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Probability and Asset Updating Using Bayesian Networks for Combinatorial Prediction Markets

Probability and Asset Updating Using Bayesian Networks for Combinatorial Prediction Markets. Wei Sun, C4I Center Robin Hanson, Economics Kathryn Laskey, Systems Engineering Charles Twardy, C4I Center GE ORGE MASON UNIVERSITY. “Pays $1 if Obama wins”. Will price rise or fall?. sell.

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Probability and Asset Updating Using Bayesian Networks for Combinatorial Prediction Markets

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  1. Probability and Asset Updating Using Bayesian Networks for Combinatorial Prediction Markets Wei Sun, C4I Center Robin Hanson, Economics Kathryn Laskey, Systems Engineering Charles Twardy, C4I Center GE ORGE MASON UNIVERSITY

  2. “Pays $1 if Obama wins” Will price rise or fall? sell E[ price change | ?? ] buy price sell Lots of ?? get tried, price includes all! buy Buy Low, Sell High

  3. Current Event Prices 52-65% August US Unemployment 8.3% or more 56-57% Obama is re-elected in November 19-22% Greece defaults by 2013 12-15% Portugal defaults by 2013 6-8% Spain defaults by 2013 46-48% Syria’s al-Assad out by 2013 22-30% US or Israel strike Iran by 2013 12-40% Category 4 or bigger storm hits US by 2013 6-8% Cap & trade for US emissions by 2014 53-58% Nation using Euro drops it by 2014 68-85% Global ave temp higher in 2019 than 2009 InTrade.com

  4. Beats Alternatives • Vs. Public Opinion • I.E.M. beat presidential election polls 709/964 (Berg et al ‘08) • Re NFL, beat ave., rank 7 vs. 39 of 1947 (Pennock et al ’04) • Vs. Public Experts • Racetrack odds beat weighed track experts (Figlewski ‘79) • If anything, track odds weigh experts too much! • OJ futures improve weather forecast (Roll ‘84) • Stocks beat Challenger panel (Maloney & Mulherin ‘03) • Gas demand markets beat experts (Spencer ‘04) • Econ stat markets beat experts 2/3 (Wolfers & Zitzewitz ‘04) • Vs. Private Experts • HP market beat official forecast 6/8 (Plott ‘00) • Eli Lily markets beat official 6/9 (Servan-Schreiber ’05) • Microsoft project markets beat managers (Proebsting ’05) • XPree beat corp error, 3.5 vs 6.6%

  5. Simple Info Markets Market Scoring Rules Scoring Rules opinion pool problem thin market problem 100 .001 .01 .1 1 10 Best of Both Accuracy Estimates per trader

  6. $ ex if x $ s(1)-s(0) Market Scoring Rule (MSR) • Scoring rule: if report r, state is x, get sx(r) • Proper if: p in argmaxrΣx px sx(r) • MSR: user t gets change Δsx = sx(pt) - sx(pt-1) “Anyone can use scoring rule if pay off prior user” • Invert sx(p) for inventory market maker px(s): • Tiny sale fee: px(s) ex(sx sx+ex) • Big sale fee: 01 Σxpx(s(t)) sx´(t) dt

  7. Log Market Scoring Rule • Log MSR: sx(r)=ln(rx)/α • Cost strue(pend) - strue(p0) is N(p) = -Σxpx ln(px)/α • N(pall)≤Σvar N(pvar), so all combos ≤ a rule per var • Uniquely modular: • Changes p(A|B), but not p(B), p(C|B&A), p(C|B&notA), p(C|notB),I(A,B,C), I(B,A,C) • To compute, state is: probs px, assets Sxu per user u • If u edits px -> p’x , do S’xu =Sxu + ln(p’x/px)/α, if all ≥ 0 • Helps to show market value of portfolio: Su = Σxpx Sxu • PROBLEM: If many vars, way too many states x! $1 if A&B p(A|B) $1 if B

  8. DAGGRE.org • Part of IARPA funded ACE “competition” • We rank 2-3 out of 5 • Continuously live since September 2011 • 250 questions so far, 50-100 at a time • Join! • 1000 participants, 300 active in last 90 days

  9. DAGGRE Demo

  10. Prediction Market Issues • Problem: What we know depends on context • Solution: Let tell relational, conditional info • Problem: Too many combos to store/update • Solution: Bayes nets store/update probs well • Problem: Also need store/update assets, find expected assets, ensure assets not go negative • Solution: In Bayes net LMSR, ways to store/update/find-min for probs also does assets • Problem: Can’t update probs or assets exactly

  11. Edit-Based Combo System Needs • User u chooses assumptions A, target event T • Find & show to user u (who has assets Su): • Current consensus p(T|A) • Now long/short? Via: Ep[Su|A&T]-Ep [Su|A&notT] • Limits [min,max] of new p’(T|A), to ensure Su ≥ 0 • User u aborts or picks a p’(T|A) in [min,max] • Update p to reflect p(T|A) -> p’(T|A) • Update assets Su to reflect bet for p’ over p • Periodically show how Su = Ep[Su] vary with u If raise p win lose

  12. Clique Bayes/Markov Nets P(Clique | Rest of Net) = P(Clique | Its Separators) • px = ∏c pc(xc) / ∏s ps (xs) lets update, find min • Space linear in # cliques, time also if net is tree Pennock & Xia (2011): do LMSR via Bayes net • User pays cash for “Pays $Z if A=a,B=b,C=c, …” • Prior purchases can’t pay for new trades • All A,B,C … in purchase must be in same clique • Assume Bayes Net way to update prob = price • Not show long/short Ep[Su|win]-Ep[Su|lose] Separator x = <vA,vB,vC, …> BL LE T BE E SBL BLE AT TLE XE DBE

  13. Reusing Assets Belief: Supporting Trade: P(B|A1) > x $x $x $9x $x $x $1 if B&A2 $1 if B&A3 $1 if B&A9 $1 if B&A1 $1 if B $x if not A1 $x if not A9 $x if not A3 $x if not A2 P(B|A2) > x P(B|A3) > x …. …. …. …. …. P(B|A9) > x $8x

  14. Clique SBL Bayes/Markov Nets BL LE T AT TLE BLE BE E P(Clique | Rest of Net) = P(Clique | Its Separators) XE DBE Separator x = <xA,xB,xC, …> • px = ∏c pc(xc) / ∏s ps (xs) lets update p(x), find min • Our Approach:to do LMSR using Bayes nets • Let qxu= exp(Sxu/b), so q’x/qx= p’x/px, qx0 = constant • qx= ∏c qc(xc) / ∏s qs (xs) JT to update q(x), find min • Implies Sx= ΣcSc(xc) - ΣsSs(xs), S = ΣcSc - ΣsSs • If edit p(T|A) -> p’(T|A), need T,A in same clique • [min,max] = [ p/min(x in A&notT)qx ,1-((1-p)/min(x in A&T)qx)]

  15. System Needs Achieved! • User u chooses assumptions A, target event T • Find & show to user u (who has assets Su): • Current consensus p(T|A) • Now long/short? Via: Ep[Su|A&T]-Ep [Su|A&notT] • Limits [min,max] of new p’(T|A), to ensure Su ≥ 0 • User u aborts or picks a p’(T|A) in [min,max] • Update p to reflect p(T|A) -> p’(T|A) • Update assets Su to reflect bet for p’ over p • Periodically show how Su = Ep[Su] vary with u If raise p win lose

  16. Markov Engine Scalability Test

  17. Except …. • Many nets are not nearly trees • Exist approximations to update p • E.g., “loopy” propagation of JT update rule • Could users exploit predictable p errors? • Also need approximate updates to Su • Only do S’xu = S’xu + ln(p’x/px) for one clique, one u • But how figure p’ in [min,max] e.g., min(x in A&T)qx ? • Can quick find approx-min guaranteed > min

  18. What If A Is Far from T? Then Edit Assume A3 • Option 1: Find nearest changes to ideal LMSR edit of P(T|A) that fit network constraints. • Option 2: Translate far assumptions A into local clique assumptions L, let user edit P(T|L). A2 A1 T L2 L1

  19. Can Users Edit Links? • Add link => bigger cliques • Costs system more space/time to store/update • Allow if users willing to make big supporting edit? • Delete link => some old assets can’t be sold • Allow if edit creates conditional independence? • How control compute costs while allowing structure changes? • Require combine add, delete links so same cost? BL LE T BE E SBL BLE AT TLE XE DBE

  20. To learn more, come see our poster. • Within a month, see live combo market at DAGGRE.org (beta test available now)

  21. An Asset Scenario C B AB BC User wants to raise p(A|B). Will gain “$x if A&B”, lose “$y if B&notA”. System must fast figure max feasible y can lose. 2. draw from separator x 1. use cash B B B y A A A cash cash cash C C C B B B

  22. Asset Scenario Cont. C B AB BC now can lose up to y = 3 3. draw from neighbor cliques, separators B B B A A A cash cash cash C C C B B B

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