

4. Every nation*quarter: • Political stability • Military activity • Economic growth • US aid/trade • US military activity But: # Parameters > # Nations * #Quarters * #Variables # Probabilities > 2 #Parameters

5. Problems Making Many Prices • Thin markets if # traders << # prices • Solution: Combinatorial market makers • Exponential explosion, can’t store prices • Solution: Overlapping market makers • Arbitrage updates are not modular • Solution: Bayesian network technology

6. Accuracy Simple Info Markets Market Scoring Rules Scoring Rules opinion pool problem thin market problem 100 .001 .01 .1 1 10 Estimates per trader New Tech + Old

7. Old Tech: Proper Scoring Rules • Assume: states i from ex post verifiable vars • When report r, state is i, reward is si(r) p = argmaxrSi pi si(r), Si pi si(p)  0 • E.g., log rule (Good 1952) si = a log(ri) • Long used in weather/business forecasting, student test scoring, economics experiments

8. $ ei if i $ s(1)-s(0) Market Scoring Rules • MSRs combine scoring rules, info markets • User t faces $ rule: Dsi = si(pt) - si(pt-1) “Anyone can use scoring rule if pay off last user” • Is auto market maker, price from net sales s • Tiny sale fee:  pi(s) ei (sisi+ei) • Big sale fee: 01 Sipi(s(t)) si´(t) dt • Log MSR is: pi(s) = exp(si) / Sk exp(sk) • This has cost, modularity advantages

9. $1 if A&B p(A|B) $1 if B A Simple Implementation • Make a state per variable-value combo • Per state, store N+1 nums: prob, user cash • Users browse marginals (e.g., p(A),p(B),…) • Can make assumptions (e.g, B true, C not, …) • Also see cash if event (max, min, ave, …) • For each trade: • Raise prob, user’s cash in states in A & B • Lower prob, user’s cash in states in not-A & B LISP: http://hanson.gmu.edu/mktscore-prototype.html

10. A Scaleable Implementation D A • Overlapping variable patches • A simple MSR per patch • If consistent, is Markov network • Var independent of rest given neighbors • Allow trade if all vars in same patch • Arbitrage overlapping patches • Sure to eventually agree, robust to gaming C G F B E H

11. .1 A 1.000 .734 A B B C .065 .9 .2 B B C .6 .4 Arbitraging Patches A B C .02 .08 .3 .1 .2 .7 .3 .3 Cash extracted

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

13. A H But Arbitrage Is Not Modular D A C G F B E H • Everyone agrees on prices • Expert on A gets new info, trades • Arbitrage updates all prices • Expert on H has no new info, but must trade to restore old info!

14. Bayesian/Markov Networks • Local info trades not require distant corrections if updates follow Bayes’ rule • Bayesian/Markov net tech does this • Off the shelf exact tech if net forms a tree • Many approx. techs made for non-trees • I.E.T. will pick a tech for this purpose • Must update user assets as well as prices • Want robust to gaming on errors

15. Netica Hugin MSBNx Problem size Optimized for a particular query & observation pattern SPI Speed JSPI: A Scalable BN Solution Finer-grained structures (intra-node factorizations) Local Exp Runs forever with streaming inputs Temporal factorizations CPCS: medical diagnosis 422-node noisy-max BN PDBN Maximum utilization of the given resource Approx MIT-LL: missile discrimination 39-node/time Dynamic BN OO modeling JPF IA: cyberspace situation assessment 792-node/time Dynamic BN in JPF Optimized embeddable Compiler Future: Parallel Distributed Collaborated DDB: battleground level 2,3 fusion 20,000-node BN in JPF No one else can solve any of these BNs.

16. The Plan • I.E.T., Laskey pick Bayes update tech • N.E. provides interface, arbitrage update • Hanson picks test environment(s) • Smith, Porter, Hanson, N.E., run lab experiments comparing update techs • See: Does modular/Bayes updating speed/improve information aggregation?

17. Environments: Goals, Training (Actually: X Z Y ) 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 -- -- -- • Want in Environment: • Many variables, few directly related • Few people, each not see all variables • Can compute rational group estimates • Explainable, fast, neutral • Training Environment: • 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

18. Experiment Environment (Really: W V X S U Z Y T ) 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 -- -- -- -- -- -- -- -- … • 8 binary vars: STUVWXYZ • 28 = 256 combinations • 20% = P(S=0) = P(S=T) = P(T=U) = P(U=V) = … = P(X=Y) = P(Y=Z) • 6 people, each see 10 cases: ABCD, EFGH, ABEF, CDGH, ACEG, BDFH • random map STUVWXYZ to ABCDEFGH

19. Experiment Results – 3 variables

20. Experiment Results – 8 Variables

21. KL(prices,group) 1- KL(uniform,group) Time to Aggregate Info 1 % Info Agg. = 0 0 5 10 15 Minutes -1