a new understanding of prediction markets via no-regret learning
TRANSCRIPT
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
ti
ttj
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?