A New Understanding of Prediction Markets via No-Regret Learning

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

Prediction Markets Cost functions

Prediction Markets QiQi Cost of Prediction

No-Regret Learning Experts i in {1,…,N} Weights w i over experts I Losses

No-Regret Learning -L i,t Loss of Algorithm due to expert i w i,t

No-Regret Learning Randomized Weighted Majority

Comparison Market Scoring RuleLearning N outcomes: 1,…,N N experts: 1,…,N Prediction by price:Prediction by weights: Price updating rule for LMSR:Weight updating rule for weighted majority:

N outcomes: 1,…,N N experts: 1,…,N

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.

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.

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

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.

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

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

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 Lower bound

Notation Change

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

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

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

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:

Questions?

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.

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

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

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.

MSR Cost Function Scoring Rule Convex Function

MSR Cost Function Scoring Rule Convex Function

MSR Cost Function Scoring Rule Convex Function

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

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

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

Recall: Can we compute B given ?

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

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:

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

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.

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

Questions?