A New Understanding of Prediction Markets via No-Regret Learning

1. A New Understanding of Prediction Markets via No-Regret Learning

2. Prediction Markets • Outcomes i in {1,…,N} • Prices pi for shares that pay off in outcome i • Market scoring rules

3. Prediction Markets • Cost functions

4. Prediction Markets Cost of Prediction Qi

5. No-Regret Learning • Experts i in {1,…,N} • Weights wi over experts I • Losses

6. No-Regret Learning wi,t Loss of Algorithm due to expert i -Li,t

7. No-Regret Learning • Randomized Weighted Majority

8. Comparison

9. N experts: 1,…,N N outcomes: 1,…,N

10. Connection-Paving the Road • Each outcome i can be interpreted as an expert, pricing contract i at $1 and other contracts at $0. • Let’s assume market run forever before any outcome realizes. When trader comes in and do short-selling, the money paid by the N experts is like a loss.

11. Connection – Paving the Road • Define the loss of an expert: at each time t, an trader comes to the market maker, and buys shares on the contract of outcome i. • Let us just assume that , i.e. only short selling happens.

12. Connection – Paving the Road The loss for expert i is: Choose a s.t.

13. Connection-Paving the Road • As a market maker, your job is to combine the opinions of your experts, and decide the price of each contract. • Your price should be set properly so that traders don’t want to trade with you at all. Your price for each outcome sums up to 1. • Still, you lose money when traders come in and sell contracts to you.

14. Connection – Paving the Road • Definition of cumulative loss of a market maker (the money market maker paid for all trades): • -stable cost function: =>

15. Connection – Paving the Road • Definition of cumulative loss of a market maker (the money market maker paid for all trades): • -stable cost function: => Actual loss for the market maker

16. Connection – Paving the Road • Definition of cumulative loss of a market maker (the money market maker paid for all trades): • -stable cost function: => Lower bound Actual loss for the market maker

17. Notation Change

18. Connection: Learning to MSR • This becomes a learning problem. Recall Weighted Majority Updating Rule: For LMSR cost function: Set the learning rate to be: =>

19. Connection: MSR to Learning • For any -stable cost function with bounded budget, we have:

20. Connection: MSR to Learning • For any -stable cost function with bounded budget, we have:

21. Connection: MSR to Learning • Recall • We set: • In Theorem 2: • If LMSR => B= b log N (the proof is waived in the paper (Lemma 5)) • Put all together into Theorem 2 we have:

22. Questions?

23. Connection • Cost Function: • Differentiability, Increasing Monotonicity and Positive Translation Invariance • Agrawal et al show that: • This paper also show that the instant price is actually the p in the expression.

24. How could we construct cost function from any market scoring rule?

25. The answer is to set: • (Theorem 3): The cost function based on the above equation is equivalent to a market scoring rule market using the scoring rule

26. Theorem 3: • Step 1: • Step 2: • Like HW2, just replace the log scoring rule and cost function with the equation above and do some KTT condition.

27. MSR Cost Function Scoring Rule Convex Function

28. MSR Cost Function Scoring Rule Convex Function

29. MSR Cost Function Scoring Rule Convex Function

30. HW2 with LMSR, but not applicable to all scoring rules MSR Cost Function Scoring Rule Convex Function

31. HW2 with LMSR, but not applicable to all scoring rules MSR Cost Function Scoring Rule Convex Function

32. Recall Theorem 2 • For any -stable cost function with bounded budget, we have:

33. Recall: • Can we compute B given ?

34. Lemma 5: B can be up-bounded by: • Let us plug this into Theorem 2: • We have a new bound:

35. Recap B: • Lemma 5: B can be up-bounded by: • Let us plug this into Theorem 2: • We have a new bound: Recall FTRL bound:

36. Can we push more to show ? • The paper doesn’t cover this.

37. Discussion • Continuous price updates versus discrete weight updates • Direction of implication • Any strictly proper market scoring rule implies corresponding FTRL algorithm with strictly convex regularizer • Any FTRL algorithm with differentiable and strictly convex regularizer implies strictly proper scoring rule.

38. Discussion • Extensive learning literature may aid progress in prediction markets. • PermELearn algorithm • Applied to combinatorial markets

39. Questions?