CS6501: T opics in Learning and Game Theory (Fall 2019) Prediction Markets and Scoring Rules Instructor: Haifeng Xu
Outline Ø Recap:Scoring Rule and Information Elicitation Ø Connection to Prediction Markets Ø Manipulations in Prediction Markets 2
Information Elicitation from A Single Expert Ø We (designer) want to learn the distribution of random var 𝐹 ∈ [𝑜] • 𝐹 will be sampled in the future Ø An expert/predictor has a predicted distribution 𝜇 ∈ Δ ( Ø Want to incentivize the expert to truthfully report 𝜇 Idea: reward expert by designing a scoring rule 𝑇(𝑗; 𝑞) where: (1) 𝑞 is the expert’s report (may not equal 𝜇 ); (2) 𝑗 ∈ [𝑜] is the event realization Definition. The “scoring rule” 𝑇(𝑗; 𝑞) is [strictly] proper if truthful report 𝑞 = 𝜇 [uniquely] maximizes expected utility 𝑇(𝜇; 𝑞) . 3
Proper Scoring Rules Example 1 [Log Scoring Rule] Ø 𝑇 𝑗; 𝑞 = log 𝑞 3 Example 2 [Quadratic Scoring Rule] 8 Ø 𝑇 𝑗; 𝑞 = 2𝑞 3 − ∑ 7∈[(] 𝑞 7 Theorem. The scoring rule 𝑇(𝑗; 𝑞) is (strictly) proper if and only if there exists a (strictly) convex function 𝐻: Δ ( → ℝ such that 𝑇 𝑗; 𝑞 = 𝐻 𝑞 + ∇𝐻(𝑞)(𝑓 3 − 𝑞) basis vector 4
Information Elicitation from Many Experts 𝜇 C 𝜇 8 𝜇 A . . . Idea: sequential elicitation – experts make predictions in sequence Ø Reward for expert 𝑙 ’s prediction 𝑞 A is 𝑇 𝑗; 𝑞 A − 𝑇(𝑗; 𝑞 ABC ) • I.e., experts are paid based on how much they improved the prediction 5
Information Elicitation from Many Experts 𝜇 C 𝜇 8 𝜇 A . . . Theorem. If 𝑇 is a proper scoring rule and each expert can only predict once, then each expert maximizes utility by reporting true belief given her own knowledge. Remark Ø Each expert is expected to improve the prediction by aggregating previous predictions and then update it • Otherwise they will lose money 6
Information Elicitation from Many Experts 𝜇 C 𝜇 8 𝜇 A . . . Theorem. If 𝑇 is a proper scoring rule and each expert can only predict once, then each expert maximizes utility by reporting true belief given her own knowledge. Q1 : how does sequential elicitation relate to prediction market? Q2 : what happens is an expert can predict for multiple times? 7
Outline Ø Recap:Scoring Rule and Information Elicitation Ø Connection to Prediction Markets Ø Manipulations in Prediction Markets 8
Equivalence of PMs and Sequential Elicitation Theorem (informal). Under mild technical assumptions, efficient prediction markets are in one-to-one correspondence to sequential information elicitation using proper scoring rules. What does it mean? Ø Experts will have exactly the same incentives and receive the same return Ø Market maker’s total loss is the same Next: will informally argue using the LMSR and log-scoring rules 9
Equivalence of LMSR and Log-Scoring Rules Recall LMSR Ø Value function with current sales quantity 𝑟 : 𝑊 𝑟 = 𝑐 log ∑ 7∈[(] 𝑓 G H /J Ø To buy 𝑦 ∈ ℝ ( amount, a buyer pays: 𝑊 𝑟 + 𝑦 − 𝑊(𝑟) Ø Price function (they sum up to 1) 𝑓 G L /J ∑ 7∈[(] 𝑓 G H /J = 𝜖𝑊(𝑟) 𝑞 3 𝑟 = 𝜖𝑟 3 Fact. The optimal amount an expert purchases is the amount that moves the market price to her belief 𝜇 . Fact. Worst case market maker loses is 𝑐 log 𝑜 . 10
Equivalence of LMSR and Log-Scoring Rules Q1 : If current market price is 𝑞 ABC , what is the optimal payoff for an expert with belief 𝜇 = 𝑞 A ? Ø Let 𝑟 ABC denote the market standing corresponding to price 𝑞 ABC • That is TUV /J 𝑓 G L ABC TUV /J = 𝑞 3 ∑ 7∈[(] 𝑓 G H Crucial terms: Ø Value function 𝑊 𝑟 = 𝑐 log ∑ 7∈[(] 𝑓 G H /J N OL/P ∑ H∈[Q] N OH/P = RS(G) Ø Price function 𝑞 3 𝑟 = 11 RG L
Equivalence of LMSR and Log-Scoring Rules Q1 : If current market price is 𝑞 ABC , what is the optimal payoff for an expert with belief 𝜇 = 𝑞 A ? Ø Let 𝑟 ABC denote the market standing corresponding to price 𝑞 ABC Ø Optimal purchase for the expert is 𝑦 ∗ such that TUV XY L ∗ )/J 𝑓 (G L 𝑞 3 𝑟 ABC + 𝑦 ∗ = A ∗ )/J = 𝑞 3 TUV XY H ∑ 7∈[(] 𝑓 (G H and pays 𝑊 𝑟 ABC + 𝑦 ∗ − 𝑊(𝑟 ABC ) ∗ )/J − 𝑐 log ∑ 7∈[(] 𝑓 G H TUV XY H TUV /J = 𝑐 log ∑ 7∈[(] 𝑓 (G H Crucial terms: Ø Value function 𝑊 𝑟 = 𝑐 log ∑ 7∈[(] 𝑓 G H /J N OL/P ∑ H∈[Q] N OH/P = RS(G) Ø Price function 𝑞 3 𝑟 = 12 RG L
Equivalence of LMSR and Log-Scoring Rules Q1 : If current market price is 𝑞 ABC , what is the optimal payoff for an expert with belief 𝜇 = 𝑞 A ? Ø Let 𝑟 ABC denote the market standing corresponding to price 𝑞 ABC Ø Optimal purchase for the expert is 𝑦 ∗ such that TUV XY L ∗ )/J 𝑓 (G L 𝑞 3 𝑟 ABC + 𝑦 ∗ = A ∗ )/J = 𝑞 3 TUV XY H ∑ 7∈[(] 𝑓 (G H and pays 𝑊 𝑟 ABC + 𝑦 ∗ − 𝑊(𝑟 ABC ) ∗ )/J − 𝑐 log ∑ 7∈[(] 𝑓 G H TUV XY H TUV /J = 𝑐 log ∑ 7∈[(] 𝑓 (G H TUVZ[L ∗)/P N (OL ∗ )/J = TUV XY H ∑ 7∈[(] 𝑓 (G H T \ L 13
Equivalence of LMSR and Log-Scoring Rules Q1 : If current market price is 𝑞 ABC , what is the optimal payoff for an expert with belief 𝜇 = 𝑞 A ? Ø Let 𝑟 ABC denote the market standing corresponding to price 𝑞 ABC Ø Optimal purchase for the expert is 𝑦 ∗ such that TUV XY L ∗ )/J 𝑓 (G L 𝑞 3 𝑟 ABC + 𝑦 ∗ = A ∗ )/J = 𝑞 3 TUV XY H ∑ 7∈[(] 𝑓 (G H and pays 𝑊 𝑟 ABC + 𝑦 ∗ − 𝑊(𝑟 ABC ) ∗ )/J − 𝑐 log ∑ 7∈[(] 𝑓 G H TUV XY H TUV /J = 𝑐 log ∑ 7∈[(] 𝑓 (G H TUVZ[L ∗)/P TUV/P N (OL N OL = 𝑐 log − 𝑐 log T TUV \ L \ L 14
Equivalence of LMSR and Log-Scoring Rules Q1 : If current market price is 𝑞 ABC , what is the optimal payoff for an expert with belief 𝜇 = 𝑞 A ? Ø Let 𝑟 ABC denote the market standing corresponding to price 𝑞 ABC Ø Optimal purchase for the expert is 𝑦 ∗ such that TUV XY L ∗ )/J 𝑓 (G L 𝑞 3 𝑟 ABC + 𝑦 ∗ = A ∗ )/J = 𝑞 3 TUV XY H ∑ 7∈[(] 𝑓 (G H and pays 𝑊 𝑟 ABC + 𝑦 ∗ − 𝑊(𝑟 ABC ) ∗ )/J − 𝑐 log ∑ 7∈[(] 𝑓 G H TUV XY H TUV /J = 𝑐 log ∑ 7∈[(] 𝑓 (G H TUVZ[L ∗)/P TUV/P N (OL N OL = 𝑐 log − 𝑐 log T TUV \ L \ L Note: this holds for any 𝑗 ∗ − 𝑐(log 𝑞 3 A − log 𝑞 3 ABC ) = 𝑦 3 15
Equivalence of LMSR and Log-Scoring Rules Q1 : If current market price is 𝑞 ABC , what is the optimal payoff for an expert with belief 𝜇 = 𝑞 A ? Ø Let 𝑟 ABC denote the market standing corresponding to price 𝑞 ABC ∗ − 𝑐(log 𝑞 3 A − log 𝑞 3 ABC ) Ø Repeat our finding: expert pays 𝑦 3 𝑦 ∗ is optimal amount for purchase • Ø What is the expert utility if outcome 𝑗 is ultimately realized? ∗ − [𝑦 3 ∗ − 𝑐(log 𝑞 3 A − log 𝑞 3 ABC )] 𝑦 3 from contracts’ return 16
Equivalence of LMSR and Log-Scoring Rules Q1 : If current market price is 𝑞 ABC , what is the optimal payoff for an expert with belief 𝜇 = 𝑞 A ? Ø Let 𝑟 ABC denote the market standing corresponding to price 𝑞 ABC ∗ − 𝑐(log 𝑞 3 A − log 𝑞 3 ABC ) Ø Repeat our finding: expert pays 𝑦 3 𝑦 ∗ is optimal amount for purchase • Ø What is the expert utility if outcome 𝑗 is ultimately realized? ∗ − [𝑦 3 ∗ − 𝑐(log 𝑞 3 A − log 𝑞 3 ABC )] 𝑦 3 A − log 𝑞 3 ABC ] = 𝑐 ⋅ [log 𝑞 3 = 𝑐 ⋅ [ 𝑇 ^_` (𝑗; 𝑞 A ) − 𝑇 ^_` 𝑗; 𝑞 ABC ] = payment in the sequential elicitation (constant 𝑐 is a scalar) 17
Equivalence of LMSR and Log-Scoring Rules Q1 : If current market price is 𝑞 ABC , what is the optimal payoff for an expert with belief 𝜇 = 𝑞 A ? Ø Let 𝑟 ABC denote the market standing corresponding to price 𝑞 ABC ∗ − 𝑐(log 𝑞 3 A − log 𝑞 3 ABC ) Ø Repeat our finding: expert pays 𝑦 3 𝑦 ∗ is optimal amount for purchase • Ø What is the expert utility if outcome 𝑗 is ultimately realized? Expert achieves the same utility in LMSR and log-scoring-rule elicitation for any event realization 18
Equivalence of LMSR and Log-Scoring Rules Q2 : What is the worst case loss (i.e., maximum possible payment) when using log-scoring rule in sequential info elicitation? Ø Total payment – if event 𝑗 realized – is A − log 𝑞 3 b − log 𝑞 3 b ABC ] e ∑ AaC [log 𝑞 3 = log 𝑞 3 e ≤ 0 − log 𝑞 3 e is too small (then we need to pay Ø To avoid cases where some 𝑞 3 a lot), should choose 𝑞 e = ( C C ( , ⋯ , ( ) as uniform distribution Ø Worst-case loss is thus log 𝑜 (same as LMSR, up to constant 𝑐 ) 19
Back to Our Original Theorem… Theorem (informal). Under mild technical assumptions, efficient prediction markets are in one-to-one correspondence with sequential information elicitation using proper scoring rules. Ø Previous argument generalizes to arbitrary proper scoring rules Ø Formal proof employs duality theory • Recall, any proper scoring rule corresponds to a convex function • A prediction market is determined by a value function 𝑊 𝑟 20
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