A New Understanding of Prediction Markets via No-Regret Learning

Prediction Markets

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

Prediction Markets

• Cost functions

Prediction Markets

Qi

Cost of Prediction

No-Regret Learning

• Experts i in {1,…,N}• Weights wi over experts I• Losses

No-Regret Learning

-Li,t

Loss of Algorithmdue to expert i

wi,t

No-Regret Learning

• Randomized Weighted Majority

Comparison

Market Scoring Rule LearningN outcomes: 1,…,N

N experts: 1,…,N

Prediction by price: Prediction by weights:

Price updating rule for LMSR: Weight updating rule for weighted majority:

tip , tiw ,

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.

tir ,0, tir

Connection – Paving the Road

The loss for expert i is:

Choose a s.t.

T

tti

T

ttiTi rrL

1,

1,, )1(1

||, ,tirit

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: =>

tit

T

t

N

iiTA rqpL ,1

1 1, )(1

2)0()(1

,TCqCL TTA

Connection – Paving the Road• Definition of cumulative loss of a market

maker (the money market maker paid for all trades):

• -stable cost function: =>

tit

T

t

N

iiTA rqpL ,1

1 1, )(1

2)0()(1

,TCqCL TTA

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: =>

tit

T

t

N

iiTA rqpL ,1

1 1, )(1

2)0()(1

,TCqCL TTA

Actual loss for the market maker

Lower bound

Notation Change

T

tti

T

tti

T

ttiTi lrrL

1,

1,

1,, )1(1

T

t

N

itititit

T

t

N

iiTA lwrqpL

1 1,,,1

1 1, )(1

Connection: Learning to MSR

• This becomes a learning problem. Recall Weighted Majority Updating Rule:

For LMSR cost function: Set the learning rate to be: =>

b/

N

j

bq

bq

titj

ti

e

ew

1

/

/

,,

,

N

j

l

l

tit

tj

tti

e

ew

1

,,

,

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:

b/ )/(2 TB

• 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 )(psi

• 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

)(psi

)q(

C

MSR Cost Function

Scoring Rule Convex Function

)(psi

)q(

C

MSR Cost Function

Scoring Rule Convex Function

)(psi

)q(

C

MSR Cost Function

Scoring Rule Convex Function

)(psi

HW2 with LMSR, but not applicable to all scoring rules

)q(

C

MSR Cost Function

Scoring Rule Convex Function

)(psi

HW2 with LMSR, but not applicable to all scoring rules

)q(

C

Recall Theorem 2

• For any -stable cost function with bounded budget, we have:

• Recall:

• Can we compute B given ? )q(

C

BCqCq TTi )0()(,i

max

• 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:

BCqCq TTi )0()(,i

max

Recall FTRL bound:

• Can we push more to show ?

• The paper doesn’t cover this. 4

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?