Probability and Asset Updating Using Bayesian Networks for Combinatorial Prediction Markets

1. Probability and Asset Updating Using Bayesian Networks for Combinatorial Prediction Markets Wei Sun, Robin Hanson, Kathryn Laskey, Charles Twardy DAGGRE, George Mason University

2. Edit-Based Interface • How should users talk to a prediction market? • There are many ways to specify a trade: (First, specify assumptions, and a claim, then:) • Change in cash, or new cash amount • Change in cash win if claim true, or new amount • Change in cash lose if claim false, or new amount • Copy what another did, or undo what they did • Change in price/probability, or new probability? • Keeps user focused info: finding errors in consensus

6. 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, 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

7. 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 that this helps • Periodically show how Ep[Su] varies with u If raise p win lose

8. Log Market Scoring Rule (LMSR) • Notation: • Variable Xi has value vi, N states x = <v1,v2,…> • px is current probability, Σxpx=1, px0 = 1/N • Sxu is cash user u wins in state x, initially = constant • If u edits px -> p’x , then S’xu =Sxu + b*log(p’x/px) • Disallow edit if S’xu < 0 for any x • Max gain of all users = b*log(N) • Problem: N gets HUGE with many variables

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

10. Junction Tree Algorithm If fx = c f(C) / s f(S) … To exact update f To exact find min f 1. f(V) becomes f’(V) 1. f(V) becomes f’(V) S 2. f’(S) = min V\S f’(V) 2. f’(S) = V\S f’(V) V W 3. f’(W) = f(W) f’(S) / f(S) 3. f’(W) = f(W) f’(S) / f(S) 4. Do the rest of net 4. Do the rest of net 5 4 6 3 2 1 5. f’’(S) = min W\S f’(W) 5. f’’(S) = W\S f’(W) 6. f’’(V) = f’(V) f’’(S) / f’(S) 6. f’’(V) = f’(V) f’’(S) / f’(S)

11. 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 via Bayes net • Let qxu= exp(Sxu/b), so q’x/qx= p’x/px, qx0 = constant • qx= c qc(xc) / s qs (xs) lets update q(x), find min • Implies Sx= ΣcSc(xc)-ΣsSs(xs), S = Ep[S]= ΣcSc – ΣsSs • If edit p(T|A) -> p’(T|A), need T,A in same clique • p’ in [ p/min(x in A&notT)qx ,1-((1-p)/min(x in A&T)qx)]

12. Expected Asset Derivation

13. 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] • Edit limits [min,max] of p’(T|A), to ensure Su ≥ 0 • User u aborts or picks p’(T|A) in [min,max] • Update p to reflect p(T|A) -> p’(T|A) • Update assets Su to reflect bet that this helps • Periodically show how Ep[Su] varies with u If raise p win lose

14. 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 • Need only do qux’/qux= p’x/px for one clique, one u • But how figure p’ in [min,max] e.g., min(x in A&T)qx ? • New: can quick find approx-min guaranteed > min

15. 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

16. 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

17. 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

18. 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 • Do if edit makes a conditional independence? • Additional cost to make a change • How do users show committed interest in change? BL LE T BE E SBL BLE AT TLE XE DBE

19. Summary • Bayes nets (B.N.) can efficiently store/update 1000s of interdependent variables • So can use to handle combo pred. mkt. probs • But what about asset store/update/find-min? • In LMSR, exp transform of assets has same product form as probs, so B.N. ways work too • Expected assets (sum prob*asset) works too • New approximations for non-tree asset min