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Market Scoring Rules As Combinatorial Information Market Makers

Market Scoring Rules As Combinatorial Information Market Makers. Robin Hanson Department of Economics George Mason University. Outline: Old Tech Meets New. To gain info, elicit probs p = {p i } i , E p [x |A] Can verify state i later, N/Q = people / question

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Market Scoring Rules As Combinatorial Information Market Makers

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  1. Market Scoring Rules AsCombinatorial Information Market Makers Robin Hanson Department of Economics George Mason University

  2. Outline: Old Tech Meets New • To gain info, elicit probs p = {pi}i , Ep[x |A] • Can verify state i later, N/Q = people / question • Old tech (~1950+): Proper Scoring Rules N/Q = 1: works well, N/Q  1: hard to combine • New tech (~1990+): Information Markets N/Q  1: works well, N/Q ~ 1: thin markets • Get best of both in: Market Scoring Rules

  3. Outcomes Stock price Product sales Unemployment Economic growth Crime rate War severity Decisions Dump CEO Which ad campaign Who elected president Fed raise/lower rates More gun control Move troops nearby E[Outcome|Decision]

  4. Old Tech: Proper Scoring Rules • When report r, state is i, reward is si(r) p = argmaxrSi pi si(r), Si pi si(p)  0 • Quadratic (Brier 1950) si = 2ri – Sk rk2 • Logarithmic (Good 1952) si = log(ri) • Unique: reward via likelihood (Winkler 1969) • Long used in weather/business forecasts, student test scoring, economics experiments

  5. Problems Incentives Number shy Cognitive bias Non risk-neutral State-dependent utility Combo explosion Disagreements Solutions Proper scoring rules Prob wheel, word menu Corrections Lottery payoffs Insurance game Dependence network Dictator per Q, ?? Old Tech Issues

  6. Opinion Pool “Impossibile” • Task: pool prob. T(A) from opinions pn(A) • Any 2 of IPP, MP, EB  dictator (T= pd) ! IPP = if A,B indep. in all pn, are indep. in T MP = commutes: pool, coarsen states (-field) EB = commutes: pool, update on info • MP  T = n=0 wn pn, with wn indep. of A

  7. New Tech: Information Markets • Most markets aggregate info as side effect • Info markets beat competing institutions • OJ futures improve weather forecast (Roll 1984) • HP market beat sales forecast 6/8 (Plott 2000) • I.E.M. beat president polls 451/596 (Berg etal 2001) $1 if A p(A) $1 $ x Ep[x] $1 $1 if A&B p(A|B) $1 if B

  8. Problems Incentives Shy, complex utility Who expert on what Cognitive bias Combo explosion Thin markets (N/Q ~1) What is independent Solutions Bet with each other Same solutions Self-select Also specialists correct Combo match? Net? Market scoring rules Dictator per Q, ?? New Tech Issues

  9. Thin Market Problem • Trade requires coordination in Assets and • Time: waiting offers suffer adverse selection • Call markets, combo match, can help some, but • Most possible info markets do not exist • Most are illegal, and for most of the rest • Expect few traders, so don’t make offer • If known that only one person has opinion on a topic, price won’t reveal that opinion!

  10. Accuracy Simple Info Markets Market Scoring Rules Scoring Rules opinion pool problem thin market problem 100 .001 .01 .1 1 10 Q/N = Estimates per trader Old Tech Meet New

  11. 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) $ ei if i $ s(1)-s(0)

  12. Cost and Combos • Total cost: C = si true(pfinal) - si true(pinitial) • Expected cost: Ep[C] Sipi (si(1i) - si(p)) • For log MSR,  entropy: S(p) = - Sipi log(pi) • Let state i = combination of base var values vi • S(pall)Svar S(pvar), for pvar = {pvar value v}v • So compared to cost of log rule for each var, all var/value combos cost no more!

  13. Log Rule is Modular • Consider bet: • Changes p(A|B); only log rule keeps p(B) • Also keeps p(C|A&B), p(C|A&B), p(C|B),I(A,B,C), I(B,A,C),I(C,A,B), I(C,B,A) • A is var, one of whose values is A, etc. • I(A,B,C) iff p(A|B&C) = p(A|B) for all values • Log rule uniquely keeps changes modular $1 if A&B p(A|B) $1 if B

  14. Computational Problems • N binary vars makes 2N states • Bayes net: I(var, neighbors, others) • pi = var p(var vi | neighbor var vi) • 1000 binary vars, 10 neighbors ea., doable? • Copy of net for each trader’s assets, MSR probs • But who gets to pick/change the network? • Comparing assets across changes harder • Have several alternative nets at once?

  15. Summary • Scoring rules if N/Q = 1, info markets if 1 • Market scoring rules work for both cases • Always exists complete prob distribution that anyone can change any part of, if take risk • Log rule has combo/modularity advantages • Costs no more for all var/value combos, • If bet on p(A|B), keeps p(B), I(A,B,C), I(B,A,C) ...

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